OPTIMIZATION ON THE MATERIAL PROPERTIES PREDICTION OF STEELS USING INSTRUMENTED INDENTATION

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OPTIMIZATION ON THE MATERIAL PROPERTIES PREDICTION OF STEELS USING INSTRUMENTED INDENTATION

I N. BUDIARSA & I N. GDE ANTARA

Research Scholar, Department of Mechanical Engineering, Udayana University, Badung, Indonesia ABSTRACT

This research work aims to investigate the relationships between constitutive material parameters of elasto-plastic materials, indentation of P-h curves and hardnesses; has been systematically researched through numerical and experimental studies, using sharp and spherical indenters of indenting modeling on hardness measurements by combining representative stress analysis and Finite Element modeling. Systematic experimental work has been carried out on steel samples of various carbon contents and heat treatment including indentation, with various loads have been used to characterise the indentation size effect of the materials, and to establish the load that gives hardness values consistently. Finite Element model of Vickers hardness tests have been developed and a method of estimating the hardness values from the P-h curves based on representative stress and energy method was established and validated with experimental data. The hardness values predicted are compared with the experimental data. Based on the hardness values of materials with a wide range of properties, the relationship between the hardness values dual indenter (Hv and HRB) with representative stress was developed, which was used to evaluate the feasibility of using hardness values, thus improve the robustness of the inverse program in predicting constitutive material parameters, with a focus on uniqueness by mapping through all potential materials ranges

KEYWORDS: Vickers Hardness, Work Hardening Coefficients, Yielding Stress, H/E & P-h Curve

Received: Mar 27, 2020; Accepted: Apr 17, 2020; Published: Jun 10, 2020; Paper Id.: IJMPERDJUN202092 1. INTRODUCTION

Indenting has been used to determine the measure of material resistance to deformation in hardness testing, and are known for being fast, easy, and non-destructive (Navrátil et al., 2009; Pharr et al., 2010). Hardness on material is based on resistance of solid to local deformation values (Tabor, 2002; Efim Mazhnik et al., 2019). Hardness values can be determined through indentation curves, and are correctly obtained from the estimated P-h curve (Budiarsa et al., 2015; 2018). The advantage of predicting the hardness values from indentation P-h curves lies in the fact that it can avoid the uncertainty of material deformation around the indenter and the true contact areas, which are difficult to quantify based on the impression (Huang et al., 2000, Huang et al., 2004; Han et al., 2005). There is a recovery process associated with the unloading progress directly influencing the change of contact area, and there is no simple/commonly acceptable way for the determination of the contact area (Gubicza et al., 1996). One potential approach is, to develop a practical methodology using an energy based analysis approach. Previous works on different materials groups showed that the hardness values can be estimated using the ratio of plastic work to total work ratio (Choi et al., 2004; Kang et al., 2010). The concept is to be adapted and further developed in this work to estimate the hardness values from indentation P-h curves using representative stress analysis values, following its relationship with hardness to modulus ratio (H/E) (Choi et al., 2004; Kang et al., 2010).

Origin a l Ar ticle

International Journal of Mechanical and Production Engineering Research and Development (IJMPERD) ISSN (P): 2249–6890; ISSN (E): 2249–8001

Vol. 10, Issue 3, Jun 2020, 1061-1074

© TJPRC Pvt. Ltd.

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The indenter shape is one important factor for indentation testing (Mirshams et al., 2006). Commonly used sharp indenters (Dao et al., 2001), spherical (Budiarsa et al., 2019), pyramids, wedges, cones, cylinders or spheres (Fischer et al., 2007); this has led to different hardness testing system, typically Brinell, Vickers hardness, Knoop, Berkovich and Rockwell hardness testing. Conical indenter was not widely used as an applied standard hardness tests, but it has been used extensively as a tool in research oriented work (Ma et al., 2012). This representative stress method has been used widely in analysing indentation process and linking materials’ properties and hardness (Dao et al., 2001; Kucharski and Mroz, 2001;

Bucaille et al., 2003). There is a recovery process associated with the unloading progress directly influencing the change of contact area, and there is no simple/commonly acceptable way for the determination of the contact area (Gubicza et al., 1996). One potential approach is to develop a practical methodology using an energy based analysis approach. Previous works on different materials groups showed that the hardness values can be estimated using the ratio of work plastic to total work ratio (Choi et al, 2004; Kang et al., 2010). The concept is to be adapted and further developed in this work, to estimate the hardness values from indentation P-h curves using representative stress analysis. Using forward and reverse analisis (Oliver and Pharr, 1992; Cheng and Cheng, 1999; Bucaille et al., 2003; Chollacoop et al., 2003; Swaddiwudhipong et al., 2005a, b). In most of cases, the work is based on using the concept of representative strain and stress. Cheng and Cheng (1999) One potential improvement is introducing more measurable data into the system to enhance the robustness of the prediction such as using plural indenters (Chollacoop et al., 2003). Recent works in nanoindentation in linking dislocation behaviour of materials and Indentation Size Affect-ISE (Yang et al., 2006; Ebisu et al., 2010; Stegall et al., 2012) also suggests that the scale/extent of ISE may potentially provide additional measurable data to improve the robustness of inverse materials properties estimation from indentation/hardness tests (Kim et al., 2008).

Based on these research works, the representative stress method has been shown to be an effective tool in studying indentation of elastic–plastic materials. It is of interest to extend its use in studying the spherical indentation in both continuous indentation and the hardness tests. The potential use of single or multiple hardness value (sharp indenter + spherical indenter) and other measurable parameters (such as ISE) in estimating the value (or range) of constitutive material properties. Most of the works has been focusing on using full P-h curves while the links established between the hardness and constitutive materials properties are mostly based on empirical data, as it is difficult to quantify the contribution of the work hardening coefficient (Gaško and Rosenberg, 2011). In this study, the established relationship between material parameters and the natural P-h curve with direct hardness of material parameters (such as yield stress and work hardening coefficients) is systematically investigated. The established relationship between material properties, continuous indentation curves and the value of hardness will also provide a useful tool for studying the feasibility of predicting yield strength and hardening coefficients, based on values of hardness and building a full understanding of problems such as uniqueness that has become a major problem in identifying reverse material properties based on indentation tests.

2. EXPERIMENT

The materials used were steel. The chemical compositions of materials are listed in Table 1.

Table 1: Chemical Compositions of the Materials

Material Condition Element Composition (%)

C Mn P S Si, Ni

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Optimization on the Material Properties Prediction of Steels using Instrumented Indentation 1063

www.tjprc.org SCOPUS Indexed Journal [email protected] Carbon Steel

0.10 % C

Normalized at

900°C 0.1 0.5 <0.04 <0.05 0.100 0.01

Mild Steel N/A 0.3 0.3 0.05 0.05 0.122 490 ppm

Carbon Steel 0.54% C

Normalized at

840°C 0.54 0.9 0.055 0.014 0.19 0.014

Carbon Steel 0.85% C

Normalized at

830°C 0.85 0.9 0.04 0.04 0.35 0.015

Specimens were sectioned, mounted in resin using thermosetting (Bakelite) and polished before hardness testing.

Selected samples (0.54 % C) have been heat treated (Quenching + Tempering at different temperatures) to generate materials with different hardness-to-modulus ratio (H/E) to aid the study the ISE effect in Vickers hardness tests. The tensile tests were performed using a Lloyd LR 30K Universal material testing machine with extensometer. The machine has a maximum loading capacity of 30 kN, with the readings being accurate to 0.5% of the force. The Lloyd tester is interfaced with a microcomputer, graphical printout of the test can be obtained and test data saved. The control console of the testing machine has the digital display, where the machine can be operated either manually or remotely. Test conducted remotely using computer has all commands transmitted via the RS232 link. The Vickers hardness testers were performed on Duramin-1 Struers Vickers hardness tester while the Leits Min iload has been used to double check some hardness data.

The Duramin-1 Struers Vickers hardness tester uses a direct load method. The indenter is in the form of right pyramid with a square base and angle of 136^o between opposite face. The load range available is from 98.07 mN to 19.61 N. The device is equipped with an optical system used to set the measurement position and test sample surface height, then measure the test indentation with a combination of the 10X eye piece and 40X objective lens. Position of test sample is done with an X- Y Stage within surface area 120 x 120 mm and test sample height approximation 100 mm maximum and 140 mm deep.

The Rockwell hardness tests was performed using: Wilson Rockwell hardness tester (ACCO Wilson instrument division, USA). The test uses spherical indenter B scale with R= 0.79 mm (Diameter of the steel ball =1/16 in). In the test, the indenter is forced into the sample under a preliminary minor load (Fm) = 10 Kgf and followed by a major loading (FM)

=100 Kgf). Characteristics of indentation is determined by the penetration depth of spherical indenter loaded on the test sample. The machine is calibrated using a testing block, the variation of the hardness is within 5%.

The Finite Element (FE) model of the Vickers indenter is designed. The Vickers indenter has the form of the right pyramid with a square base and an angle of 136⁰ between opposite face. Only a quarter of the indenter and material column was simulated as a result of plane symmetric geometry. The sample size is more than 10 times the maximum indentation depth, which is sufficiently large to avoid any sample size effect or boundary effect (Johnson, 1985). The bottom face of the material volume was fixed in all degrees of freedom (DOF) and two side faces (A and B) were symmetrically fixed in y and x direction. The element type used is C3D8R (reduced integration element used in stress/displacement analysis).

Contact was defined at indenter face with a friction coefficient of 0.2. The material of interest was allowed to move and the contact between the indenter surface, and the material was maintained at all the time. The mesh in the regions with large deformation, such as that directly under the indenter tip was refined with high mesh density in order to obtain more accurate results.

3. RESULTS AND DISCUSSIONS

This work, is concerned with elastic-plastic materials behavior with using the steel as material group, the stress-strain can be represented by following the Hooke’s law and Von Misses yield criterion with isotropic power law hardening, reverse

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dependence of the true stress σ on the true strain ε. The power law description, almost used to approximate the plastic behavior of metal material (Cao et al., 2004), the dependence of the strain on stress is commonly expressed by Eq. (1).

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with 0.0 < n < 0.5 for metals. The constitutive law surely represents a good approximation for a large number of metallic materials. A few numbers of constitutive parameter, yield stress (σy) is defined at zero offset strain, the elastic modulus E, and the strain-hardening coefficient ’n’, allows developing relatively simple approach for deducing this constitutive parameter. Hence, the true strain behavior is written to be (Swaddiwudhipong et al., 2005a)

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Where, E is the Young’s modulus, ‘n’ is the strain hardening exponent and σy the initial yield stress at zero offset strain. In the plastic region, true strain can be decomposed to strain at yield and true plastic strain as ε = εy + εp (3)

The power law elasto-plastic stress-strain behaviour and representative stress concept (Cao et al., 2004), εr is a particular plastic strain point, the stress at the point (representative stress, r)can be directly linked to the hardness. The representative strain, εr, represents the mean plastic strain defined by Tabor, 2002. At this plastic strain point, the stress can also be expressed as:

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The Vickers indenter is normally made of diamond with Young’s modulus of over 1000 GPa, which is significantly stiffer than steel (E~200GPa) and other metals such as aluminium and copper (E<200 GPa). In the FE model, the indenter was considered as a rigid body to improve the modelling efficiency (in this case to reduce the computational time required to complete the model). The model can be either load controlled or displacement controlled. In a displacement controlled model, a predefined displacement was applied and the reaction force is recorded on the reference point, representing the overall load on the indenter. Alternatively, in a load controlled model, the force is applied and the reaction force can be read from sum of the reaction force of the bottom plane of the model.

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www.tjprc.org SCOPUS Indexed Journal [email protected] Figure 1: Typical Force Indentation Depth (P-h) Curve during loading and unloading of the

Vickers Indentation (E=200 GPa, σy= 350 MPa, n=0.05).

Figure 1, shows a typical P-h curve (Force vs. Indentation depth) during loading and unloading phase with a property of (E=200MPa (for steel), σy =350 MPa, n= 0.05). The loading curve represents the resistance of material to indenter penetration, while difference between the loading and unloading curve represents the energy loss (Swaddiudhipong et al, 2005). The unloading part of the P-h curve is primarily influenced by E, as it is essentially an elastic recovery process, while the loading part of the P-h curve correlates with σy and n (Taljat et al., 1998). In the indentation process of a power law elastic plastic solid, the load P is the function of many material or testing parameters expressed as:

P = P (h, E, v, Ei, vi, σy, n) (5)

Where P is the force, h is the depth, Ei is Young’s modulus of the indenter, and vi is its Poisson’s ratio. By combining elasticity effects of an elastic indenter and an elasto plastic solid as

P = P (h, E*, σy, n) (6)

Where,

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In this work, the main material group to be investigated is steel, so the E value is fixed at 200 GPa rather than using true E* value (~187 GPa with Eindenter=1220 GPa and Esteel=200 GPa) to avoid the uncertainty with the E* values from different sources. So Eq. (6) can be simplified as:

P = P (h, E, σy, n) (8)

Incorporating equation (4) and equation (8) can be written with the representative stress rather than yield stress

P = P (h, E*, σr, n) (9)

Following dimensional analysis (Dao et al., 2001), equation (9) becomes:

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P = σt h² Π1 ((E/σr), n) (10) And thus, the curvature can be written as:

Cv= P/h² = σr h² Π1 ((E/σr), n) (11)

Alternatively,

Cv/ σr =P/(h² σr )= σr h² Π1 ((E/σr), n) (12)

Figure 2(a): The Fitting Between Cv/r vs. E/r using representative Strains (εr =0.030). (b) Correlation coefficients between Cv/r and E/r with different representative Strains. The Optimum Representative Strain Identified is

0.029.

By establishing the relationship between Cv and r, the P-h curves can be determined. One way to find the optimum representative strain is by systematically varying the strain level until the best fitting is found between the measurement (target) and materials parameters. In this case, the Cv/r and the E/r. Figure 2(a) plots the Cv/r vs. E/r

using representative strains εr =0.030 as illustrated in the figures, the fitting is directly influenced by the representative strain used. As the figure, yield stress changed from 100 to 700 MPa, the correlation between the data Cv/r vs. E/r

increased significantly. Figure 2(b) shows the change of coefficients of correlation vs. the representative strain used. The best correlation coefficient is found at a representative strain of 0.29, which is slightly different from 0.33 reported (Dao et al., 2001). The curve fitting was consistent with equation as:

Cv /r = 12.972 ln (E/r) + 13.46 (13)

With equation (13), the P-h loading curve can be predicted for a material property set (y, n). Figure 3 (a), (b), (c) shows the comparison between the FE curves (solid line) and the predicted curves (point) using the representative stress based curvature equation. Only a few points have been used in the predicted P-h curves to make it easier to compare these two sets of data. In all the cases with different materials properties, the results showed good agreements. This suggests that the approach used to predict the P-h curves is accurate.

The advantage of predicting the hardness values from indentation P-h curves lies in the fact that, it can avoid the uncertainty of material deformation around the indenter and the true contact areas, which are difficult to quantify based on the impression. There is a recovery process associated with the unloading progress directly influencing the change of contact area, and there is no simple/commonly acceptable way for the determination of the contact area (Gubicza et al.,

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www.tjprc.org SCOPUS Indexed Journal [email protected] 1996). One potential approach is to develop a practical methodology using an energy based analysis approach. Previous works on different materials groups showed that the hardness values can be estimated using the ratio work of plastic to total (Choi et al., 2004; Kang et al., 2010).

Figure 3: Comparison between FE Modelling Data (Solid line) and Predicted P-h curve of Vickers Indentation using the Curvature Equation (symbols for selected points to Preserve Clarity).

The concept is to be adapted and further developed in this work, to estimate the hardness values from indentation P-h curves using representative stress analysis. In the first stage, with known materials properties (E, y, n), the representative stress is calculated, then the ratio between the residual depth (hr) and the maximum (total) indentation depth (hm) was determined using analysis over a large range of materials properties (following a similar procedure as those used to fit the curvature–properties relationship). In the next stage, the hr/hm ratio is used to calculate the ratio between elastic work (We) and total work (Wt)during an indentation process following an established framework based on dimensional analysis (Dao et al, 2001). Then, the We/Wt is used to calculate the hardness values following its relationship with hardness to modulus ratio (H/E) (Choi et al., 2004; Kang et al., 2010).

One major part of this program is to establish the relationship between residual indentation depth (hr) and the maximum (total) indentation depth (hm). Figure 4(a) illustrates the loading and unloading curves with different material properties from FE modelling. The hm refers to the maximum depth while hr is the residual indentation depth after the indenter has been removed. The hr/hm data of a range of material properties (y: 100-900 MPa, n: 0-0.3) has been computed and the relationship between r vs hr/hm is shown in Figure 4 (b). (Only a few selected data point is shown to preserve clarity). The relationship was found to follow equation:

r = -16636 (hr/hm ) + 16369 (14)

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(a) (b)

Figure 4: Typical Loading and unloading Curves used to develop the relationship between hr/hm and representative Stress, (b) Selected Data showing the relationship: r vs. hr/hm.

Once the hr/hm ratio is established through this simple equation, it can be used to calculate the ratio between plastic work and the total work (We/Wt) following equation (Dao et al., 2001):

(15) Then the ratio between elastic work (We) and total work (Wt) can be calculated by:

(16) The hardness values can be estimated with the following relationship (Choi et al, 2004):

(17) Figure 5(a) plots the Hv data against the constitutive material properties (yield stress and the work hardening coefficient, there is no equation can be directly determined with a reasonable number of constants and correlation accuracy.

Figures 5(b) shows the correlation between Hv vs σr respectively. In the case of Hv vs σr the correlation is reasonable. The curve could be simplified as a linear line with an equation of

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Hv = 0.3115 σr + 11.186 (18)

The correlation coefficients between Eq. (18) and the data are over 99%. In the case of the Hv/r vs E/r; the correlation is less good; while the Hv vs. E/r is clearly not acceptable. The better fitting found with the Hv vs r is physically reasonable as both have a unit of N/mm².

Figure 5(a): Surface plot of Hv data over a Wide Range of material properties ( y, n), 5(b) relationship between Hv and representative Stress, 5(c) Surface plot of the HRB data over a Wide Range of Material properties (y, n), and

5(d) relationship between HRB and representative Stress.

Figures 5(c) shows the surface plot of HRB vs material properties (y and n). Similar to the case for the Vickers hardness (Figure 5(a)), there is no suitable equation to describe this relationship directly between HRB and (y, n). Figure 5(d) shows the correlation between the HRB vs r respectively. In this case, the HRB has shown a reasonable correction with all the three terms used, while the best fitting is found to be between HRB/r vs E/r with an effective representative strain of 0.033.

HRB/r = 0.0748 ln (E/r) - 0.2945 (19)

These relationships (Eqs.18 and 19) established allow direct hardness prediction from material properties. This is assessed using the two steel materials as example, the predicted Hv and HRB showed a similar level of agreement with the experimental data. In the case of the 0.1 C Steel, the hardness value is 98.368 % of the measured value; the HRB values is

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within 107% of the measured value; In the case of Mild steel, the predicted HV value is within 98.611% of the measured value, HRB is within 102% of the measured value. Similar agreement has been found in other materials (within 5% error range). This suggests that these can be used to predict the hardness values with sufficient accuracy with the measurement error ranges. In this study, the inverse analysis approach was developed by using the value of hardness to predict constitutive material properties (such as work hardening coefficient, yield stress) based on the methodology and the uniqueness of the established equation (having a good correlation with a 5% error range) through the equation 18 and 19.

This approach works through an objective function that is determined to make a comparison between experimental results data and Finite element model simulation results data. The simulation space specified covers a wide range of material properties. In this work, the yield stress is designed from 100 MPa to 1000 MPa with an increase of 10 MPa with E = 200 GPa and is expected to represent the entire range of steel properties as a material group. While the range of strain hardening coefficient used is 0.01-0.3 with an increase of 0.01. Through the simulation of objective function optimization which includes more than 2.400 hardness data for each Hv and HRB. The simulation results are a collection of possible pairs of material properties (yield stress and work hardening coefficients) that are very close to the experimental result data (in this case, the value of hardness). Optimization determined by mapping the objective function (Equation 20) (Bolzon et al., 2004).

(20) Objective Function (G) is a linear function that is useful for influencing the optimum results of a simulation or in other words minimizing errors. In this case, the data set of material properties with lower objective values will potentially be the target material. The advantage of this approach is that, in a short time, it can be used in determining the optimum flexible solution of various sets of potential material properties as a result of predicting material properties. Thus could be a useful feature in feasibility studies in particular, to establish the uniqueness of the inverse method with confidence. This is very important for practical applications, where uniqueness is crucial; otherwise, the results may converge to a wrong property sets. In this work, the approach has been applied to both single indenter approach and the dual indenters approach.

In the single indenter method, the hardness values of one indenter type (Hv or HRB) was used. Another approach explored is to use dual hardness method, in which the hardness values of Hv and HRB were jointly used to predict the properties.

This may potentially improve the accuracy and the uniqueness.

Figure 6(a) shows the plot of the objective function surface on the Hv and HRB usage data together compared to the stress yield parameter and the work hardening coefficient material, showing that there are many sets of material properties that have the same minimum objective function values marked by solid colors in the valley, indicating there are groups of materials that have variations in yield stress and work hardening, able to provide the same Hv, HRB value for both cases. Figure 6(b) shows the plot of the data set obtained from the objective surface function valley plot, where the property data set has a value of Hv, HRB with similarity below 1% for the experimental data of mild steel as a material sample. Seen in the data plot between work hardening coefficient vs yield stress, it is found that HRB hardness has the same minimum objective function value while Hv hardness is spread out, regularly. This shows with the objective function, the found characteristic material spread very close in a straight line on the relationship between work hardening coefficient versus yield stress, despite the fact that hardness is measured by different techniques and shape of the indenters,

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www.tjprc.org SCOPUS Indexed Journal [email protected] namely Hv and HRB.

Figure 6(a): Surface plot of the objective function of the combined Hv and HRB mild Steel ; 6(b) Mild steel Materials sets with an objective function within 1%.

This work also explains that no improvement can be given with the dual indenters approach, on the other hand it clearly highlights that there is a unique uncertainty in the inspection of set property, which is the main challenge for the application of inverse modeling, which has been the main concern of inverse modeling based on the indentation approach (Dao et al., 2001; Bucaille et al, 2003; Chen et al, 2007), if physical processes are not unique, in a search-based method, the results can converge to a point (set of material) at the local minimum point rather than the minimum point of the globe, thereby identifying the wrong properties (Li B., 1992). In this work, the identification of the material properties of elesto plastic is by a representative stress approach rather than repeating a limited number of FE modeling. Identification of all potential identified ingredients will effectively narrow the material search range, and provide the possibility to pre-identify based on actual material or other measurable data.

4. CONCLUSIONS

In this work, a parametric Vickers and spherical indentation model was developed based on the validated FE model, which allows the change of materials properties in the file, and then re-run the simulation. The parametric model was used to simulate the Vickers and spherical indentation over a wide range of material properties relevant to steel. The curvature of the P-h curves was then determined. The curvature was firstly correlated to the constitutive materials properties (yield stress and work hardening coefficients) to assess the possibility of establishing an equation directly. Then, the relationship between the curvatures and representative stress was determined by identifying the best representative strain. The accuracy of using the relationship established in predicting the P-h curves was investigated, which could provide a frame work for predicting hardness values from material properties and for evaluation of the uniqueness of inverse materials property estimation from hardness values.

ACKNOWLEDGEMENTS

This work was supported by Faculty of Engineering, Udayana University, Bali, Indonesia, through PNBP. The authors would also like thank to all the parties, who have helped and supported the implementation of the study from the beginning, till the publication of this paper

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