Journal of Materials Science

, Volume 48, Issue 6, pp 2454–2461

Finite element analysis of plastic deformation in variable lead axisymmetric forward spiral extrusion

Authors

  • A. Farhoumand
    • Department of Mechanical and Aerospace EngineeringMonash University
  • P. D. Hodgson
    • Institute for Technology Research and InnovationDeakin University
    • Department of Mechanical and Aerospace EngineeringMonash University
Article

DOI: 10.1007/s10853-012-7033-7

Cite this article as:
Farhoumand, A., Hodgson, P.D. & Khoddam, S. J Mater Sci (2013) 48: 2454. doi:10.1007/s10853-012-7033-7

Abstract

A modified axisymmetric forward spiral extrusion (AFSE) has been proposed recently to enhance the strain accumulation during the process. The new technique is called variable lead axisymmetric forward spiral extrusion (VLAFSE) that features a variable lead along the extrusion direction. To assess the effect of design modification on plastic deformation, a comprehensive study has been performed here using a 3D transient finite element (FE) model. The FE results established the shear deformation as the dominant mode of deformation which has been confirmed experimentally. The variable lead die extends strain accumulation in the radial and longitudinal directions over the entire grooved section of the die and eliminates the rigid body rotation which occurs in the case of a constant lead die, AFSE. A comparison of forming loads for VLAFSE and AFSE proved the advantages of the former design in the reduction of the forming load which is more pronounced under higher frictional coefficients. This finding proves that the efficiency of VLAFSE is higher than that of AFSE. Besides, the significant amount of accumulated shear strain in VLAFSE along with non-axisymmetric distribution of friction creates a surface feature in the processed sample called zipper effect that has been investigated.

Introduction

Severe plastic deformation (SPD) processes have been widely used to produce ultrafine grained (UFG) materials that possess a unique combination of superior mechanical and physical properties [1]. Equal channel angular extrusion [2], high-pressure torsion (HPT) [3], twist extrusion (TE) [4] and groove pressing [5] are among SPD processes to produce UFG materials. The non-symmetric nature of the TE process results into unfavourable heterogeneous material property changes. To overcome these disadvantages, some efforts have been made on the modifications of these methods such as dual equal channel lateral extrusion [6], torsional-equal channel angular pressing [7], twist-channel angular pressing [8] and continuous HPT [9]. As the samples geometry is unchanged after deformation by most SPD techniques, it is possible to repeat the process to impose a desired amount of strain to the sample.

However, it would be more favourable if such amount of strain could be achieved during single/continuous pass operation to increase the process efficiency while decreasing the process cost. As a result, there have been some investigations on applying an intense strain on a material in a single-processing step such as continuous high-pressure torsion [9], torsion extrusion [10], cone on cone [11] and shear extrusion [12]. In the aforementioned processes, rotation of the die facilitates strain accumulation. These processes are complicated and expensive because of the requirement for special rotating die. Besides, the slippage between die and material is inevitable in those processes, which reduces the efficiency of the strain accumulation. Axisymmetric forward spiral extrusion (AFSE) has been proposed recently [13] to overcome these problems. FEM analysis of the process [14] indicates that AFSE deformation is limited in a narrow zone at the vicinity of a plane normal to the extrusion axis; where the container and spiral die meet. The samples experience rigid body motion only before and after passing this narrow zone demanding several passes of AFSE to implement the desired amount of deformation.

Variable lead axisymmetric forward spiral extrusion (VLAFSE) has been recently proposed to overcome the requirements for multipass AFSE using a modified die design concept [15]. A detailed solution of VLAFSE is needed to understand the strain distribution, the required forming loads and the process efficiency to implement the VLAFSE into practical applications. In this research, the FEM analysis of VLAFSE will be performed to investigate the strain distribution and relevant process requirements.

FE modelling of VLAFSE

A 3D model of VLAFSE was developed using ABAQUS [16] software. The FE model consists of different components as shown in Fig. 1. The VLAFSE die was constructed in NX software [17] and then imported into ABAQUS for the simulations. The punch, container and VLAFSE die were chosen as discrete rigid parts, while the specimen was defined as deformable and assigned material properties.
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Fig. 1

FE model of VLAFSE in ABAQUS software (a) and VLAFSE die (b)

The VLAFSE die parameters, namely chamfer length, \( L_{I} \) and twist length, \( L_{II} \), were selected as 3 and 17 mm, respectively, while the helix angle, \( \beta \), varied between 13.35° and 34.59°. Details of VLAFSE can be found elsewhere [15]. The material used in this study was an ultra-low-carbon Ti-IF steel (0.005 % C, 0.003 % N, 0.13 % Mn, 0.084 % Ti, 0.042 % Al (in wt%)). The torsion test was conducted at room temperature to acquire the stress–strain relation the result of which is presented in Fig. 2.
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Fig. 2

Stress–strain relation for Ti-IF steel specimen

The power law relation was assumed for the corresponding constitutive model of Ti-IF steel specimen as follows:
$$ \sigma \, = \,398 \varepsilon^{0.28}. $$
(1)

The reasonable agreement between the experimental results with those of Eq. 1 predictions, shown in Fig. 2, suggests the power law as a reasonable constitutive behaviour for Ti-IF steel.

During VLAFSE, a back pressure was applied to ensure the penetration of the specimen into the die grooves. The back pressure is comparable to the yield strength of the material being deformed. Therefore, the pressure created on the die wall by the specimen, \( \sigma_{y} \), becomes so high that the friction pressure, \( \tau \), can be described by Zibel’s law as
$$ \tau \, = \,m\frac{{\sigma_{y} }}{\sqrt 3 },$$
(2)
where \( m \) is the friction factor.

The process involves large deformations and therefore an explicit method with dynamic displacement was employed in the model. The specimen contact with die and container were modelled with the surface to surface finite sliding contact pair algorithm. The separable contact algorithm was used to describe the specimen-die and specimen-chamber frictional conditions, while the inseparable algorithm was used to describe punch–specimen interaction. All degrees of freedom of VLAFSE die and container were fixed. Also, all degrees of freedom for the punch are closed apart from that of translation, which is in the z direction (Fig. 1b). A backpressure of 200 MPa was applied to specimen that was meshed with 3D stress elements with reduced integration scheme (C3D8R). The accuracy of the FE model was studied followed by the simulation results.

Verification of FE model

According to a kinematic investigation of VLAFSE [15], the accumulated plastic strain for a typical point located at a radius of r and a longitudinal distance of z measured from point O, as shown in Fig. 1b, can be calculated as
$$ \varepsilon \, = \, \frac{1}{\sqrt 3 } \left( {{\raise0.7ex\hbox{$r$} \!\mathord{\left/ {\vphantom {r {r_{0}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${r_{0}^{2} }$}}} \right)R_{c} \tan \beta , \,R_{c} \, = \,\sqrt {z^{2} \, + \,r^{2} }, $$
(3)
where \( r_{0} \) and \( \beta \) are specimen radius and instantaneous helix angle, respectively. In order to compare the FE results with analytical predictions by Eq. 3, the cross-section of a specimen that was located at \( z\, = \,5 \,{\text{mm}} \) was selected which corresponds to \( \beta \, = \,20^\circ \). Given \( r_{0} \, = \,6.5 \,{\text{mm}} \), the strain was calculated by Eq. 3 at different radii. Due to the presence of grooves in VLAFSE and its corresponding effect on localized deformation on specimen’s surface in FE model, the average values of effective strain from the FE results were extracted and compared to Eq. 3 predictions; the results of which are depicted in Fig. 3.
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Fig. 3

The strain in VLAFSE predicted by kinematics and FE results

According to Fig. 3, the maximum strain is achieved at the interface of the die and specimen. The peripheral regions experience a higher degree of rotation while moving along the extrusion direction than those in the vicinity of the specimen centre. The derivative of rotation along the extrusion direction corresponds to the plastic strain increment [15], while its corresponding effective plastic strain can be calculated by integration. As a result, the regions that are closer to the specimen surface experience higher accumulated plastic strain than those in the vicinity of specimen centre. Furthermore, the predicted plastic strain using Eq. 3 is in good agreement with those of FE results within a cylindrical zone radius of 4 mm radius with deviation from the FE results outside the zone. This can be explained by noting that only the tangential and longitudinal components of velocity were considered in the derivation of strain by Eq. 3, while the radial components of velocity due to the grooves were overlooked [15]. In the vicinity of the grooves, a radial component of velocity contributes significantly to the total tri-axial effective strain. As the radius increases, the effect of grooves becomes more pronounced and hence the deviation between the FE result and analytical calculation increases. Therefore, application of Eq. 3 is most accurate for the specified cylindrical zone which does not see the groove effect.

To clarify the effect of the new design on strain distribution and process requirements, a comparison between AFSE and VLAFSE was made, and the results of which follow.

Results and discussion

Strain distribution

FE analysis was performed to compare the strain distribution in VLAFSE and AFSE for a special case with a helix angle of 50°. (Details of the FE analysis for AFSE can be found elsewhere [14].) The equivalent plastic strain (PEEQ) in the longitudinal cross-section of specimen has been shown for VLAFSE and AFSE in Figs. 4 and 5, respectively.
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Fig. 4

Equivalent plastic strain in longitudinal cross-section of specimen inside VLAFSE die

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Fig. 5

Equivalent plastic strain in longitudinal cross-section of specimen inside AFSE die

Figures 4 and 5 show that specimen enters the chamfer section, zone \( I \), the strain starts to accumulate in both VLAFSE and AFSE. Having passed zone \( I \), the strain accumulation terminates for AFSE while it still continues for VLAFSE which asserts the contribution of zone \( II \), twist zone, in deformation during VLAFSE. The equivalent plastic strain for three different elements that are located at \( 2 \), \( 4 \) and 6 mm from specimen centre in the end of the twist zone are indicated in Figs. 6 and 7 for VLAFSE and AFSE, respectively.
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Fig. 6

Equivalent plastic strain in VLAFSE for different elements

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Fig. 7

Equivalent plastic strain in AFSE for different elements

Considering the die geometry that was shown in Figs. 4 and 5, the specimen enters twist zone, zone \( II \), having passed the chamfer one, zone \( I \), by moving along z direction. In Fig. 7, the strain rises in zone \( I \) and remains constant in zone \( II, \) which clearly indicates that the material experiences deformation in the chamfer section, while rigid body rotation occurs in the twist section. In contrast, Fig. 6 represents that strain continues to rise in zone \( II \) after passing through zone \( I \). Therefore, one can see that in VLAFSE, both die zones, namely chamfer and twist, contribute towards deformation, while in AFSE deformation only takes place in chamfer zone. Furthermore, for the same helix angle (in this case 50°), VLAFSE and AFSE result in the same amount of accumulated plastic strain at the end of the process. In VLAFSE, gradual strain accumulation occurs in both zones in contrast to AFSE in that strain accumulation occurs in zone \( I \). The benefit of this is conversion of the harmful effect of friction in zone \( II \) of AFSE to useful impetus for shear deformation in the new design of VLAFSE.

Deformation mode in VLAFSE and its experimental investigation

Figure 8 represents the shear strain \( \varepsilon_{z\theta } \), normal strains \( \varepsilon_{zz} \) and \( \varepsilon_{rr} \) during VLAFSE for an element that is 6 mm apart from specimen centre in VLAFSE.
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Fig. 8

The shear and normal strains in VLAFSE

Simulation results indicate that at the initial stage of VLAFSE normal strain along extrusion direction, \( \varepsilon_{zz} \), is positive while for that of radial direction, \( \varepsilon_{rr} , \) is negative. This means that the element is extended in the extrusion direction and is compressed in the radial direction which can be attributed to the presence of the chamfer zone at the beginning of VLAFSE die. But as the specimen proceeds into the twist zone of the die, the normal strains disappear which gives rise to a shear strain, \( \varepsilon_{z\theta } .\) Therefore, based on the FE results the dominant mode of deformation in VLAFSE is shear deformation. Experimental VLAFSE [15] was implemented to verify the FE result. Here, the VLAFSE was performed on a lead sample the product of which has been shown in Fig. 9a, while the shear planes are highlighted by the arrow in Fig. 9b. The shear planes in Fig. 9b are perpendicular to the extrusion direction, z. This finding is in agreement with the FE results which predicted the shear deformation to be the dominant mode of deformation in VLAFSE. Besides, as one can see from Fig. 9b, a surface feature has been formed on the sample periphery along shear planes which is called the zipper effect.
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Fig. 9

The sample processed by VLAFSE (a), the shear planes and zipper effect on periphery of sample (b)

To study the zipper effect in detail, FE simulation was performed for the lead specimen during VLAFSE. The constitutive behaviour of lead was acquired by torsion tests [10] at room temperature and imported into FE model as depicted in Fig. 10. FE analysis was performed, the result of which is presented in Fig. 11. As highlighted in Fig. 11, in the vicinity of the grooves, the effective plastic strain is significantly higher in comparison to that of its neighbouring area. This can be attributed to the localized plastic deformation due to the groove effect. The plastic strain in vicinity of grooves in Fig. 11 is 4.18, which is higher than the fracture strain of lead in Fig. 10 which can be estimated as 3.5. Therefore, it can be concluded that localized deformation in vicinity of grooves causes the plastic strain to exceed the fracture strain of lead and resulting in material failure. Figure 9b reveals that only some parts of the fractured shear planes are elongated in the extrusion direction, z. This behaviour can be attributed to the effect of friction in VLAFSE which is non-axisymmetric due to the presence of grooves. The friction between the VLAFSE die and the specimen surface causes some area of fractured shear planes to elongate with respect to extrusion direction which leads to zipper effect in Fig. 9b. Therefore, it can be concluded that the zipper is the result of direct and indirect effects from the groove that are localized plastic deformation and non-axisymmetric friction, respectively.
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Fig. 10

The stress–strain behaviour of lead from torsion test

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Fig. 11

The strain distribution in lead specimen during VLAFSE and localized deformation in the vicinity of grooves

The presence of grooves in VLAFSE also causes the accumulated strain to be heterogeneously distributed in the specimen cross-section; the effect of which will be explained next.

Strain heterogeneity in VLAFSE

Figure 12 shows the longitudinal cross-section of the specimen inside the VLAFSE die. The strain heterogeneity can be identified in three directions which are radial, longitudinal and circumferential and designated by \( r,\;z \) and \( \theta \), respectively. The heterogeneity in the \( r \) direction is due to the nature of VLAFSE process which imposes higher strain to periphery of the specimen according to Eq. 3. Besides, the strain accumulation starts at the onset of zone \( I \) and is continuous till the exit point of zone \( II \) which leads to strain heterogeneity in \( z \) direction. After specimen passed through the VLAFSE die, the deformation terminates which leads to even distribution of strain in the specimen longitudinal cross-section as shown in Fig. 13.
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Fig. 12

The longitudinal cross-section of specimen inside VLAFSE die

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Fig. 13

The longitudinal cross-section of specimen outside VLAFSE die

Therefore, the strain heterogeneity in the \( z \) direction disappears considerably after the VLAFSE process has finalized. One can notice that even after VLAFSE, Fig. 13, the strain distribution in the core of specimen is not even. Two non-homogeneous regions in Fig. 13 can be identified as the first of which is located at the interface of zones \( I \) and \( II \). This region is located at the proximity of die entrance. At the entrance of the die, the specimen experiences a contact with punch which exerts normal pressure and frictional force. In contrary, the second non-homogeneous region is located at the end of zone \( II \) in Fig. 13 which contains a free surface. Thus, it can be assumed that there is no active normal/shear stress or frictional force acting on that free surface. Therefore, it can be concluded that the difference in stress state in aforementioned regions give rise to heterogeneity of strain in z direction after VLAFSE completion.

Presence of grooves results in strain heterogeneity normal to cross-section of the specimen even after processing. To clarify this, quantitative assessment of the effect of groove geometry, such as groove depth and width, on strain heterogeneity in the θ direction was performed. A circular path in the VLAFSE sample cross-section at an arbitrary radius of r was defined and mapped on the cross-section of the processed specimen.

The number of sampling points was selected to be proportional to the number of grooves (8) in the VLAFSE die and their positions were measured with respect to the middle of a groove. Therefore, 16 equally spaced sampling points were selected for each circular path and the strain components were calculated at different radii. The strain heterogeneity index (SHI) was calculated as
$$ {\text{SHI}}\, = \, \frac{{\left( {\varepsilon_{ \hbox{max} } \, - \, \varepsilon_{ \hbox{min} } } \right)}}{{\varepsilon_{\text{ave}} }}, $$
(4)
where \( \varepsilon_{ \hbox{max} } \), \( \varepsilon_{ \hbox{min} } \) and \( \varepsilon_{\text{ave}} \) denote the maximum, minimum and average effective plastic strain, respectively, over the circular path evaluated at r.

A normalized distance was defined as the ratio of the circular path radius to the sample radius and designated by \( \psi \). Calculations for SHI were performed for several radii and its variations with \( \psi \) are shown in Fig. 15 for different groove geometries.

According to Fig. 14, both the groove width and depth affect the strain heterogeneity. A maximum SHI of 0.7 at the sample surface can be seen in Fig. 14, while groove depth and width are 0.15 and 1 mm, respectively. Increasing the groove width from 1 to 2 mm at a constant groove depth of 0.5 mm, increases SHI from 0.5 to 0.6, while increasing the groove depth from 0.05 to 0.15 at 1 mm groove width, increases SHI from 0.5 to 0.65. This finding proposes that the effect of groove depth on SHI is more prominent than that of groove width. Besides, a change in groove width from 1 to 2 mm at constant grove depth of 0.05 mm, leads to an increase in SHI for points that are located near to the specimen surface \( ( \psi \, > \,0.7). \) In contrast, an increase in groove depth from 0.05 to 0.15 mm at a constant groove width of 1 mm, shifts the total SHI graph to higher values by more than 0.1.
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Fig. 14

Variation of SHI vs. ψ for different groove geometries

Although, changing the groove width has only a surface effect on strain heterogeneity, changing the groove depth in VLAFSE extends the strain heterogeneity towards the specimen centre.

A comparison was made between VLAFSE and AFSE to clarify the effect of modified design on the process requirements.

Forming load and efficiency

Figure 15 shows the required forming load in AFSE and VLAFSE for the helix angle of 50° under different friction conditions.
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Fig. 15

The forming load in AFSE and VLAFSE underdifferent frictional conditions

As Fig. 15 indicates, the forming load in the frictionless condition (\( m\, = \,0 \)) is equal for AFSE and VLAFSE. Forming load in VLASE can be considered to be comprised of deformational and frictional load that are \( F_{\text{dif}} \) and \( F_{\text{fri}} \), respectively. Under frictionless conditions, \( F_{\text{fri}} \) is equal to zero and only \( F_{\text{dif}} \) contributes towards the forming load. Considering the identical helix angle (50°) in simulations for both AFSE and VLAFSE, the corresponding \( F_{\text{dif}} \) is equal which leads to a similar forming load trend in Fig. 15. An increase in friction factor, \( m \), increases the forming loads of VLAFSE. This leads to VLAFSE and AFSE to deviate from each other. For instance, an increase in \( m \) from 0.1 to 0.2, increases the required AFSE load by 185 % which is in contrast to VLAFSE with only 65 %. This is due to change in \( F_{\text{fri}} \) and one may conclude that AFSE is more sensitive to frictional conditions than VLAFSE. This behaviour can be attributed to the elimination of the rigid body rotation in zone \( II \) in VLAFSE and as a result less power consumption to overcome friction. Therefore, the same amount of deformation can be achieved in VLAFSE at lower applied loads. This indicates that VLAFSE provides higher efficiency in comparison to other similar processes. Considering that \( F_{\text{fri}} \) is applied to overcome the frictional forces while \( F_{\text{dif}} \) is applied to induce the favourable deformation during the process, the efficiency of deformation, \( \eta \), can be expressed as
$$ \eta \, = \, \frac{{F_{\text{dif}} }}{{F_{\text{dif}} + F_{\text{fri}} }}\, \times \,100, $$
(5)
where \( F_{\text{dif}} \) is the forming load that has been presented in Fig. 15 where a frictionless condition applies (\( m = 0 \)) and \( (F_{\text{dif}} \, + \,F_{\text{fri}} ) \) is the forming load due to presence of friction. The maximum forming loads for frictionless and different frictional conditions were extracted from Fig. 15 and \( \eta \) was calculated according to Eq. 5, the result of which is illustrated in Fig. 16.
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Fig. 16

The efficiency of VLAFSE and AFSE under different frictional conditions

For both processes, the efficiency is inversely proportional to the friction factor, \( m \). An increase in \( m \) increases \( F_{\text{fri}} \) at constant value for \( F_{\text{dif}} \) which results in an increase in value of \( \eta \) (Eq. 5). This suggest that efficiency of VLAFSE is always higher than that of AFSE which is more prominent for higher values of \( m \). From this, it can be implied that AFSE is much more sensitive to the friction conditions than VLAFSE. This makes VLAFSE an attractive option for industrial applications due to less demand in force and dies design requirements than similar processes such as AFSE.

Conclusions

The consistency between the developed FE model results and those of kinematics verified the accuracy of the proposed FE model. The FE results show the dominance of shear deformation during VLAFSE which has been confirmed by experimental investigation. Based on the FE results, the rigid body rotation of zone \( II \) in VLAFSE was eliminated which results in a required forming load lower than that of AFSE. The difference in required forming load becomes more substantial as the friction factor increases. It was concluded that VLAFSE is less sensitive to frictional conditions than AFSE. Furthermore, the improved efficiency of VLAFSE over its counterpart AFSE suggest the former as a better candidate for applications concerning less sophisticated die design requirements. Also, the observed zipper effect on the processed sample periphery can be attributed to localized plastic deformation in the vicinity of grooves along with non-axisymmetrical friction.

Copyright information

© Springer Science+Business Media New York 2012