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Research Papers

# Evaluation of Elastoplasticity-Dependent Creep Property of Magnesium Alloy With Indentation Method: A Reverse Numerical Algorithm and Experimental ValidationOPEN ACCESS

[+] Author and Article Information
Shoichi Fujisawa

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

Masafumi Noda

Magnesium Division,
Gonda Metal Industry Co., Ltd.,
1-1-16 Miyashimo, Chuo,
Sagamihara, Kanagawa 252-0212, 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 20, 2016; final manuscript received August 29, 2016; published online February 1, 2017. Assoc. Editor: Xi Chen.

J. Eng. Mater. Technol 139(2), 021004 (Feb 01, 2017) (9 pages) Paper No: MATS-16-1147; doi: 10.1115/1.4035280 History: Received May 20, 2016; Revised August 29, 2016

## Abstract

Magnesium (Mg) alloys have been widely used in automotive and aerospace industries due to its merits of exceptional lightweight, super strong specific strength, and high corrosion-resistance, where intermetallic compounds with a small volume are very critical to achieve these excellent performance. This study proposes a reverse analysis that can be employed to extract elastoplasticity-dependent creep property of commercial die-cast Mg alloys and their intermetallic compounds from instrumented indentation with two sharp indenters. First, the creep deformation that obeys the Norton's law ($ε˙$  = A$σn$) is studied, and the parameters of A and n are determined from two indentation experiments conducted with different sharp indenters. Then, a numerical algorithm and dimensional function developed is extended to extract the elastoplasticity of various metallic materials by focusing on the loading stage of indentation experiments. By considering the full loading history with both linear increase and holding stages of loads, we propose a framework of reverse analysis to determine both elastoplasticity and creep properties simultaneously. Parallel indentation experiments on pure magnesium and aluminum and Mg alloys are performed, and the results agree well with the numerical predictions.

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## Introduction

Die-casting technique is often used in the manufacturing of magnesium (Mg) alloys, especially production with complex configurations. To improve the mechanical properties (i.e., yield stress and tensile strength) of Mg alloys, alloy processing by adding other atomic elements such as aluminum (Al) is very important. Intermetallic compounds will form during this processing and are critical to macroscopic mechanical performance of Mg alloys [17]. For example, AZ91 is the most popular material among casting Mg alloys. It contains a small fraction (at micron-scale) of intermetallic compound of β-Mg17Al12 phase and usually distributes along the grain boundary of $α$-Mg phase (matrix) [3,8]. Therefore, investigation of microscopic mechanical properties (i.e., elastoplastic and creep property of microstructural compounds) becomes important. However, each intermetallic compound is usually only several 10 μm in size, and it is quite challenging to evaluate mechanical properties using the conventional tensile/compressive techniques.

Instrumented indentation emerges as an attractive technique for probing the mechanical properties of materials with small volumes. It is convenient, quick, and inherently simple without extensive efforts of sample preparation in comparison with conventional tensile or compressive experiments [914]. Using the indentation method, several evaluation methods for extracting elastoplastic and creep properties have been proposed [1517]. For the creep property, under the framework of Norton's law, $ε˙=Aσn$, where $ε˙$ is the strain rate, σ is the applied constant stress, and A and n are the creep constant and stress exponent, respectively, Mahmudi et al. estimated the creep parameters A and n of magnesium alloy using indentation creep test with a flat indenter [3]. During their estimation, the elastic and plastic properties are assumed to be known prior, and they did not consider the effect of indentation elastoplastic deformation on the creep properties determination, whereas the creep properties of magnesium alloy are typical elastoplastic dependent, and the effect of elastoplastic deformation needs to be probed.

This study1 aims to establish a new method to determine both creep and elastoplastic properties from indentation experiments. Dimensional analysis is conducted to develop dimensionless function through which the indentation creep deformation is correlated with the creep parameters (A and n), and the indentation loading curvature is bridged with elastoplastic properties. This dimensionless function is also employed to perform a simple reverse analysis that can directly estimate the elastoplastic-dependent creep properties from indentation response. Extensive finite element method (FEM) computations are carried out to establish the method with a wide range of materials properties. This proposed method is built on the basis of dimensionless functions with simple formula and it is thus expected that it can readily apply to the estimation of microstructural properties in magnesium alloy, especially small intermetallic compounds.

## Materials

A commercial die-cast magnesium alloy, AZ91D (Mg: 90%, Al: 9%, and Zn: 1%), was used in this experiment. The original shape is round bar with the diameter of 10 mm. In order to stabilize the microstructure, the heat treatment was carried out, including solution heat treatment and aging heat treatment at 200 °C for about 10 h.

For the microindentation test, the disk type specimen was taken from the round bar. The size is 10 mm in diameter and 2 mm in thickness. After the mechanical polishing, the specimen surface was immersed in the etching solution (distillated water: 24 ml, 60% nitric acid: 1 ml, and ethylene glycol: 75 ml). The microstructure observed by scanning electron microscope (SEM) is shown in Fig. 1. An intermetallic compound (eutectic $β$-Mg17Al12 phase) with a white island shape is clearly observed. This structure is quite different from that of $α$-Mg matrix. The $β$-Mg17Al12 phase precipitates along the grain boundary of $α$-Mg phase. This study observes the large area of more than 1 mm2 in the microstructure. It reveals that the $β$-Mg17Al12 distributes uniformly. We measured the area proportion of each microstructure of $α$-Mg and $β$-Mg17Al12, indicating that the area ratio of α-phase to β-phase is 95/5.

In parallel, we also prepared pure Mg and pure Al to further verify our estimation method in a broad range of materials. We purchased the round bars of pure Mg (99.99%) and pure Al (99.99%), with a diameter of 9.5 mm and 10 mm, respectively. For both materials, the heat treatment was conducted, including aging heat treatments at 400 °C for about 24 h for pure Mg and at 350 °C for about 1 h for pure Al. This study conducted two types of experiments: microindentation test and uniaxial compressive test. In the uniaxial compressive test, a cylinder type specimen was prepared. Note that these specimens were taken from the same round bar with that for indentation experiment.

These mechanical properties are shown in Table 1. In this table, the elastoplastic properties of AZ91D ($α$-Mg and -Mg17Al12) were previously estimated by microindentation test [18]. Pure magnesium and pure aluminum were estimated by compressive test. The plastic property is described as constitutive equation, showing the functional relationship between the flow stress σflow and plastic strain εp. In addition, the values of representative stress σr are described in Table 1, which will be explained later.

## Experiment

###### Experimental Setup.

Indentation test was performed using Dynamic Ultra micro Hardness Tester (DUH-510 S, Shimadzu Corp., Kyoto, Japan) with two diamond triangle indenters (apex angles: 115 deg and 100 deg). The indenter with 115 deg is a Berkovich indenter, and the one with 100 deg is a sharper indenter. For the experiment, a loading control was used, and the loading rate was set to be between 5.53 and 7.74 mN/s. When it reaches the maximum load, the load is sustained to be constant for 1000 s at room temperature. During the test, the indentation force F and the resultant penetration depth of indenter change as a function of loading time. The indentation curve (loading, holding, and unloading) is continuously measured.

Uniaxial compression test and creep test were conducted for pure Mg and pure Al to estimate creep parameter (Norton's law: $ε˙=Aσn$) as references, such that our indentation-based method can be verified. These uniaxial compression tests were performed using hydraulic fatigue testing equipment (electrohydraulic servo testing equipment, Shimadzu Corp. EHF-EB50KN-10 L). Compression creep measurements were carried out, and the applied stress is between 60 and 140 MPa with dwell time of 200,000 s. During the test, the creep strain was measured by two displacement sensors (eddy current sensor, EX-201, Keyence).

###### Experimental Result.

Figure 2 shows the indentation creep curve (i.e., indentation creep displacement and time curve during the holding period of loading). The inserted figure shows the indentation curve, and the constant loading period corresponds to indentation creep curve. As shown in Fig. 2, the indentation creep curve consists of two deformation regions: primary (transient) creep deformation and secondary (steady-state) creep. In the primary creep region, the measured indentation creep depth is significantly changed. There is a high penetration rate at the initial stage, and then it moderates with time. Subsequently, in the secondary region, the variation of creep depth becomes approximately constant, which agrees with previous reports [19,20]. As mentioned above, this study focuses on the Norton's law that primarily describes the secondary (steady-state) creep deformation. Thus, this study uses a constant rate of indentation creep depth as shown in this figure. The rate is designated as .

Similarly, uniaxial compression creep tests were conducted. When the applied compressive stress was fixed, we measured creep deformation, exhibiting primary (transient) creep and secondary (steady-state) creep. Since this study focuses on secondary (steady-state) creep region, the obtained results are expressed by Norton's law ($ε˙=Aσn$), and we extracted the creep parameters n and A. These values will be explained later, when the present indentation method is verified.

## Estimation Method of Determining Creep Property and Applications

###### Dimensionless Function.

This study assumes that creep deformation obeys the Norton's law ($ε˙=Aσn$), and the creep parameters A and n will be evaluated by our indentation method. As shown in Fig. 2, we focus on the indentation creep depth rate, $h˙$, when it becomes constant in the steady-state. In order to connect the depth rate $h˙$ with creep parameters (A and n), we explore various materials with a wide range of creep parameters. In this study, the dimensionless function, which correlates the creep depth rate $h˙$ with the parameters to be identified (A and n), is established through the parametric FEM study. The final goal would be that the experimentally obtained depth rate is substituted to the established dimensionless, yielding the creep parameters (A and n).

The depth rate $h˙$ depends on materials properties, such as Young's modulus E, representative stress2$σr$, and creep parameter (A and n). In addition, it depends on testing condition, i.e., the maximum indentation depth hmax and indenter tip angle $α$, and can be expressed as Display Formula

(1)$h˙=f(E*,σr,A,n,α,hmax)$

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. Dimensionless analysis with $Π$ theory dictates the following relationship: Display Formula

(2)$h˙E*nAhmax=Π(σrE*,α,n)$

Here, we attempt to reduce the variables that are dependent each other in the dimensionless function of Eq. (2), because we aim to establish a simple dimensionless function. As the initial investigation, the angle of indenter tip $α$ is fixed, and plastic property (σr) of the target material is known prior. Equation (2) can change to Display Formula

(3)$h˙E*nAhmax=Π(n)$

This function is dependent on the creep parameters (A and n). When E and the depth rate $h˙$ are known, Eq. (3) yields to the relationship of A and n.

###### Application to Pure Magnesium.

To establish the detailed function of Eq. (3), numerical experiments with FEM were conducted. As shown in Fig. 3, this study created two-dimensional axisymmetric FEM model. For simplicity, the indenters of two triangle indenters were modeled as conical indenter with rigid body. The half apex angles are 70.3 deg and 50.7 deg for Berkovich indenter (115 deg triangle indenter) and sharp triangle indenter (100 deg triangle indenter). For the size of axisymmetric model in Fig. 3, both radius and height are 300 $μm$, which is very large enough compared with the indentation deformation area. The model comprised more than 20,000 four-node elements, wherein fine meshes were created around the contact region and a mesh converge test was carried out. Since the maximum indentation depth is a few micrometers, the plastic deformation zone due to the indenter penetration does not reach the bottom edge of the model. Thus, the FEM model can be considered as a semi-infinite model against the impression (plastic zone), i.e., indentation curve is not influenced by the bottom edge and boundary condition. The friction coefficient between indenter and material is 0.15. For the indentation computation, the loading process was computed with elastoplastic analysis, and the load holding process was computed with creep analysis. The mechanical properties for pure magnesium are shown in Table 1.

To establish the dimensionless functions of Eq. (3), various creep properties are considered by performing parametric FEM study. Table 2 shows the wide range of creep property for FEM computation, so that we investigate their indentation responses. As an example case, Fig. 4 shows the indentation creep depth as a function of creep time, i.e., indentation creep curve. The indenter is Berkovich one. We assume that the elastoplastic property is referred to pure Mg (Table 1) and investigate the dependency of various creep parameters, having A = 2.5 $×$ 10−16 (fixed) and various values of n (n = 4, 4.5, and 5). As shown in Fig. 4, the indentation creep depth is significantly changed dependent on the n-value. In other words, the depth rate $h˙$ changes as creep parameter (A and n). The computed result of depth rate $h˙$ is substituted to Eq. (3), and we establish a dimensionless function as shown in Fig. 5. It is found that the function is monotonically decreasing as a function of the creep parameter n, in which, it is strongly dependent on the parameter n and slightly depends on the parameter of A. For each A, these curves can be fitted by the following equation:

Display Formula

(4)$h˙E*nAhmax=exp(an+b)$

This polynomial fitting function has two coefficients, i.e., a and b, where a is the slope and b is the intersection. These coefficients are dependent of the parameter A. Similarly, the parametric study was carried out with 100 deg triangle indenter, so that we establish the dimensionless functions of Eq. (4).

To verify our above method, the actual experiment of pure magnesium was conducted. Indentation creep tests were carried out for pure magnesium by using two triangle indenters (100 deg and 115 deg). From the experimental results, we obtained the experimental data of $h˙$ and $hmax$, such that these were substituted to the dimensionless function of $h˙/E*nAhmax$ (see Eq. (3)). Indeed, we computed $h˙/E*nAhmax$ for each parameter A (i.e., A = 2.5 $×$ 10−8, 2.5 $×$ 10−10, 2.5 $×$ 10−12, 2.5 $×$ 10−14, 2.5 $×$ 10−16, and 2.5 $×$ 10−18), as shown in Fig. 6, yielding the relationship between $h˙/E*nAhmax$ and n (i.e., $n−h˙/E*nAhmax$ curve). Note that this is the result obtained with the 115 deg triangle indenter (Berkovich indenter).

We next compare the experimental $n−h˙/E*nAhmax$ curve (Fig. 6) with the established dimensionless function (Fig. 5), as shown in Fig. 7. This figure separately shows different parameter A, and solid lines with symbols indicate the established function (Fig. 5), while the dot lines indicate the experimental data. It is observed that each curve for different A intersects at a certain point. The intersection point is unique in this figure. From these intersections, the combinations of creep parameter (A and n) are extracted, so that A–n curve for 115 deg triangle indenter can be readily obtained. The result was shown in Fig. 8.

To identify unique solution of A and n, we need other independent combination of A–n. As a similar way, the data obtained from different indenter (100 deg triangle indenter) were used, and we extract the combination A and n (A–n curve). Figure 8 shows two combinations of A and n (A–n curves) from both indenters, indicating that A decreases with the increase of n. However, the slope of the curve is slightly different. They intersects at the certain point as shown in Fig. 8 (A =  $9.22×10−10$ and n = 1.82). This intersection is the estimated solution, which may be identified uniquely.3

To verify our estimation, uniaxial creep compression test was conducted. The result is (A =  $1.42×10−10$ and n = 1.20), which is close to our estimation (A =  $9.22×10−10$ and n = 1.82). Therefore, the present method enables us to estimate creep parameters (A and n) with a high accuracy.

###### Application to AZ91D ($α$-Mg and $β$-Mg17Al12 Phases).

As mentioned earlier, we establish a fundamental method to evaluate creep property. By using a similar concept, creep property of the microstructure in AZ91D can also be identified. As shown in Fig. 1, AZ91D has several phases, i.e., $α$-Mg phase and $β$- Mg17Al12 phase. The elastoplastic property of $α$-Mg phase and $β$-Mg17Al12 phase is shown in Table 1. Based on this property, parametric FEM studies were carried out, covering various creep properties (see Table 2). The computational results are employed to establish the dimensionless function of Eq. (4), whose results are similar with Fig. 5. The dimensionless functions are developed based on Berkovich indenter and 100 deg triangle indenter.

To estimate creep the parameter of $α$-Mg and $β$-Mg17Al12 phases, indentation creep tests were conducted for each phase. Figure 9 shows indentation creep curve for $α$-Mg phase for both indenters. As a similar way, the estimated creep parameters of $α$-Mg phase are A = 4.25 $×10−10$ and n = 1.74. Next, Fig. 10 shows an indentation creep curve and its impression. Note that it is found that the indenter impression made inside of the $β$-phase. This indicates that these tests can extract the property of such a small intermetallic compound. Using a similar process, creep parameters for $β$-Mg17Al12 phase are estimated: A = 1.69 $×10−10$ and n = 1.83. In summary, Table 3 shows the plastic property and the estimated creep parameters for AZ91D (including $α$-Mg phase and $β$-Mg17Al12 phase). This reveals that the $β$-phase is harder than that of the $α$-phase. However, creep parameter of $β$-phase is very similar to that of $α$-phase (matrix) at room temperature. This suggests that intermetallic compound of β-phase strengthens, whereas it does not affect the creep deformation resistance at room temperature.

## Extension to Various Materials

###### Estimation Method of Determining Elastoplastic Property.

As mentioned previously, we have shown the fundamental concept to evaluate the creep parameters (A and n) from two indentation creep tests with different sharp indenters. To establish the method, however, the elastoplastic property was fixed. Thus, the dimensionless function (Eq. (4)), $n−h˙/E*nAhmax$ curve, may change depending on the elastoplastic property. In other words, our method may be applicable to materials whose elastoplastic property is known, leading to limitation in the practical applications. In fact, since high temperature affects both creep property and plastic property, the evaluation of plastic property is also important. Therefore, we will additionally propose a dimensionless function for estimating elastoplastic property and then combine with the above creep evaluation method.

This part focuses on loading process in indentation curve, such that we will establish dimensionless function to evaluate plastic property. During the loading process, indentation force F depends on elastoplastic property of the target material and testing condition, i.e., indentation depth h and indenter tip angle $α$. It can be expressed as follows: Display Formula

(5)$F=f(E*,σr,h,α)$

Dimensionless analysis with $Π$ theory dictates the following relationship: Display Formula

(6)$FE*h2=CE*=Π(σrE*,α)$
where C is the loading curvature (Kick's law: $F=Ch2$). To determine the detailed functional formula of Eq. (6), several materials are considered. This study employed four materials (in Table 1) and three other materials, referred to as Mat. 1, Mat. 2, and Mat. 3 (in Table 4). Note that $σr$ is the representative stress and corresponds to the representative strain εr. This is dependent on an indenter angle for sharp indenter, and it is reported that the value of εr is expressed by 0.105 cot(θ), where θ is a tip angle of conical indenter [21]. For the present 115 deg and 100 deg triangle indenters, we calculate the representative strain of $εr115$  = 0.038 and $εr100$  = 0.086. With the constitutive equation of plastic property, the representative stresses σr115 and σr100 are calculated. As shown in Tables 1 and 4, seven different materials (regarding different elastoplastic properties) are prepared for FEM computation. From the computations, the loading curvature C was obtained. With reference to Eq. (6), functional relationship between $C/E*$ and $σr/E*$ is obtained among seven materials. This result is shown in Fig. 11,4 indicating the independent relationship for 115 deg and 100 deg triangle indenters.

For the estimation process, we first extract loading curvatures (C115 and C100) from each indentation curves using 115 deg and 100 deg triangle indenters. Next, the values of C115, C100, and E* are substituted to Fig. 11 and obtain the plastic property of $σr115$ and $σr100$. Note that the E* is determined prior from conventional Oliver–Pharr method [22]. Finally, we can describe the stress–strain curve by using two combinations of representative stress–strain point ($σr115−εr115$ and $σr100−εr100$), and then it is concluded that we can estimate plastic property from the loading curvature in indentation curve. In the next step, this evaluation method connects to that of creep property. Therefore, our method for estimating creep parameters is developed for various materials.

###### Combination of Plastic and Creep Evaluations.

As shown in Fig. 5 and Eq. (4), a simple function is established to evaluate creep property (A and n). It is expected that the function is dependent on elastoplastic property, $σr/E*$. As described in Eq. (4), $n−h˙/E*nAhmax$ curve is fitted by exponential function, in which a and b are the coefficients of fitted curve (i.e., slope of the curve is a, and the intercept is b). Thus, here we investigate the relationship between elastoplastic property $σr/E*$ and their coefficients (a and b). To do this, four materials (Table 1) and three materials (Mat. 1, Mat. 2, and Mat. 3 in Table 4) were investigated.

As heretofore in Sec. 4.1, parametric FEM study was conducted to develop the dimensionless function of $n−h˙/E*nAhmax$ curve for seven materials. Subsequently, each $n−h˙/E*nAhmax$ curve is fitted by the exponential function of Eq. (4), and their coefficients (a and b) are connected to the parameter A (A = 2.5 $×10−8$, 2.5 $×10−10$, 2.5 $×10−12$, 2.5 $×10−14$, 2.5 $×10−16$, and 2.5 $×10−18$) and elastoplastic property $σr/E*$. As a representative case, the relationship between the coefficients (a and b) and elastoplastic property $σr/E*$ is shown in Fig. 12. This figure shows the curves of $σr/E*−$a and $σr/E*−$b at A = 2.5 $×10−10$. These curves are fitted by the below equation

Display Formula

(7)$a,b=αln(σrE*)2+βln(σrE*)+γ$

Three coefficients α, β, and γ are involved. Similar to Eq. (7), other cases of A (from 2.5 × 10−8 to 2.5 × 10−18 as total of six) are investigated. In addition, we investigate them for two indenters. These coefficients of α, β, and γ are described in Table 7 in the Appendix. From these relationships with coefficients (α, β, γ), we can estimate the coefficient of a and b in Eq. (5) for a given elastoplastic property $σr/E*$, yielding dimensionless function to evaluate creep property.

###### Flowchart and Verification of the Proposed Method.

Figure 13 shows the flowchart of our estimation method for creep property and plastic property. At first, indentation tests with the 115 deg triangle indenter and 100 deg triangle indenter are carried out. We extract the values of E*, C, hmax, and $h˙$ from the indentation curve. As indicated in step 1, we substitute E* and C to the dimensionless functions (Fig. 11) in order to estimate elastoplastic property. Next, as indicated in step 2, we substitute elastoplastic property to the dimensionless function (Fig. 12) and obtain complete $n−h˙/E*nAhmax$ curve (e.g., see Fig. 5). Finally, we compare the experimental results with the obtained $n−h˙/E*nAhmax$ curve, yielding A–n curve for two indenters. These A–n curves intersect at one point (e.g., see Fig. 8). This intersection is the estimated solution, which may be identified uniquely. Therefore, we can estimate plastic property and creep parameter (A and n).

Next, we verified our method by estimating five materials, including pure Mg, pure Al in experiment, and three virtual materials (Mat. 4, Mat. 5, and Mat. 6) in FEM. For these virtual materials, the material properties are shown in Table 5. Thus, the target seven materials have different properties (Tables 1 and 5). Indeed, Mat. 4, Mat. 5, and Mat. 6 are not used to establish the present estimation method. For these materials, indentation tests were conducted experimentally (for pure Mg and pure Al) and numerically (for Mat. 4, 5, and 6). Based on the obtained results, elastoplastic property and creep property are evaluated in Figs. 14 and 15, respectively. As shown in these figures, estimation results showed good agreement each other.

Another concern for estimations is the perturbation of indentation response. When we experimentally conducted 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 probe how the input data affect these properties. In this study, we briefly conducted sensitivity analysis for three representative materials (in Table 5). The perturbation cases of the indentation response (input data) were set as shown in Table 6. In fact, for the indentation curve, the perturbation values are generally set to around 5 $%$ [23]. Here, we conducted sensitivity analysis for Mat. 4, Mat. 5, and Mat. 6. Table 6 shows the result of sensitivity analysis. The deviations (errors in the estimated values compared to the input values) are also shown in Table 6. The estimations show reasonable agreement and robust for all the perturbation cases. Therefore, our method using two sharp indenters can estimate plastic property and creep property simultaneously.

## Conclusion

The present study proposes a reverse indentation analysis conducted with two sharp indenters to estimate both creep and elastoplastic property. The creep deformation is assumed to obey the Norton's law ($ε˙$ = A$σn$), where two creep parameters (A and n) are included. Two indentation tests with different sharp indenters are carried out in a loading control manner, and once the maximum force reaches, it keeps for a long period of time, where the indentation loading process predicts the elastoplastic property, and the duration stage of loading associated with the a stable indentation depth rate $h˙$ predicts the creep property. A new reverse analysis algorithm is established based on the parametric FEM computations. The method consists of several dimensionless functions that can directly evaluate the material properties. Through these functions, the loading curvature in indentation curves is correlated with the elastoplastic property, and indentation creep depth rate $h˙$ is related with the creep property. Finally, both functions are bridged, yielding a new method to evaluate both elastoplastic property and creep property simultaneously. We verify our method by investigating various materials, including pure magnesium and pure aluminum materials. Finally, the properties of microstructures in magnesium alloy are investigated using our method. Since this method can be used to estimate elastoplastic and creep properties of small intermetallic compounds, it may be useful to guide the synthesis of various Mg alloys by probing mechanical property of intermetallic compounds.

## Acknowledgements

This work was supported in part by JSPS KAKENHI (Grant No. 26420025) from the Japan Society for the Promotion of Science (JSPS) and Research Grant from Osawa Scientific Studies Grants Foundation (No. 27-22).

## Appendices

###### Appendix

As described in Eq. (4), the coefficients (a and b) are required to estimate creep property (A and n for the Norton's law). This study further investigates the dependency of elastoplasticity for creep property estimations as described in Sec. 5.2. In other words, the coefficients (a and b) in Eq. (4) are dependent on elastoplasticity. Here, the relationship between the coefficients and elastoplasticity is explained. Note that the elastoplastic property is described in σr/E* (representative stress/reduced modulus).

The representative relationship of between the coefficients and σr/E* is shown in Fig. 12, where the data are for A = 2.5 × 10−10 of creep parameter and are obtained by 115 deg indenter experiment. In this figure, the data are approximated by polynomial function (α ln(σr/E*)2 + β ln (σr/E*) + γ) in Eq. (7). Their coefficients (α, β, γ) are shown in Table 7. Similarly, other cases (A = 2.5 $×10−8$, 2.5 $×10−12$, 2.5 $×10−14$, 2.5 $×10−16$, and 2.5 $×10−18$) are also investigated and are shown in Table 7. Furthermore, the data of different indenter (100 deg triangle indenter) experiments are investigated. Their coefficients are shown in Table 8. Therefore, after the estimation of elastoplastic property σr/E* from Eq. (6) (Fig. 11; Table 9), Tables 7 and 8 are used to estimate the coefficient of a and b in Eq. (4) for a given elastoplastic property σr/E*, so that we can determine dimensionless function to evaluate creep property.

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Ogasawara, N. , Chiba, N. , and Chen, X. , 2009, “ A Simple Framework of Spherical Indentation for Measuring Elastoplastic Properties,” Mech. Mater., 41(9), pp. 1025–1033.
Yonezu, A. , Akimoto, H. , Fujisawa, S. , and Chen, X. , 2013, “ Spherical Indentation Method for Measuring Local Mechanical Properties of Welded Stainless Steel at High Temperature,” Mater. Des., 52, pp. 812–820.
Xu, B. , Yue, Z. , and Chen, X. , 2010, “ Characterization of Strain Rate Sensitivity and Activation Volume Using the Indentation Relaxation Test,” J. Phys. D, 43(24), p. 245401.
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.
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.
Samadi-Dooki, A. , Malekmotiei, L. , and Voyiadjis, G. Z. , 2016, “ Characterizing Shear Transformation Zones in Polycarbonate Using Nanoindentation,” Polymer, 82, pp. 238–245.
Meza, L. R. , and Greer, J. R. , 2014, “ Mechanical Characterization of Hollow Ceramic Nanolattices,” J. Mater. Sci., 9(6), pp. 2496–2508.
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.
Xu, B. , Eggler, G. , and Yue, Z. , 2007, “ A Numerical Procedure for Retrieving Material Creep Properties From Bending Creep Tests,” Acta Mater., 55(18), pp. 6275–6283.
Liu, T. , Deng, Z. C. , and Lu, T. J. , 2007, “ Minimum Weights of Pressurized Hollow Sandwich Cylinders With Ultralight Cellular Cores,” Int. J. Solids Struct., 44(10), pp. 3231–3266.
Deana, J. , Campbell, J. , Aldrich-Smith, G. , and Clyne, T. W. , 2014, “ A critical assessment of the “stable indenter velocity” method for obtaining the creep stress exponent from indentation data,” Acta Mater., 80, pp. 56–66.
Nautiyal, P. , Jain, J. , and Agarwal, A. , 2015, “ A comparative study of indentation induced creep in pure magnesium and AZ61 alloy,” Mater. Sci. Eng. A, 630, pp. 131–138.
Bucaille, J. L. , Stauss, S. , Felder, E. , and Michler, J. , 2003, “ Determination of Plastic Properties of Metals by Instrumented Indentation Using Different Sharp Indenters,” Acta Mater., 51(6), pp. 1663–1678.
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.
Inoue, N. , Yonezu, A. , Watanabe, Y. , Okamura, T. , Yoneda, K. , and Xu, B. , 2015, “ Prediction of Viscoplastic Properties of Polymeric Materials Using Sharp Indentation,” Comput. Mater. Sci., 110, pp. 321–330.
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## References

Huang, Z. W. , Zhao, Y. H. , Hou, H. , and Han, P. D. , 2012, “ Electronic Structural, Elastic Properties and Thermodynamics of Mg17Al12, Mg2Si and Al2Y Phases From First-Principles Calculation,” Physica B, 407(7), pp. 1075–1081.
Guomin, H. , Zhiqiang, H. , Luo, A. , Sachdev, A. K. , and Liu, B. , 2013, “ A Phase Field Model for Simulating the Precipitation of Multi-Variant β-Mg17Al12 in Mg-Al-Based Alloys,” Scrip. Mater., 68(9), pp. 691–694.
Mahmudi, R. , and Moeendarbari, S. , 2013, “ Effects of Sn Additions on the Microstructure and Impression Creep Behavior of AZ91 Magnesium Alloy,” Mater. Sci. Eng. A, 566, pp. 30–39.
Hiraia, K. , Somekawa, H. , Takigawa, Y. , and Higashi, K. , 2005, “ Effects of Ca and Sr Addition on Mechanical Properties of a Cast AZ91 Magnesium Alloy at Room and Elevated Temperature,” Mater. Sci. Eng. A, 403(1–2), pp. 276–280.
Lin, L. , Wang, F. , Yang, L. , Chen, L. J. , Liu, Z. , and Wang, Y. M. , 2011, “ Microstructure Investigation and First-Principle Analysis of Die-Cast AZ91 Alloy With Calcium Addition,” Mater. Sci. Eng. A, 528(15), pp. 5283–5288.
Hort, N. , Huang, Y. , and Kainer, K. U. , 2006, “ Intermetallics in Magnesium Alloys,” Adv. Eng. Mater., 8(4), pp. 235–240.
Mu, M. , Zhi-min, Z. , Bao-Hong, Z. , and Jin, D. , 2012, “ Flow Behavior and Processing Maps of As-Cast and As-Homogenized AZ91 Alloy,” J. Alloys Compd., 513, pp. 112–117.
Nagasekhar, A. V. , Cáceres, C. H. , and Kong, C. , 2010, “ 3D Characterization of Intermetallics in a High Pressure Die Cast Mg Alloy Using Focused Ion Beam Tomography,” Mater. Charcterization, 61(11), pp. 1035–1042.
Ogasawara, N. , Chiba, N. , and Chen, X. , 2009, “ A Simple Framework of Spherical Indentation for Measuring Elastoplastic Properties,” Mech. Mater., 41(9), pp. 1025–1033.
Yonezu, A. , Akimoto, H. , Fujisawa, S. , and Chen, X. , 2013, “ Spherical Indentation Method for Measuring Local Mechanical Properties of Welded Stainless Steel at High Temperature,” Mater. Des., 52, pp. 812–820.
Xu, B. , Yue, Z. , and Chen, X. , 2010, “ Characterization of Strain Rate Sensitivity and Activation Volume Using the Indentation Relaxation Test,” J. Phys. D, 43(24), p. 245401.
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.
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.
Samadi-Dooki, A. , Malekmotiei, L. , and Voyiadjis, G. Z. , 2016, “ Characterizing Shear Transformation Zones in Polycarbonate Using Nanoindentation,” Polymer, 82, pp. 238–245.
Meza, L. R. , and Greer, J. R. , 2014, “ Mechanical Characterization of Hollow Ceramic Nanolattices,” J. Mater. Sci., 9(6), pp. 2496–2508.
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.
Xu, B. , Eggler, G. , and Yue, Z. , 2007, “ A Numerical Procedure for Retrieving Material Creep Properties From Bending Creep Tests,” Acta Mater., 55(18), pp. 6275–6283.
Liu, T. , Deng, Z. C. , and Lu, T. J. , 2007, “ Minimum Weights of Pressurized Hollow Sandwich Cylinders With Ultralight Cellular Cores,” Int. J. Solids Struct., 44(10), pp. 3231–3266.
Deana, J. , Campbell, J. , Aldrich-Smith, G. , and Clyne, T. W. , 2014, “ A critical assessment of the “stable indenter velocity” method for obtaining the creep stress exponent from indentation data,” Acta Mater., 80, pp. 56–66.
Nautiyal, P. , Jain, J. , and Agarwal, A. , 2015, “ A comparative study of indentation induced creep in pure magnesium and AZ61 alloy,” Mater. Sci. Eng. A, 630, pp. 131–138.
Bucaille, J. L. , Stauss, S. , Felder, E. , and Michler, J. , 2003, “ Determination of Plastic Properties of Metals by Instrumented Indentation Using Different Sharp Indenters,” Acta Mater., 51(6), pp. 1663–1678.
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.
Inoue, N. , Yonezu, A. , Watanabe, Y. , Okamura, T. , Yoneda, K. , and Xu, B. , 2015, “ Prediction of Viscoplastic Properties of Polymeric Materials Using Sharp Indentation,” Comput. Mater. Sci., 110, pp. 321–330.

## Figures

Fig. 1

Microstructure of α-Mg and β-Mg17Al12 in die-cast magnesium alloy (AZ91D)

Fig. 7

Dimensionless function of pure Mg obtained by 115 deg triangle indenter. This compares with n−h˙/E*nAhmax curve estimated by indentation creep test for pure Mg.

Fig. 8

Relationship between creep parameter A and n for 115 deg and 100 deg triangle indenters

Fig. 5

Dimensionless function of pure Mg obtained by 115 deg triangle indenter

Fig. 4

Relationship between indentation creep depth and creep time by changing the creep parameter, n

Fig. 3

Two-dimensional axisymmetric FEM model of indentation test with triangle indenter and input creep parameter

Fig. 2

Relationship between indentation creep depth and creep time and indentation curve

Fig. 6

Relationship between h˙/E*nAhmax and n estimated by indentation creep test of Fig. 2

Fig. 10

Relationship between indentation creep depth and creep time for β-phase

Fig. 9

Relationship between indentation creep depth and creep time for α-phase

Fig. 11

Relationship between C/E* and σr/E* for 115 deg and 100 deg triangle indenters

Fig. 14

Comparison of estimated σr/E* with input ones for 115 deg and 100 deg triangle indenter experiments

Fig. 15

Comparison of estimated creep parameter with input ones

Fig. 12

Comparison of σr/E* estimated by experiment and σr/E*—slope and intercept of n−h˙/E*nAhmax curve for 115 deg triangle indenter (A = 2.5 × 10−10)

Fig. 13

Flowchart of estimation process

## Tables

Table 1 Material property of specimens used in this study
Table 2 Input creep parameter for FEM
Table 3 Comparison creep parameter of $α$ -Mg and $β$ -Mg17Al12
Table 4 Material property of example material used in this study
Table 5 Material property of virtual material used in this study
Table 6 Result of sensitivity analysis for representative cases
Table 7 Coefficients (α, β, γ) of Eq. (7) for each creep parameter A for Berkovich indenter (115 deg triangle indenter)
Table 8 Coefficients (α, β, γ) of Eq. (7) for each creep parameter A for 100 deg triangle indenter
Table 9 Coefficients ($α,β,γ$) for Eq. (6) in Fig. 11, in which $C/E*-σr/E*$ curve is fitted by α ln(σr/E*)2 + β ln (σr/E*) + γ

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