12.1 Evaluating the performance of selected constitutive laws in the modeling of friction stir processing of Mg Alloy AZ31b – Toward a more sustainable

process

A. H. Ammouri, A. H. Kheireddine, R. F. Hamade

Department of Mechanical Engineering, American University of Beirut (AUB), Beirut, Lebanon

Abstract

In modeling friction stir processes (FSP), the choice of material constitutive law directly influences the state variables output which, in turn, is critical in producing uniform metal sheets. This is especially true in AZ31b due to the temperature sensitivity of magnesium. Different constitutive laws tend to produce wide variations in the values of predicted flow stress as well as in temperature profiles especially in the stir zone. Capturing accurate state variables would improve the controllability of friction stir processes by providing suitable control models and, thus, contributing to enhanced sustainability of this process. Two constitutive laws widely used in FSP modeling of AZ31b are assessed in this work. We utilize a robust finite element model with fine-tuned boundary conditions. Comparing the output state variables with those from experiments provided for an objective assessment of the capabilities and limitations both constitutive laws over variable ranges of interest. Keywords: AZ31b, Material constitutive laws, FEM, Friction stir processing, Sustainability

1 INTRODUCTION

Friction stir processing (FSP) is a microstructure modification processing technique initially developed in 2001 by Mishra and Mahoney [1] and is based on similar principles to those of friction stir welding (FSW). As in FSW, a rotating tool is plunged into the workpiece to be processed and traversed across areas of interest to be modified. Figure 1 illustrates FSP and describes the different zones of the process. The stirred zone (SZ) is the area with the most severe mechanical deformation and frictional heating where the high strain rate and temperatures initiate dynamic recrystallization (DRX). The thermo-mechanically affected zone (TMAZ) is the area surrounding the stirred zone and is subject to both thermal effects and mechanical deformation which deform the original microstructure. The heat-affected zone (HAZ) is subject to thermal effects from the nearby welding zone.

Figure 1: FSP illustration showing affected zones.

Friction stir processes are associated with large strains, moderate to high strain rates, and elevated deformation temperatures [2] with reported values were either results of direct sensor measurements or predictions of numerical

simulations. Temperature is the only state variable that can be directly measured and, thus, the values in the literature for temperatures are of high confidence. Such reported temperatures reach up to 95% of the material’s melting point

[3]. Strains and strain rates are quite difficult to measure directly due to the complexity of the process. Thus, numerical models are typically used for the prediction of these variables. Estimated values of strain are reported to reach values of up to 125.1% [4] and maximum strain rates values to range from 101 1/s [5] upwards to 103 1/s [6] (assuming identical processing parameters). Most of the reported values were produced after tuning the tool/workpiece interface boundary conditions to produce temperature profiles match those experimentally measured. Different constitutive laws have been used in the literature to model the material behavior in friction stir processes but with each law suffering from drawbacks and with steep state variables’ gradients it was proven difficult for most of these constitutive laws to predict the variables values over the entire range of variation. Evaluated in this work is the performance of two of the more popular material models in friction stir processes. Specifically, the Sellars-Tegart [7] and Johnson-Cook [8] constitutive laws were extensively used in FSW and FSP modeling with each law having advantages in capturing accurate state variables at different regimes of interest . Both are empirical-based laws that fit experimental mechanical behavior data to mathematical equations. The validity of the models and their applicability to friction stir processing are as good as the ranges of strain, strain rate, and temperature parameters over which the laws were originally fit. The wider the range, the better, supposedly, is the applicability but this would require extensive mechanical testing ranging from tensile/compressive tests on universal testing machines to impact testing (e.g., split Hopkinson bar tests). The reason for adopting these two models is due to their ease of

G. Seliger (Ed.), Proceedings of the 11th Global Conference on Sustainable Manufacturing - Innovative Solutions ISBN 978-3-7983-2609-5 © Universitätsverlag der TU Berlin 2013

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A. H. Ammouri, A. H. Kheireddine, R. F. Hamade

implementation in numerical solvers and the availability of their material constants in the literature for many metals of interest. This paper presents a numerical-based evaluation for the performance of the Johnson-Cook and Sellars-Tegart material models for FSP of magnesium alloy AZ31b. The importance of this comparison is to provide the reader with a reference on the accuracy of each model for capturing the state variables at different areas of interest in the friction processed zone. This would yield to better numerical simulation results that would, ultimately, enhance the sustainability of the process. 2 THE FE MODEL

A 3D thermo-mechanically coupled FE model was developed using the commercial FEA software DEFORM-3D™ (Scientific

Forming Technologies Corporation, 2545 Farmers Drive, Suite 200, Columbus, Ohio 43235 [9]). The meshed model shown in Figure 2 consists of a tool, a workpiece, and backing plate. Both the tool and the backing plate were modeled as rigid un-deformable bodies where only heat transfer was accounted for while the workpiece was modeled as a plastic body subject to both deformation and heat transfer.

Figure 2: The meshed FEM model used in the simulations

The considered tool had an 18 mm cylindrical shoulder with a 6 mm diameter smooth unthreaded pin that extrudes 6 mm from the bottom of the shoulder. The tool was tilted 3° about the vertical axis in the processing direction to further improve material flow. Both the workpiece and the backing plate had an area of 80x54 mm2 and a height of 10 mm.

Materials used in the FEM model were H13 steel for the tool, AISI-1025 steel for the backing plate and AZ31b for the workpiece. Table 1 summarizes the different mechanical and thermal properties of AZ31b obtained from literature [10-11].

Tetrahedral elements were used in the FEM model with active local re-meshing triggered by a relative interference ratio of 70% between contacting edges. The tool and the backing plate were meshed for thermal analysis purposes with each containing around 6000 and 5000 elements respectively while the workpiece had around 16000 elements. To further capture the state variables at the tool-workpiece interface, a rectangular mesh control window was applied around the processing area of interest where finer mesh elements were created.

Heat transfer with the environment was accounted for all the three meshed objects with a convective heat coefficient of 20 W/(m2 ºC) at a constant temperature of 293K. The heat transfer

coefficient between the tool-workpiece and backing plate-workpiece interfaces was set to 11 kW/(m2 ºC) [12].

Friction at the tool-workpiece interface is a significant factor in any FSP/FSW simulation. It is determined that 86% of the heat generated is due to frictional forces [13]. Determination of the friction factor is a daunting challenge due to the variation of temperature, strain rate, and stress. A literature search for identifying suitable expressions of friction coefficient in magnesium alloys revealed that ring upsetting and compressions tests are used for determining the coefficient of friction [14-15]. It is found that the friction factor increases with temperature [16]. However, this increase of friction factor with temperature is valid until the liquidus temperature of AZ31b (630°C) is reached at which the friction drops drastically. Experimental values of the coefficient [17] were entered to the model and then extrapolated by tuning different runs and analyzing state variables. The friction coefficient vs. temperature behavior that was arrived at by tuning in this work is shown in Figure 3 co-plotted on which are the experimental values from [17]. The performance of the utilized FE model along with the assumed friction coefficient was previously validated by the authors [18-19].

Figure 3: Friction coefficient VS Temperature as used in the FE model; shown compared with experimental [17]

Table 1: Material properties of the utilized AZ31b

Property Value

Elastic Modulus [10] 44830 MPa Poisson’s ratio [10] 0.35 Coefficient of thermal expansion [10] 2.65x10-5

Thermal conductivity [10] 96 N/(s K) Heat capacity [11] 2.43 N/(mm2 C) Emissivity [11] 0.12

The constitutive laws considered for modeling the material behavior were the Sellars-Tegart and the Johnson-Cook laws.

The Sellars-Tegart material model considers the material to be rigid visco-plasic where flow stress (�̅�) is related to temperature (T) and strain rate (𝜀̅̇) according to

1

11 1sinh

H n

RTeA

(1)

Where A, α, and n are material constants, ΔH being the activation energy, and R the universal gas constant.

0

0,1

0,2

0,3

0,4

200 400 600 800 1000

Fric

tio

n C

oe

ffic

ien

t

Temperature (K)

As used in FE modelExperimental data [17]

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Evaluating the performance of selected constitutive laws in the modeling of friction stir processing of Mg Alloy AZ31b – Toward a more sustainable process

The Johnson-Cook model accounts for the effects of strain (𝜀), strain rate (𝜀̅̇) and temperature (T) according to

0

1 ln 1

m

n room

melt room

T TA B C

T T

(2)

Where A, B, C, n, and m are material constants. Troom and Tmelt

being the room and melting temperatures. 𝜀̅0̇ is used to normalize the strain rate and is equal to 1s-1

For preserving the authenticity of the comparison, the material constants of both equations (1) and (2) were determined by fitting the equations to experimentally collected stress strain curves of tensile tests carried out for this purpose. The ranges of temperatures and strain rates of the performed tests were 25-300 °C and 10-4-10-1 s-1 and the specimen used was according to ASTM E2448-11.

The material constants of AZ31b for the Sellars-Tegart and Johnson-Cook models are described in tables 2 and 3 respectively.

Table 2: Constants of the Sellars-Tegart model for AZ31b [13]

A α n

27.5 0.052 1.8

Table 3: Constants of the Johnson-Cook model for AZ31b [20]

A B n C m

224 380 0.761 0.012 1.554

The simulation process parameters for the traverse phase were 1000 RPM for the tool rotational speed and 90 mm/min for the tool traverse speed. These parameters were selected according to optimum processing conditions for AZ31b which were determined by the authors [18]. For each constitutive model, simulations were run and the state variables were determined and logged for comparison.

3 RESULTS AND OBSERVATIONS

For each of the constitutive models, a complete traverse FSP pass was simulated where the tool starts from a pre-set position inside a groove in the workpiece at the final position of the plunging phase. Plunging phase wasn’t simulated in this

work since our only interest is in the traverse phase which will have the most significant effects on the state variables in the processed area. After the conclusion of the simulations, strains, temperatures and stresses of a cross-section passing through the center of the tool were captured and the results were plotted accordingly. For systematic comparison all the differences between the predicted state variables will be calculated as percentage of the Sellars-Tegart results. 3.1 Strain comparison

Strain output values for both constitutive models are shown in Figure 3. The trend and levels of strain are of the same order of magnitude for both cases as one would expect given that

FEM solves initially for deformations and their derivatives, the strains independently of the stress. The average reported total Von Mises strain values for the Johnson-Cook and Sellars-Tegart models were 0.91 and 1.08 mm/mm respectively. These values are of the same order of magnitude of those reported in the literature [5].

Figure 3: Effective strain (mm/mm) for the (a) Sellars-Tegart

and (b) Johnson-Cook; 1000 RPM, 90 mm/min The ~16% difference between the maximum reported values of both constitutive models can be due to the inter-object boundary condition variation during the re-meshing process. Appreciable values of strain was only observed in the stir zone and at the interface between the TMAZ and the stir zone. The maximum strain value was observed on the advancing side of the processed area (the left side) where the tool rotation and workpiece translation oppose each other causing rise in state variables at the tool/workpiece interface. 3.2 Temperature comparison

Temperature snapshots of the processed area cross-section are shown in Figure 4. The trend for both models differed slightly for the temperature state variables. Higher temperature profile values can be noticed for the Johnson-cook model. The maximum reported value for the temperature were 452 ⁰C at the tool/workpiece for the Sellars-Tegart model and 596 ⁰C for the Johnson-Cook model (which is close to AZ31b liquidus temperature). The Johnson-Cook model values of the maximum reported temperature exceeded that of the Sellars-Tegart model by 31.8%. The temperature profile however seems to extend more towards the TMAZ and HAZ in the case of Johnson-Cook. It was also observed that the thermal effects in the Johnson-Cook model reached the bottom of the workpiece where values of 420 ⁰C were measured compared to values of 329 ⁰C for the Sellars-Tegart model. This discrepancy will be discussed in the conclusion section.

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A. H. Ammouri, A. H. Kheireddine, R. F. Hamade

Figure 4: Temperature (⁰C) for the (a) Sellars-Tegart and (b)

Johnson-Cook; 1000 RPM, 90 mm/min 3.3 Stress comparison

The reported total Von Mises stress values for the simulations are described in this section where major discrepancies were observed between the model predictions. The maximum reported values for the Sellars-Tegart model were 303 MPa compared to 524 MPa for those of the Johnson-Cook model. For proper comparison the color scales of both models was set at the ultimate stress of AZ31b at 25⁰C (330 MPa). The Johnson-Cook model over predicted stress values of those of the Sellars-Tegart by 73% in the stir zone. A different trend of stress distribution was observed for the two test cases which can be clearly seen in Figure 5.

Figure 5: Effective stress (MPa) for the (a) Sellars-Tegart and

(b) Johnson-Cook; 1000 RPM, 90 mm/min 4 CONCLUSIONS

The following findings may be concluded from the results and observations described in Section 3:

1. Strain values are found to be practically independent of whether the Sellars-Tegart constitutive material law (equation (1)) or the Johnson-Cook law (equation (2)) is used. This is due mainly to how FEM calculates strains from differentiating deformations independently of the constitutive laws used. Strain values are a strong function of other important factors that need to be defined such as the boundary conditions set between tool and work.

2. The Johnson-Cook law (equation 2) appears to over predict the values of strain in the stir zone where strains exceed saturation values whereas the strain-independent Sellars-Tegart law (equation 1) accounts for saturation. This increased stress estimates would also result in increased temperature predictions in the case of the Johnson-Cook model compared with the Sellars-Tegart law.

3. The Sellars-Tegart constitutive material law (equation 1) is incapable of capturing the process parameters outside the stir zone especially due to its inability to capture strain hardening in areas with low strains such as the TMAZ.

One can deduce from the above findings that each constitutive law has its advantages over the other in specific areas of the processed zone. If the main interest is in capturing the stir zone state variables then the Sellars-Tegart would be the model to work. Johnson-Cook on the other hand should be used for assessing the TMAZ and HAZ since it captures better state variables in these zones. 5 SUMMARY

A robust FE numerical model was used to compare the performances of two different material constitutive laws used in the friction stir processing of magnesium alloy AZ31b. Both laws resulted in marked variations in the estimated values of the output state variables namely stress and temperature. Such variations were especially noted across the different regions within the friction processed areas. Each of the two law shows advantages over the other depending on the processing conditions. The Sellars-Tegart model yielded more representative values of state variables in the stir zone whereas the Johnson-Cook model yielded more representative values in the TMAZ. The high strain values observed in the stir zone resulted in stress over-prediction when using the Johnson-Cook law due to the strain hardening constituent of the model. The lack of strain hardening in the Sellars-Tegart law results in constant stress values at different strains which results in large errors in areas with low strains. Care should be taken when choosing which constitutive law to be used while modeling friction stir processes depending on the accuracy of the state variable prediction required at a desired area of interest. The accuracy of the captured state variables is a key factor to improve the controllability of friction stir processing by providing suitable control models which contribute to enhanced sustainability of the process. 6 ACKNOWLEDGMENTS

This publication was made possible by the National Priorities Research Program (NPRP) grant # 09-611-2-236 from the Qatar National Research Fund (a member of The Qatar Foundation). The statements made herein are solely the responsibility of the authors. The first author gratefully acknowledges the support of Consolidated Contractors

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Evaluating the performance of selected constitutive laws in the modeling of friction stir processing of Mg Alloy AZ31b – Toward a more sustainable process

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