- Email: [email protected]
Strain distributions in plane strain deformation through wedge-shaped dies
Journal of Materials Processing Technology 125–126 (2002) 317–322
Strain distributions in plane strain deformation through wedge-shaped dies M. Salimi Department of Mechanical Engineering, Isfahan University of Technology, P.O. Box 84154, Isfahan, Iran Received 2 December 2001; received in revised form 2 March 2002; accepted 2 March 2002
Abstract In this paper, a theoretical model for plane strain deformation based on a slip line field (SLF) solution is developed to provide the strain distribution across a section of a product in plane strain drawing. The results obtained from the SLF solutions were compared to those of the finite element method. From both solutions, it is shown that the shear strain at a position close to the contact area is greater than that of the centre of the product. Ratios of strain at the surface to strain at the centre against the ratio of the semi-height to the contact length were obtained. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Strain distribution; Plane strain; Slip line field; Drawing; Extrusion
1. Introduction In the process of forming of metals, the material properties and surface quality of the product are influenced by different parameters. An understanding of how the different parameters influence the final product can help to avoid production defects and improve product quality. The problem of deforming a sheet through wedge-shaped dies (drawing/extrusion) was investigated by many investigators under the assumption that the material is in the state of plane plastic flow and obeys the Saint Venant–Mises yield condition [1–3]. Exact stress analysis of the deformation was first proposed by Hill in 1950 [4] using slip line field (SLF) theory developed by Hencky [5] under idealised conditions, i.e. plane strain of a rigid–perfectly plastic solid. Subsequent work has consolidated this, though without anything new in principle. The stress state in the deformation zone is usually determined by the method of stress discontinuity surfaces [6], the upper bound technique [7], the SLF method [4], the finite element method (FEM) [8–10] and other techniques such as dual stream functions [11]. In recent years attempts were made to extend slip line theory to axi-symmetric deformation [12,13]. However, apparently, little theoretical information has been published to describe the strain distribution. From a manufacturing point of view, the strain distribution and the heat developed in the process determines the maxiE-mail address: [email protected] (M. Salimi).
mum deformation speed that can be used for producing sound products and influence lubrication conditions and tool life.
2. Analysis In sheet drawing processes, progressive reduction in thickness is caused by pulling the sheet through a die (Fig. 1). Except at the start and the end of the deformation, processes such as extrusion and drawing are usually considered as steady state. Steady state solutions in these processes are needed for equipment and die design and for controlling product properties. In the present investigation an attempt is made to predict the strain distribution theoretically by reference to SLF. A detailed discussion of the slip line theory and its application can be found in [4]. 2.1. Construction of SLF The SLF solution for strain distribution across the section is obtained using the following simplifying features: 1. Reduction per pass is assumed to be small. In drawing, however, this cannot be very large, since the maximum amount of reduction per pass to avoid rupture is generally small. 2. Equal and small projected contact lengths between the workpiece and the two die surfaces were assumed so that the plastic region joins across the centre of the slab.
0924-0136/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 0 3 4 1 - 2
318
M. Salimi / Journal of Materials Processing Technology 125–126 (2002) 317–322
Fig. 1. The deformation zone in plane wedge or sheet drawing assuming no friction at the interface (a < 10; r ¼ 5%).
3. The projected contact lengths are treated as flat surfaces. Rigid material enters the plastic region which is fixed in space on one side, it is deformed while passing through and leaves on the other in a uniform stream which becomes rigid again as it is unloaded. The slip line boundary must be constructed to satisfy all conditions in stress and velocity that directly concern the zone of plastically deforming material. The proposed SLF for the deformation is shown in Fig. 2(a) and the corresponding hodograph in Fig. 2(b). The SLF satisfies boundary conditions directly affecting the plastic zone. The maximum reduction for which the present field is valid, for geometrical reasons alone, is r¼
Hh 2 sin a ¼ H 1 þ 2 sin a
(1)
The slip line c0,0cn,0 and c0,0c0,m are circular arcs of equal radius. By applying Hencky’s first theorem, it is apparent that angles cn;0 Ac0;0 ¼ l1 and c0;0 Bc0;m ¼ l2 must satisfy the relation l 1 l2 ¼ a
(2)
The following equations are used in the numerical calculation of SLFs:
p þ 2kj ¼ const: ðalong an a lineÞ; (3)
where j is the angle between the a line and x-direction of a Cartesian co-ordinate system (x, y), p the hydrostatic pressure and k the maximum shear stress. Thus, with respect to the (x, y) co-ordinates the following relations must be satisfied: dy ¼ tan f ðalong an a lineÞ; dx dy ¼ cot f ðalong a b lineÞ dx
Geiringer equations: du v ¼ 0 ðalong an a lineÞ; df dv þ u ¼ 0 ðalong a b lineÞ df
(5)
where u and v are velocities in a- and b-directions, respectively. From Hencky’s second theorem:
Hencky’s equations: p 2kj ¼ const: ðalong a b lineÞ
Fig. 2. (a) Assumed SLF for half of the section (a < 10; r ¼ 5%); (b) and (c) hodographs of the assumed SLF shown in (a).
(4)
@S ¼ þ1; @Sa
@R ¼ 1 @Sb
(6)
where R is the radius of curvature of the a line and S the radius of curvature of the b line, and @S R ¼ 0; @f
@R þS¼0 @f
(7)
The method of solving problems by approximate step by step procedures has been fully described by Hill [4]. In general, for any die shape the procedures consist of replacing the foregoing equations by their finite difference relations,
M. Salimi / Journal of Materials Processing Technology 125–126 (2002) 317–322
and replacing each small arc of the field by a chord whose slope is the mean of the terminal slopes. For a particular problem, the number of nodal points cannot be determined a priori and therefore the number of equations involved in the solutions is not known. To solve the numerical problem, the non-linear algebraic equations must be uncoupled. The uncoupling is achieved by assuming a form of SLF and assuming the values of the unknown functions at a sufficient number of points so that the defining equations and the boundary conditions can be used to evaluate the functions at the remaining nodal points. Once the pressure on the die is found a force balance on the die wall may be used to determine the drawing force. 2.2. Strain distribution In passing through the die the plate is progressively reduced in thickness from 2H to 2h. oaL in Fig. 2(b) represents the unit velocity of particles in the rigid region before entering the deformation zone. Particles from this region encounter the velocity discontinuity Acn,0cm,n in Fig. 2(a) and undergo a sudden change in velocity. At exit from the plastic region (Bc0,mcm,n), they undergo another sudden change in velocity parallel to their former direction, so that a velocity at exit H/h is achieved. For any element traversing the plastic zone in the gap there are pairs of slip lines, which are almost symmetrical for small reduction per pass. The path for a particle passing through point cm,n (centreline) is aL cm;n bR on the hodograph. The path for a particle traversing Ac0,0 (close to the surface of plate) is aL cn;0 c0;0 c0;m bR and the path for a particle crossing the plastic region at point ‘e’ is aLefgbR. The strain distribution can be determined using the input work per unit volume of the material crossing the plastic region at different positions. Consider a stream tube of thickness t entering the plastic region at position y1 (Fig. 1). The increment input energy per unit volume d(W/V) in passing from a to b and from b0 to a0 in Fig. 2(a) is given by [3]: W du1 du2 d þ ¼k (8) V vp1 vp2
319
so that du1 du2 2du v 2du þ ¼ 2 2 v vp1 vp2 v Dvp
(9)
It can be seen from Fig. 2(b) and (c) that v¼
hþH cos y and 2H
du ¼ dZ cos y
thus 2du 4h ¼ dZ v Hþh
(10)
The work per unit volume for a streamline crossing the plastic region is Z Z W W h ¼ d ¼ 4k dZ (11) V V Hþh R where dZ is integration of the corresponding curve as defined by Z Z du (12) dZ ¼ cos y At the centre layer in Fig. 3 we have Z H dZ ¼ 1 ¼ d1 d10 h
(13)
R and at the outer layer (close to the surface): dZ ¼ d3 d30 . The work rate per unit volume at different positions can be represented by W ¼ 2keH b V
(14)
where eH is the thickness strain or eH ¼ DH=H. Hence at the centreline W 2ðH hÞ ¼ 2k 2keH V Hþh so that at the centreline b ¼ 1.
where du1 and du2 refer to tangent velocity increments and vp1 and vp1 are the normal velocities to the field at those points (see Fig. 2(b)). du=vp for streamlines entering the plastic region at the centre of the plate or very close to its surface, are directly obtained from the hodograph (as the path and the corresponding curve on the hodograph are known). Referring to Fig. 2(b), it can be argued that for a symmetrical SLF, du1 du2 ¼ du and 2Dvp the difference in normal velocities and vp1 and vp2 is small. Two normal velocities vp1 and vp2 are related to a mean normal velocity v by vp1 þ Dvp vp2 Dvp v
Fig. 3. Calculation of
R
dZ.
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M. Salimi / Journal of Materials Processing Technology 125–126 (2002) 317–322
Fig. 4. Variation of strain ratio, b0 against H/l.
The strain ratio b0 (ratio of strain at surface to strain at centre against H/l is shown in Fig. 4, where 2l is the length of contact and 2H the initial thickness. As indicated the variation in straining increases with an increase in ratio H/l. To determine the strain distribution across the section (for points other than the centre and the surface) we have to go to a more complicated solution. As the path and the corresponding velocity curve at an intermediate point on the hodograph are not the same and dy ¼ du=cos y at points close to the minimum of the curve (point f in Fig. 2(b)) are indefinite. A numerical solution, which can be offered, is to consider a velocity discontinuityRpattern, which corresponds to the SLF in order to enable dZ to be determined. Corresponding elements of the SLF and the hodograph are orthogonal and therefore corresponding lines of the pattern shown in Fig. 2(a) and (b) are perpendicular. As a particle is crossing a number of a and b lines shown in Fig. 2(a); the material is considered to be sheared in a number of steps as shown in Fig. 3 (in reality these would be infinitesimal).
Fig. 6. SLF and FEM solutions for normalised extrusion pressure.
R For particles entering at positions y1, y2, etc. dZ obtained numerically using the graphical method shown in Fig. 3. By increasing the number of increments more accurate results will be obtained. The input work per unit volume of material passing through the whole section can be represented by W ¼ 2kgeH V
(15)
where g is known as the pressure factor or g ¼ p=2k. Hill [4] provided the first correct plot of the frictionless indentation by two opposing indenters. The strain distribution from Eq. (14) for H=l ¼ 7:40 is shown in Fig. 5. The pressure factor g (¼p/2k) from the SLF. solution is also shown in the same figure. Normalised indentation pressure (pressure on the die surface) versus H/l is shown for several values of H/l. Sheet drawing with low angle dies similar to indentation. The variation in the normalised extrusion pressure for frictionless plane strain extrusion against reduction from slip line solution is shown in Fig. 6.
Fig. 5. Strain distribution ratio across the section of product for H=l ¼ 7:4.
M. Salimi / Journal of Materials Processing Technology 125–126 (2002) 317–322
321
Fig. 7. A typical effective strain distribution in extrusion of rigid–perfectly plastic material.
In this investigation strain ratio versus H/l, which was not already considered by other investigators, was obtained and is clearly shown that with an increase in reduction the variation in strain distribution becomes smaller.
3. FE simulation Plane strain extrusion through wedge-shaped die was also analysed by using a rigid–plastic FEM. This model is one of the most powerful numerical techniques for metal forming analysis. The variational form as a basis for finite element discretisation was used. The rigid–plastic material model in which elastic deformation is ignored was adapted. The numerical solutions for the deformation through wedgeshaped dies are obtained by a finite element approximation of the displacement fields. The variational equation of the rigid–plastic material model is given by the following [14]: Z Z Z de_ dV þ K e_ v d_e dV fi dui dS ¼ 0 (16) s v
v
Sf
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð3=2Þs0ij s0ij , e_ ¼ ð2=3Þ_eij e_ ij , e_ v ¼ e_ ii and K, where s s0ij , e_ ij , fi and e_ v are the penalty constant, deviatoric stress, strain rate, surface traction components and volumetric strain rate, respectively. In plane strain deformation e_ includes only non-zero strain rate components, namely, e_ x , e_ y and g_ xy . To obtain the exact solution, the following functional F for the deforming material is minimised: # Z "Z e_ Z _ de dV þ tf Dv dS F¼ s v
0
Predicted results were obtained for normal pressure distribution on the die and for the velocity. From FEM solution the effective strain distribution for a rigid–perfectly plastic material and H=l ¼ 3:5 is shown in Fig. 7, which is a typical strain distribution. Variation in normalised extrusion pressure against reduction from FEM is shown in Fig. 6 which is in good agreement with SLF solution. The strain distribution ratio form the FE solution of non-work-hardening material at the exit section is also shown is in Fig. 5. The strain has the highest value near the surface and lowest value near the axis. The FEM solution is in good agreement with that of the SLF solution. If the deformation is rapid enough, temperature gradients caused by inhomogeneous deformation upon subsequent cooling will cause the surface to undergo more contraction than the interior so that residual tension stress will appear on the surface. In practice, this is only one cause of the residual stress. As the SLF theory assumes rigid–perfectly plastic behaviour and allows for no plastic deformation after material leaves the field it does not adequately explain residual stress patterns.
Sf
where tf is the frictional shear stress and Dv the relative velocity between the workpiece and the die. The equations of equilibrium are given in terms of the principle of virtual work with body force neglected and a linear incremental solution procedure is employed. The isoparametric elements are used in the plate simulation for model plane strain deformation. The workpiece was divided into 500 elements. The friction factor is taken to be zero throughout the computation. The initial entry of the workpiece into the die gap and its continuing deformation simulated until steady state is reached. The sharp corner on the die was slightly modified to avoid singularities.
4. Conclusion In this paper, a theoretical model for plane strain deformation based on an SLF solution is developed and the strain distribution across the section of product in plane strain drawing was obtained. The results from FE solution were compared to that of the SLF solution, which was shown to be in a good agreement. Ratios of strain at surface to strain at the centre against the ratio of the semi-height to the contact length were obtained. From the theory, it was concluded that shear strains at positions close to the contact areas are greater than at the centre of the plate. This effect will be more significant for small reductions. It is clear that efficiency decreases sharply at low reduction because of the redundant work. It is clearly shown that the shape of the deformation zone exerts strong influence upon dissipated energy and metal flow.
References [1] W. Johnson, Extrusion through wedge-shaped dies, Part II, J. Mech. Phys. Solids 3 (1) (1975) 218–226.
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[2] J.M. Alexander, On complete solution for frictionless extrusion in plane strain, Quart. Appl. Math. 19 (1961) 31–40. [3] C.C. Chen, S.I. Oh, S. Kobayashi, Deformation mechanics of extrusion and drawing, Trans. ASME J. Eng. Ind. 101 (1979) 23–32. [4] R. Hill, Mathematical Theory of Plasticity, Oxford University Press, Oxford, 1950. [5] H. Hencky, Concerning a few statically determinant cases in plastic bodies, Z. Angew. Mech. 3 (1923) 241–254. [6] F. Ellis, Stress discontinuities in plane plastic flow, J. Strain Anal. 2 (1967) 52–59. [7] M.J.M.B. Barata, P.A.F. Martins, A solution to plane strain extrusion by the upper bound approach and the weighted residual method, Int. J. Mech. Sci. 31 (1989) 395–404. [8] Z.G. Voyiadjis, M. Frouzesh, A finite strain, total Lagrangian finite element solution for metal extrusion problems, Comp. Meth. Appl. Mech. Eng. 86 (1991) 337–370. [9] J. Huetink, Analysis of metal forming process based on a combined
[10] [11]
[12]
[13] [14]
Eulerian–Lagrangian finite element formulation, in: Numerical Method in Industrial Forming Process, Pineridge Press, Swansea, 1982, pp. 501–510. M.S. Gadala, J. Wang, ALE formulation and its application in solid mechanics, Comp. Meth. Appl. Mech. Eng. 167 (1998) 33–55. V. Nagpal, T. Altan, Analysis of the three-dimensional metal flow in extrusion of shapes with the use of dual stream functions, in: Proceedings of the Third NAMRC, Carnegie-Mellor University, Pittsburgh, PA, May 1975. N.R. Chitkara, M.A. Butt, A general numerical method of construction of axisymmetric slip-line fields, Int. J. Mech. Sci. 34 (1992) 833–848. J. Chakrabarty, The analytical solution of some boundary value problems in plane plastic strain, Int. J. Mech. Sci. 33 (1991) 89–99. J.H. Yoon, D.Y. Yang, Rigid–plastic finite element analysis of threedimensional forging by considering friction on continuous curved dies with initial guess generation, Int. J. Mech. Sci. 30 (1988) 887.
Strain distributions in plane strain deformation through wedge-shaped dies M. Salimi Department of Mechanical Engineering, Isfahan University of Technology, P.O. Box 84154, Isfahan, Iran Received 2 December 2001; received in revised form 2 March 2002; accepted 2 March 2002
Abstract In this paper, a theoretical model for plane strain deformation based on a slip line field (SLF) solution is developed to provide the strain distribution across a section of a product in plane strain drawing. The results obtained from the SLF solutions were compared to those of the finite element method. From both solutions, it is shown that the shear strain at a position close to the contact area is greater than that of the centre of the product. Ratios of strain at the surface to strain at the centre against the ratio of the semi-height to the contact length were obtained. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Strain distribution; Plane strain; Slip line field; Drawing; Extrusion
1. Introduction In the process of forming of metals, the material properties and surface quality of the product are influenced by different parameters. An understanding of how the different parameters influence the final product can help to avoid production defects and improve product quality. The problem of deforming a sheet through wedge-shaped dies (drawing/extrusion) was investigated by many investigators under the assumption that the material is in the state of plane plastic flow and obeys the Saint Venant–Mises yield condition [1–3]. Exact stress analysis of the deformation was first proposed by Hill in 1950 [4] using slip line field (SLF) theory developed by Hencky [5] under idealised conditions, i.e. plane strain of a rigid–perfectly plastic solid. Subsequent work has consolidated this, though without anything new in principle. The stress state in the deformation zone is usually determined by the method of stress discontinuity surfaces [6], the upper bound technique [7], the SLF method [4], the finite element method (FEM) [8–10] and other techniques such as dual stream functions [11]. In recent years attempts were made to extend slip line theory to axi-symmetric deformation [12,13]. However, apparently, little theoretical information has been published to describe the strain distribution. From a manufacturing point of view, the strain distribution and the heat developed in the process determines the maxiE-mail address: [email protected] (M. Salimi).
mum deformation speed that can be used for producing sound products and influence lubrication conditions and tool life.
2. Analysis In sheet drawing processes, progressive reduction in thickness is caused by pulling the sheet through a die (Fig. 1). Except at the start and the end of the deformation, processes such as extrusion and drawing are usually considered as steady state. Steady state solutions in these processes are needed for equipment and die design and for controlling product properties. In the present investigation an attempt is made to predict the strain distribution theoretically by reference to SLF. A detailed discussion of the slip line theory and its application can be found in [4]. 2.1. Construction of SLF The SLF solution for strain distribution across the section is obtained using the following simplifying features: 1. Reduction per pass is assumed to be small. In drawing, however, this cannot be very large, since the maximum amount of reduction per pass to avoid rupture is generally small. 2. Equal and small projected contact lengths between the workpiece and the two die surfaces were assumed so that the plastic region joins across the centre of the slab.
0924-0136/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 0 3 4 1 - 2
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M. Salimi / Journal of Materials Processing Technology 125–126 (2002) 317–322
Fig. 1. The deformation zone in plane wedge or sheet drawing assuming no friction at the interface (a < 10; r ¼ 5%).
3. The projected contact lengths are treated as flat surfaces. Rigid material enters the plastic region which is fixed in space on one side, it is deformed while passing through and leaves on the other in a uniform stream which becomes rigid again as it is unloaded. The slip line boundary must be constructed to satisfy all conditions in stress and velocity that directly concern the zone of plastically deforming material. The proposed SLF for the deformation is shown in Fig. 2(a) and the corresponding hodograph in Fig. 2(b). The SLF satisfies boundary conditions directly affecting the plastic zone. The maximum reduction for which the present field is valid, for geometrical reasons alone, is r¼
Hh 2 sin a ¼ H 1 þ 2 sin a
(1)
The slip line c0,0cn,0 and c0,0c0,m are circular arcs of equal radius. By applying Hencky’s first theorem, it is apparent that angles cn;0 Ac0;0 ¼ l1 and c0;0 Bc0;m ¼ l2 must satisfy the relation l 1 l2 ¼ a
(2)
The following equations are used in the numerical calculation of SLFs:
p þ 2kj ¼ const: ðalong an a lineÞ; (3)
where j is the angle between the a line and x-direction of a Cartesian co-ordinate system (x, y), p the hydrostatic pressure and k the maximum shear stress. Thus, with respect to the (x, y) co-ordinates the following relations must be satisfied: dy ¼ tan f ðalong an a lineÞ; dx dy ¼ cot f ðalong a b lineÞ dx
Geiringer equations: du v ¼ 0 ðalong an a lineÞ; df dv þ u ¼ 0 ðalong a b lineÞ df
(5)
where u and v are velocities in a- and b-directions, respectively. From Hencky’s second theorem:
Hencky’s equations: p 2kj ¼ const: ðalong a b lineÞ
Fig. 2. (a) Assumed SLF for half of the section (a < 10; r ¼ 5%); (b) and (c) hodographs of the assumed SLF shown in (a).
(4)
@S ¼ þ1; @Sa
@R ¼ 1 @Sb
(6)
where R is the radius of curvature of the a line and S the radius of curvature of the b line, and @S R ¼ 0; @f
@R þS¼0 @f
(7)
The method of solving problems by approximate step by step procedures has been fully described by Hill [4]. In general, for any die shape the procedures consist of replacing the foregoing equations by their finite difference relations,
M. Salimi / Journal of Materials Processing Technology 125–126 (2002) 317–322
and replacing each small arc of the field by a chord whose slope is the mean of the terminal slopes. For a particular problem, the number of nodal points cannot be determined a priori and therefore the number of equations involved in the solutions is not known. To solve the numerical problem, the non-linear algebraic equations must be uncoupled. The uncoupling is achieved by assuming a form of SLF and assuming the values of the unknown functions at a sufficient number of points so that the defining equations and the boundary conditions can be used to evaluate the functions at the remaining nodal points. Once the pressure on the die is found a force balance on the die wall may be used to determine the drawing force. 2.2. Strain distribution In passing through the die the plate is progressively reduced in thickness from 2H to 2h. oaL in Fig. 2(b) represents the unit velocity of particles in the rigid region before entering the deformation zone. Particles from this region encounter the velocity discontinuity Acn,0cm,n in Fig. 2(a) and undergo a sudden change in velocity. At exit from the plastic region (Bc0,mcm,n), they undergo another sudden change in velocity parallel to their former direction, so that a velocity at exit H/h is achieved. For any element traversing the plastic zone in the gap there are pairs of slip lines, which are almost symmetrical for small reduction per pass. The path for a particle passing through point cm,n (centreline) is aL cm;n bR on the hodograph. The path for a particle traversing Ac0,0 (close to the surface of plate) is aL cn;0 c0;0 c0;m bR and the path for a particle crossing the plastic region at point ‘e’ is aLefgbR. The strain distribution can be determined using the input work per unit volume of the material crossing the plastic region at different positions. Consider a stream tube of thickness t entering the plastic region at position y1 (Fig. 1). The increment input energy per unit volume d(W/V) in passing from a to b and from b0 to a0 in Fig. 2(a) is given by [3]: W du1 du2 d þ ¼k (8) V vp1 vp2
319
so that du1 du2 2du v 2du þ ¼ 2 2 v vp1 vp2 v Dvp
(9)
It can be seen from Fig. 2(b) and (c) that v¼
hþH cos y and 2H
du ¼ dZ cos y
thus 2du 4h ¼ dZ v Hþh
(10)
The work per unit volume for a streamline crossing the plastic region is Z Z W W h ¼ d ¼ 4k dZ (11) V V Hþh R where dZ is integration of the corresponding curve as defined by Z Z du (12) dZ ¼ cos y At the centre layer in Fig. 3 we have Z H dZ ¼ 1 ¼ d1 d10 h
(13)
R and at the outer layer (close to the surface): dZ ¼ d3 d30 . The work rate per unit volume at different positions can be represented by W ¼ 2keH b V
(14)
where eH is the thickness strain or eH ¼ DH=H. Hence at the centreline W 2ðH hÞ ¼ 2k 2keH V Hþh so that at the centreline b ¼ 1.
where du1 and du2 refer to tangent velocity increments and vp1 and vp1 are the normal velocities to the field at those points (see Fig. 2(b)). du=vp for streamlines entering the plastic region at the centre of the plate or very close to its surface, are directly obtained from the hodograph (as the path and the corresponding curve on the hodograph are known). Referring to Fig. 2(b), it can be argued that for a symmetrical SLF, du1 du2 ¼ du and 2Dvp the difference in normal velocities and vp1 and vp2 is small. Two normal velocities vp1 and vp2 are related to a mean normal velocity v by vp1 þ Dvp vp2 Dvp v
Fig. 3. Calculation of
R
dZ.
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M. Salimi / Journal of Materials Processing Technology 125–126 (2002) 317–322
Fig. 4. Variation of strain ratio, b0 against H/l.
The strain ratio b0 (ratio of strain at surface to strain at centre against H/l is shown in Fig. 4, where 2l is the length of contact and 2H the initial thickness. As indicated the variation in straining increases with an increase in ratio H/l. To determine the strain distribution across the section (for points other than the centre and the surface) we have to go to a more complicated solution. As the path and the corresponding velocity curve at an intermediate point on the hodograph are not the same and dy ¼ du=cos y at points close to the minimum of the curve (point f in Fig. 2(b)) are indefinite. A numerical solution, which can be offered, is to consider a velocity discontinuityRpattern, which corresponds to the SLF in order to enable dZ to be determined. Corresponding elements of the SLF and the hodograph are orthogonal and therefore corresponding lines of the pattern shown in Fig. 2(a) and (b) are perpendicular. As a particle is crossing a number of a and b lines shown in Fig. 2(a); the material is considered to be sheared in a number of steps as shown in Fig. 3 (in reality these would be infinitesimal).
Fig. 6. SLF and FEM solutions for normalised extrusion pressure.
R For particles entering at positions y1, y2, etc. dZ obtained numerically using the graphical method shown in Fig. 3. By increasing the number of increments more accurate results will be obtained. The input work per unit volume of material passing through the whole section can be represented by W ¼ 2kgeH V
(15)
where g is known as the pressure factor or g ¼ p=2k. Hill [4] provided the first correct plot of the frictionless indentation by two opposing indenters. The strain distribution from Eq. (14) for H=l ¼ 7:40 is shown in Fig. 5. The pressure factor g (¼p/2k) from the SLF. solution is also shown in the same figure. Normalised indentation pressure (pressure on the die surface) versus H/l is shown for several values of H/l. Sheet drawing with low angle dies similar to indentation. The variation in the normalised extrusion pressure for frictionless plane strain extrusion against reduction from slip line solution is shown in Fig. 6.
Fig. 5. Strain distribution ratio across the section of product for H=l ¼ 7:4.
M. Salimi / Journal of Materials Processing Technology 125–126 (2002) 317–322
321
Fig. 7. A typical effective strain distribution in extrusion of rigid–perfectly plastic material.
In this investigation strain ratio versus H/l, which was not already considered by other investigators, was obtained and is clearly shown that with an increase in reduction the variation in strain distribution becomes smaller.
3. FE simulation Plane strain extrusion through wedge-shaped die was also analysed by using a rigid–plastic FEM. This model is one of the most powerful numerical techniques for metal forming analysis. The variational form as a basis for finite element discretisation was used. The rigid–plastic material model in which elastic deformation is ignored was adapted. The numerical solutions for the deformation through wedgeshaped dies are obtained by a finite element approximation of the displacement fields. The variational equation of the rigid–plastic material model is given by the following [14]: Z Z Z de_ dV þ K e_ v d_e dV fi dui dS ¼ 0 (16) s v
v
Sf
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð3=2Þs0ij s0ij , e_ ¼ ð2=3Þ_eij e_ ij , e_ v ¼ e_ ii and K, where s s0ij , e_ ij , fi and e_ v are the penalty constant, deviatoric stress, strain rate, surface traction components and volumetric strain rate, respectively. In plane strain deformation e_ includes only non-zero strain rate components, namely, e_ x , e_ y and g_ xy . To obtain the exact solution, the following functional F for the deforming material is minimised: # Z "Z e_ Z _ de dV þ tf Dv dS F¼ s v
0
Predicted results were obtained for normal pressure distribution on the die and for the velocity. From FEM solution the effective strain distribution for a rigid–perfectly plastic material and H=l ¼ 3:5 is shown in Fig. 7, which is a typical strain distribution. Variation in normalised extrusion pressure against reduction from FEM is shown in Fig. 6 which is in good agreement with SLF solution. The strain distribution ratio form the FE solution of non-work-hardening material at the exit section is also shown is in Fig. 5. The strain has the highest value near the surface and lowest value near the axis. The FEM solution is in good agreement with that of the SLF solution. If the deformation is rapid enough, temperature gradients caused by inhomogeneous deformation upon subsequent cooling will cause the surface to undergo more contraction than the interior so that residual tension stress will appear on the surface. In practice, this is only one cause of the residual stress. As the SLF theory assumes rigid–perfectly plastic behaviour and allows for no plastic deformation after material leaves the field it does not adequately explain residual stress patterns.
Sf
where tf is the frictional shear stress and Dv the relative velocity between the workpiece and the die. The equations of equilibrium are given in terms of the principle of virtual work with body force neglected and a linear incremental solution procedure is employed. The isoparametric elements are used in the plate simulation for model plane strain deformation. The workpiece was divided into 500 elements. The friction factor is taken to be zero throughout the computation. The initial entry of the workpiece into the die gap and its continuing deformation simulated until steady state is reached. The sharp corner on the die was slightly modified to avoid singularities.
4. Conclusion In this paper, a theoretical model for plane strain deformation based on an SLF solution is developed and the strain distribution across the section of product in plane strain drawing was obtained. The results from FE solution were compared to that of the SLF solution, which was shown to be in a good agreement. Ratios of strain at surface to strain at the centre against the ratio of the semi-height to the contact length were obtained. From the theory, it was concluded that shear strains at positions close to the contact areas are greater than at the centre of the plate. This effect will be more significant for small reductions. It is clear that efficiency decreases sharply at low reduction because of the redundant work. It is clearly shown that the shape of the deformation zone exerts strong influence upon dissipated energy and metal flow.
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