- Email: [email protected]
Rigid-plastic finite element analysis of plastic deformation of porous metal sheets containing internal void defects
Journal of Materials Processing Technology 180 (2006) 193–200
Rigid-plastic finite element analysis of plastic deformation of porous metal sheets containing internal void defects Dyi-Cheng Chen Department of Industrial Education and Technology, National Changhua University of Education, Changhua 500, Taiwan, ROC Received 24 September 2004; received in revised form 7 September 2005; accepted 6 June 2006
Abstract Using rigid-plastic finite element DEFORM-2D and -3D software, this study simulates the plastic deformation of metal sheets at the roll gap during the sheet rolling process. The study focuses specifically upon the deformation of porous metal sheets containing internal void defects. The present numerical analysis investigates the relative density distributions, the void closure behavior, the deformation mechanisms and the stress–strain distributions around the internal voids for various rolling conditions. The influences on the dimensions of the final void of the thickness reduction, the initial internal void dimensions, the friction factors and the relative density are systematically discussed. The critical rolling conditions also investigated. A series of sheet rolling experiments are performed in order to verify the validity of the simulation results. The current numerical results provide a valuable source of reference for the design of pass schedules for porous metals undergoing rolling processes. © 2006 Elsevier B.V. All rights reserved. Keywords: Rigid-plastic finite element; Porous metals rolling; Void defect
1. Introduction Understanding the deformation behavior and density distribution of porous metals during forming is very important in achieving good quality powder metallurgy parks. Moriya et al. [1–3] showed that in porous materials, the slip lines in two families one another at an angle which varies as a function of the relative density of the material. Furthermore, the yield condition of the material is influenced by the hydrostatic stress component and by the relative density. The study also established an upperbound theory to estimate the load which causes deformation of porous materials. Hwang and Kobayashi [4] developed a plasticity theory for powdered metals in compaction. In their study, the yield criterion used for sintered powdered metals was modified to describe the asymmetric behavior of powdered metal compacts in tension and in compression. Han et al. [5,6] applied the elastoplastic finite element method to the deformation of porous metals, the density of the porous rings and the compression loads were calculated as a function of the height reduction. Nagaki et al. [7] investigated the influence of void distribution on the yielding of elastic–plastic porous solids and compared the yield loci derived from the yield function with those deter-
E-mail address: [email protected]. 0924-0136/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2006.06.004
mined from a FE model. Cho et al. [8] studied the thermoplastic response and densification behavior of porous alloy steel under P/M forging for various initial relative densities, strain rates and temperatures. Wang and Karabin [9] used a pressure dependent constitutive model and a FE rolling process model to analyze the evolution of porosity during the hot rolling of powder-based porous aluminum plates. Shi et al. [10] investigated the rolling of metal sheets and metal powders of composite materials to be an asymmetric rolling process for the workpiece. Hirohata et al. [11] analyzed the differential speed rolling of electrolytic copper under the conditions of a carefully controlled powder feed rate. A review of the literature reveals that few researchers have investigated the effects of various dimensions of the void defective upon the critical reductions which result in complete closure of the void during the porous metal sheet rolling process. Consequently, this study employs the finite element method of continuum mechanics to consider the void closure behavior inside a porous metal sheet during a sheet rolling process. The effects of various rolling conditions on the final dimensions of the void defect after rolling are discussed systematically. These conditions include the ratio of the roll radius to the sheet thickness, the diameter of the void defect, the thickness reduction, the friction factor between the porous metal and the rolls, and the initial relative density of the porous metal sheet.
194
D.-C. Chen / Journal of Materials Processing Technology 180 (2006) 193–200
Fig. 1. Schematic illustration of porous metal rolling with void inside sheet.
Fig. 2. Two dimension mesh configuration of sheet before rolling. R = 100 mm, V = 5 mm/s, d0 /t0 = 0.2, m = 0.6, and D (initial relative density) = 0.8.
Fig. 4. Relative density distributions at various reductions, (a) r = 10% and (b) r = 20%, during rolling. R = 100 mm, V = 5 mm/s, d0 /t0 = 0.2, m = 0.6, and D (initial relative density) = 0.8.
the fact that in porous metals, a volumetric strain results in a significant change in porosity. Kuhn and Downey [12] and Green [13] were the first researchers to present a suitable yield criterion and stress–strain relationship for porous solids. The theory developed by these researchers was subsequently extended by Shima and Oyane [14] to a general stress state, leading to a yield criterion of the form: 2 1/2 {(σ1 −σ2 )2 + (σ2 −σ3 )2 + (σ3 − σ1 )2 } σm f σ¯ = + 2 f (1)
Fig. 3. Three dimension mesh configuration of sheet before rolling. R = 100 mm, V = 5 mm/s, t0 = 10 mm, w0 = 40 mm, d0 /t0 = 0.2, m = 0.6, and D (initial relative density) = 0.8.
2. Plasticity theory for porous metals The onset of plastic deformation in porous metals is governed by yield criteria and flow rules which are fundamentally different from those applicable to fully dense materials. This is due to
where σ m is the hydrostatic stress, f a material-dependent factor and f is the ratio between the apparent stress applied to the porous solid and the effective stress exerted on the metal matrix. From Eq. (1), the following stress–strain relationship can be derived: 3 ρ d¯ε dεi = 2 f 2 σ¯
2 σi − 1 − 9f 2
σm
for i = 1, 2, 3 (2a)
D.-C. Chen / Journal of Materials Processing Technology 180 (2006) 193–200
195
and f = ρC where A, B and C are material constants. The relative density can be written as: 2 2 q 2 1.028 p + 2.49 (1 − D) − D5 = 0 σm σm
(3b)
(4)
where p is the hydrostatic pressure, q the Mises equivalent stress, is the matrix yielding stress. D the relative density, and σm The yield criterion in Eq. (1) is treated with the damage value during DEFORM software numerical simulation. The present study considers the damage values presented below: ε¯ f σmax Normalized C and L criterion : (5) d¯ε = Cd σ¯ 0 where σ max is the maximum ductility stress, ε¯ f the effective strain of fracture, and Cd is the damage value of the material. 3. Method of analysis The finite element method has been widely applied to simulate plastic flow in workpiece materials as they undergo forming processes. For the plastic deformation which occurs during the rolling of porous metal sheets, the governing
Fig. 5. Effective stress distributions at various reductions, (a) r = 10% and (b) r = 20%, during rolling. R = 100 mm, V = 5 mm/s, d0 /t0 = 0.2, m = 0.6, and D (initial relative density) = 0.8.
where d¯ε =
f 2 ((dε1 − dε2 )2 + (dε2 − dε3 )2 ρ 9 1/2 2 2 + (dε3 − dε1 ) + (f dεv ) )
(2b)
and dεv = dε1 + dε2 + dε3
(2c)
In the equations above, σ¯ and ε¯ refer to the effective stress and the cumulative effective strain, respectively. In cold rolling, strain-hardening effects permit the deforming material to be characterized by a relationship in which σ¯ is specified as a function of ε¯ . In general, the factors f and f are functions of the relative density, ρ. For sintered porous metals, Shima and Oyane [14] suggested correlations of the form: f =
1 A(1 − ρ)B
(3a)
Fig. 6. Effective strain distributions at various reductions, (a) r = 10% and (b) r = 20%, during rolling. R = 100 mm, V = 5 mm/s, d0 /t0 = 0.2, m = 0.6, and D (initial relative density) = 0.8.
196
D.-C. Chen / Journal of Materials Processing Technology 180 (2006) 193–200
Fig. 7. Relative density distributions at various friction factors, (a) m = 0.3 and (b) m = 0.9, during rolling. R = 100 mm, V = 5 mm/s, d0 /t0 = 0.2, r = 20%, and D (initial relative density) = 0.8.
equations for the plastic deformation of rigid-plastic and rigid-viscoplastic materials involve equilibrium equations, yield criteria, constitutive equations and compatibility conditions. The duality of the boundary value problem and the variation problem can be seen clearly by considering the construction of the function [15]:
π=
σ¯ ε¯˙ dV − v
Fi ui dS
(6)
SF
where π is a function of the total energy and work, σ¯ the effective stress, ε¯˙ the effective strain rate, Fi represents the surface tractions and ui is the velocity component. The incompressibility constraint on admissible velocity fields in Eq. (6) can be removed by using the penalized form of the incompressibility. Hence, the variational form used as the basis for finite element discretization is given by:
␦π =
v
ε˙ v ␦˙εv dV −
σ¯ ␦ε¯˙ dV + K v
Fi ␦ui dS = 0
(7)
Fig. 8. Relative density distributions at various roll diameters and speed, (a) R = 50 mm, V = 2.5 mm/s and (b) R = 200 mm, V = 10 mm/s, during rolling. d0 /t0 = 0.2, m = 0.6, r = 20%, and D (initial relative density) = 0.8. the flow formulation approach using an updated Lagrange procedure. In the finite element discretization procedure, Eq. (7) is converted into a set of nonlinear algebraic equations. The Newton–Raphson method and direct iteration methods are adopted to solve the non-linear equations (see previous question) in DEFORM-2D and DEFORM-3D. The direct iteration method is used to generate a suitable initial estimate for the Newton–Raphson method, which is then employed to obtain a rapid final convergence. The convergence criteria for the iteration are the velocity error norm ||v||/||v|| 0.005, and the force 1/2 error norm ||F||/||F|| 0.05, where ||v|| is defined as (vT v) . For a detailed description of the finite element theory and the modeling formulations, readers are referred to Ref. [15]. Fig. 1 depicts the rolling process for a porous metal containing an internal void. In this figure, R and V are the radius and peripheral velocity of the roll, respectively, L is the contact length between the sheet and the roll, X is the distance from the internal void to the exit of the roll gap, and t0 and t1 are the
Table 1 The stress–strain relationship of the aluminum A6062 Strain
SF
where ␦ε¯˙ and ␦˙εv are the variations in strain rate and volumetric strain rate, respectively, and K is a penalty constant having a very large positive value. Eq. (6) or (7) represents the basic equation for the finite element formulation. This study applies the commercial finite element codes DEFORM-2D and DEFORM-3D to simulate the closure behavior of voids inside porous metal sheets during the sheet rolling process. The finite element code is based on
0 0.2 1 2
Strain rate 1
100
138 168.312 204.828 208
139 170.312 206.828 210
Flow stress = f (temperature, strain rate, strain) (MPa); temperature = 20 ◦ C.
D.-C. Chen / Journal of Materials Processing Technology 180 (2006) 193–200
197
Fig. 9. Relative density distributions at various void length and initial relative density, (a) Lv0 /w0 = 1, D = 0.8, (b) Lv0 /w0 = 0.5, D = 0.8, (c) Lv0 /w0 = 0.25, D = 0.8, and (d) Lv0 /w0 = 1, D = 0.9, during rolling. R = 100 mm, V = 5 mm/s, t0 = 10 mm, w0 = 40 mm, d0 /t0 = 0.2, m = 0.6. initial and final thickness of the porous metal sheet, respectively. The internal void is assumed to have an initial circular cross-section with a diameter of d0 . The final height and length of the internal void are given by h1 and l1 , respectively. The present simulations apply the following assumptions: (1) the rolls are rigid, (2) the sheet is a porous material, (3) the deformation of the porous metal sheet occurs as the result of plane strain, (4) the properties of the porous metal sheet are isotropic, and (5) a constant friction factor exists at the interface between the rolls and the porous metal sheet.
4. Mesh configurations of porous metal sheet As shown in Fig. 2, the present two-dimensional model consists of two objects: the roll and the porous metal sheet.
Isoparametric four-node elements are used. The porous metal sheet is divided into approximately 1000 elements. In the threedimensional model shown in Fig. 3, the porous metal sheet is divided into 6060 nodes, 25,500 elements and 6850 surface polygons. Although the setting of the mesh density in the deformation zones can be varied according to the strain or strain rate distributions, a uniform mesh density is chosen here. The friction factor, m, between the sheet and the rolls is specified as 0.6 to reflect dry friction conditions. The material of the sheet used in the present simulation is porous aluminum A6062. The stress–strain relationship of the aluminum A6062 is presented in Table 1. The material con-
198
D.-C. Chen / Journal of Materials Processing Technology 180 (2006) 193–200
stants are A = 2.27, B = 0.49, and C = 2.30 for porous metal sheet. Fig. 4(a and b) shows the relative density distributions of the porous metal at the roll gap during rolling with sheet reductions of r = 10 and 20%, respectively. In both cases, R = 100 mm, d0 /t0 = 0.2, m = 0.6, and D (initial relative density) = 0.8. It is clear that the relative density is greatest at the front and rear regions of the internal void in both cases. Furthermore, it is observed that the increased rolling force associated with a greater reduction percentage increases the magnitude of the relative density. Figs. 5 and 6 show the effective stress and effective strain distributions, respectively, at the roll gap during rolling of the porous metal sheet under reductions of r = 10 and 20%. The rolling conditions are as presented above for Fig. 3. The results indicate that the larger the reduction and front–rear of voids are, the larger the effective stress and the effective strain. It is apparent that a larger reduction yields a greater relative density in porous metals. Fig. 7(a and b) shows the relative density distributions of the porous metal at the roll gap during rolling with friction factors of m = 0.3 and 0.9, respectively. The rolling conditions are specified as R = 100 mm, d0 /t0 = 0.2, r = 20%, and D = 0.8 in both cases. The relative density is clearer greater at the front and rear regions of the internal void. Furthermore, it is apparent that a higher friction factor yields a slight increase in the relative density. Fig. 8(a and b) illustrates the relative density distributions of the porous metal at the roll gap during rolling with roll diameters of R = 50 mm (V = 2.5 mm/s) and 200 mm (V = 10 mm/s), respectively. The rolling conditions are given by d0 /t0 = 0.2, m = 0.6, r = 20%, D = 0.8 in both figures. As in the case above, the relative density increases at the front and rear regions of the internal voids. Additionally, a larger roll diameter and speed is seen to increase the relative density slightly since the larger diameter and speed generates a higher rolling force. Fig. 9 presents the relative density distributions of the porous metal at the roll gap during rolling for various void lengths, Lv0 , and initial relative densities, D. The rolling conditions are R = 100 mm, V = 5 mm/s, t0 = 10 mm, w0 (width of sheet) = 40 mm, d0 /t0 = 0.2, m = 0.6, and D = 0.8–0.9. It can be seen that a longer void length, Lv0 /w0 , results in a slightly higher final relative density. This result is to be expected since when the value of Lv0 /w0 is high, then necessarily a reduced area of porous metal sheet remains. It is noted that when the initial relative density has a value of 0.9, the final density is 1.0.
Fig. 10. Effects of roll diameter upon final void aspect ratio.
Fig. 11 shows the critical reduction, i.e. the reduction which achieves complete closure of the voids, for various roll radii. In this figure, the region above each curve represents complete void closure. It can be seen that the critical reduction percentage decreases with increasing roll radius, but increases with increasing d0 /t0 . The critical reductions obtained for 0.04 d0 /t0 0.20 and 5 R/t0 20 are found to be 15% rc (%) 30%.
5. Results and discussion The rolling conditions used in the current numerical simulations are presented in Table 2. Fig. 10 illustrates the effects of ratio of the roll radius to the sheet thickness upon the aspect ratio of the final void at the exit for rolling conditions of d0 = 2.0 mm and D = 0.8. It can be seen that as the roll diameter increases, the final aspect ratio of the internal void decreases. This result is reasonable since a larger roll diameter generates larger flow stresses in porous metals, and these stresses suppress the complete closure of the voids.
Fig. 11. Effects of roll diameter on critical reductions.
D.-C. Chen / Journal of Materials Processing Technology 180 (2006) 193–200
199
Table 2 Rolling conditions for porous metal simulations No.
Void diameter, d0 /t0
Friction factor, m
Initial relative density
Roll radius, R/t0
Reduction, r (%)
Case 1 Case 2 Case 3
0.2 0.1 0.04
0.3, 0.6, 0.9 0.3, 0.6, 0.9 0.3, 0.6, 0.9
0.8, 0.9 0.8 0.8
5, 10, 20 5, 10, 20 5, 10, 20
5–25 2.5–20 2.5–10
6. Experiments on porous metal sheet rolling with internal voids In sheet rolling processes, the lubricity at the interface between the porous metal sheet and the rolls plays a key role in determining various rolling results, including void closure. The lubricity is expressed by the constant shear friction parameter τ s = mk, where k is the yield shear stress of the deforming material and m is the friction factor (0 m 1). In the present study, the friction factor between the sheet and the rolls was determined by the so-called ring compression test. The aspect ratio of the test piece was 6:3:2. The surface of the test piece was polished slightly with sandpaper and then cleaned using a cloth with an acetone solution to ensure the presence of dry friction. The experimental measurements of the internal ring diameter
reductions were then compared with the analytically obtained results. It was found that the two sets of values correlated most closely for a friction factor of 0.6. Fig. 12 shows the appearance of the rolled product and the internal voids for different reductions under rolling conditions of R = 100 mm, V = 5 mm/s, t0 = 10 mm, w0 = 40 mm, m = 0.6, and d0 /t0 = 0.15. It can be seen that at a reduction of 27%, the final void height decreases to zero, hence indicating that the void is completely closed. 7. Conclusions This paper has used the finite element method of continuum mechanics to investigate the void closure behavior inside a porous metal sheet during the sheet rolling process. The effects of various rolling conditions on the final dimensions of the void defect have been discussed systematically. These rolling conditions include the ratio of the roll radius to the sheet thickness, the diameter of the defective void, the thickness reduction, the friction factor between the porous metals and rolls, the initial relative density of the porous metal sheet, etc. From the present simulation results, it can be concluded that: (1) the relative density is greatest at the front and rear regions of the internal void. Furthermore, the larger the values of the reduction percentage, the friction factor and the roll diameter, the higher the relative density; (2) the larger the reduction and front–rear of voids are, the larger the effective stress and the effective strain; (3) a longer void length, Lv0 /w0 , results in a slightly higher relative density; (4) the critical reduction decreases with increasing roll radius, but increases with increasing d0 /t0 . The current simulation results provide a valuable source of reference for the rolling of porous metal sheets with internal void defects. Acknowledgement The authors gratefully acknowledge the support provided to this study by the National Science Council of the Republic of China under Grant No. NSC 92-2212-E-274-005. References
Fig. 12. Appearance of rolled product and void: (a) r = 8%, (b) r = 18%, and (c) r = 27%. R = 100 mm, V = 5 mm/s, t0 = 10 mm, w0 = 40 mm, m = 0.6, d0 /t0 = 0.15.
[1] O. Moriya, T. Tsuyoshi, Ductile fracture of porous material, J. Jpn. Soc. Technol. Plast. 14 (149) (1973) 439–445. [2] O. Moriya, T. Tsuyoshi, Slip-line field theory and upper-bound theory for porous materials, J. Jpn. Soc. Technol. Plast. 15 (156) (1974) 43–51. [3] O. Moriya, S. Susumu, O. Kunio, Ductile fracture of sintered iron compacts, J. Jpn. Soc. Technol. Plast. 20 (226) (1979) 1037–1044. [4] B.B. Hwang, S. Kobayashi, Deformation characterization of powdered metals in compaction, Int. J. Mach. Tools Manuf. 30 (2) (1990) 309–323.
200
D.-C. Chen / Journal of Materials Processing Technology 180 (2006) 193–200
[5] H.N. Han, H.S. Kim, K.H. Oh, D.N. Lee, Elastoplastic finite element analysis for porous metals, Powder Metall. 37 (2) (1994) 140–146. [6] H.N. Han, H.S. Kim, K.H. Oh, D.N. Lee, Analysis of coefficient of friction in compression of porous metal rings, Powder Metall. 37 (4) (1994) 259–264. [7] S. Nagaki, M. Goya, R. Sowerby, The influence of void distribution on the yielding of an elastic–plastic porous solid, Int. J. Plast. 9 (4) (1993) 199–211. [8] H.K. Cho, J. Suh, K.T. Kim, Densification of porous alloy steel preforms at high temperature, Int. J. Mech. Sci. 36 (4) (1994) 317–328. [9] P.T. Wang, M.E. Karabin, Evolution of porosity during thin plate rolling of powder-based porous aluminum, Powder Technol. 78 (1994) 67–76. [10] Q. Shi, S. Zhang, Y. Sun, D. Zhang, The characteristics of the deformation and flow process in the rolling of metal sheets and metal pow-
[11]
[12]
[13] [14] [15]
ders of composite materials, J. Mater. Process. Technol. 63 (1997) 421– 425. T. Hirohata, S. Masaki, S. Shima, Experiment on metal powder compaction by differential speed rolling, J. Mater. Process. Technol. 111 (2001) 113–117. H.A. Kuhn, C.L. Downey Jr., Deformation characteristics and plasticity theory of sintered powdered materials, Int. J. Powder Met. 7 (1971) 15– 25. R.J. Green, A plasticity theory for porous solids, Int. J. Mech. Sci. 14 (1972) 215–224. S. Shima, M. Oyane, Plasticity theory for porous metals, Int. J. Mech. Sci. 18 (1976) 285–291. S. Kobayashi, S.I. Oh, T. Altan, Metal Forming and the Finite Element Method, Oxford University Press, New York, 1989.
Rigid-plastic finite element analysis of plastic deformation of porous metal sheets containing internal void defects Dyi-Cheng Chen Department of Industrial Education and Technology, National Changhua University of Education, Changhua 500, Taiwan, ROC Received 24 September 2004; received in revised form 7 September 2005; accepted 6 June 2006
Abstract Using rigid-plastic finite element DEFORM-2D and -3D software, this study simulates the plastic deformation of metal sheets at the roll gap during the sheet rolling process. The study focuses specifically upon the deformation of porous metal sheets containing internal void defects. The present numerical analysis investigates the relative density distributions, the void closure behavior, the deformation mechanisms and the stress–strain distributions around the internal voids for various rolling conditions. The influences on the dimensions of the final void of the thickness reduction, the initial internal void dimensions, the friction factors and the relative density are systematically discussed. The critical rolling conditions also investigated. A series of sheet rolling experiments are performed in order to verify the validity of the simulation results. The current numerical results provide a valuable source of reference for the design of pass schedules for porous metals undergoing rolling processes. © 2006 Elsevier B.V. All rights reserved. Keywords: Rigid-plastic finite element; Porous metals rolling; Void defect
1. Introduction Understanding the deformation behavior and density distribution of porous metals during forming is very important in achieving good quality powder metallurgy parks. Moriya et al. [1–3] showed that in porous materials, the slip lines in two families one another at an angle which varies as a function of the relative density of the material. Furthermore, the yield condition of the material is influenced by the hydrostatic stress component and by the relative density. The study also established an upperbound theory to estimate the load which causes deformation of porous materials. Hwang and Kobayashi [4] developed a plasticity theory for powdered metals in compaction. In their study, the yield criterion used for sintered powdered metals was modified to describe the asymmetric behavior of powdered metal compacts in tension and in compression. Han et al. [5,6] applied the elastoplastic finite element method to the deformation of porous metals, the density of the porous rings and the compression loads were calculated as a function of the height reduction. Nagaki et al. [7] investigated the influence of void distribution on the yielding of elastic–plastic porous solids and compared the yield loci derived from the yield function with those deter-
E-mail address: [email protected]. 0924-0136/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2006.06.004
mined from a FE model. Cho et al. [8] studied the thermoplastic response and densification behavior of porous alloy steel under P/M forging for various initial relative densities, strain rates and temperatures. Wang and Karabin [9] used a pressure dependent constitutive model and a FE rolling process model to analyze the evolution of porosity during the hot rolling of powder-based porous aluminum plates. Shi et al. [10] investigated the rolling of metal sheets and metal powders of composite materials to be an asymmetric rolling process for the workpiece. Hirohata et al. [11] analyzed the differential speed rolling of electrolytic copper under the conditions of a carefully controlled powder feed rate. A review of the literature reveals that few researchers have investigated the effects of various dimensions of the void defective upon the critical reductions which result in complete closure of the void during the porous metal sheet rolling process. Consequently, this study employs the finite element method of continuum mechanics to consider the void closure behavior inside a porous metal sheet during a sheet rolling process. The effects of various rolling conditions on the final dimensions of the void defect after rolling are discussed systematically. These conditions include the ratio of the roll radius to the sheet thickness, the diameter of the void defect, the thickness reduction, the friction factor between the porous metal and the rolls, and the initial relative density of the porous metal sheet.
194
D.-C. Chen / Journal of Materials Processing Technology 180 (2006) 193–200
Fig. 1. Schematic illustration of porous metal rolling with void inside sheet.
Fig. 2. Two dimension mesh configuration of sheet before rolling. R = 100 mm, V = 5 mm/s, d0 /t0 = 0.2, m = 0.6, and D (initial relative density) = 0.8.
Fig. 4. Relative density distributions at various reductions, (a) r = 10% and (b) r = 20%, during rolling. R = 100 mm, V = 5 mm/s, d0 /t0 = 0.2, m = 0.6, and D (initial relative density) = 0.8.
the fact that in porous metals, a volumetric strain results in a significant change in porosity. Kuhn and Downey [12] and Green [13] were the first researchers to present a suitable yield criterion and stress–strain relationship for porous solids. The theory developed by these researchers was subsequently extended by Shima and Oyane [14] to a general stress state, leading to a yield criterion of the form: 2 1/2 {(σ1 −σ2 )2 + (σ2 −σ3 )2 + (σ3 − σ1 )2 } σm f σ¯ = + 2 f (1)
Fig. 3. Three dimension mesh configuration of sheet before rolling. R = 100 mm, V = 5 mm/s, t0 = 10 mm, w0 = 40 mm, d0 /t0 = 0.2, m = 0.6, and D (initial relative density) = 0.8.
2. Plasticity theory for porous metals The onset of plastic deformation in porous metals is governed by yield criteria and flow rules which are fundamentally different from those applicable to fully dense materials. This is due to
where σ m is the hydrostatic stress, f a material-dependent factor and f is the ratio between the apparent stress applied to the porous solid and the effective stress exerted on the metal matrix. From Eq. (1), the following stress–strain relationship can be derived: 3 ρ d¯ε dεi = 2 f 2 σ¯
2 σi − 1 − 9f 2
σm
for i = 1, 2, 3 (2a)
D.-C. Chen / Journal of Materials Processing Technology 180 (2006) 193–200
195
and f = ρC where A, B and C are material constants. The relative density can be written as: 2 2 q 2 1.028 p + 2.49 (1 − D) − D5 = 0 σm σm
(3b)
(4)
where p is the hydrostatic pressure, q the Mises equivalent stress, is the matrix yielding stress. D the relative density, and σm The yield criterion in Eq. (1) is treated with the damage value during DEFORM software numerical simulation. The present study considers the damage values presented below: ε¯ f σmax Normalized C and L criterion : (5) d¯ε = Cd σ¯ 0 where σ max is the maximum ductility stress, ε¯ f the effective strain of fracture, and Cd is the damage value of the material. 3. Method of analysis The finite element method has been widely applied to simulate plastic flow in workpiece materials as they undergo forming processes. For the plastic deformation which occurs during the rolling of porous metal sheets, the governing
Fig. 5. Effective stress distributions at various reductions, (a) r = 10% and (b) r = 20%, during rolling. R = 100 mm, V = 5 mm/s, d0 /t0 = 0.2, m = 0.6, and D (initial relative density) = 0.8.
where d¯ε =
f 2 ((dε1 − dε2 )2 + (dε2 − dε3 )2 ρ 9 1/2 2 2 + (dε3 − dε1 ) + (f dεv ) )
(2b)
and dεv = dε1 + dε2 + dε3
(2c)
In the equations above, σ¯ and ε¯ refer to the effective stress and the cumulative effective strain, respectively. In cold rolling, strain-hardening effects permit the deforming material to be characterized by a relationship in which σ¯ is specified as a function of ε¯ . In general, the factors f and f are functions of the relative density, ρ. For sintered porous metals, Shima and Oyane [14] suggested correlations of the form: f =
1 A(1 − ρ)B
(3a)
Fig. 6. Effective strain distributions at various reductions, (a) r = 10% and (b) r = 20%, during rolling. R = 100 mm, V = 5 mm/s, d0 /t0 = 0.2, m = 0.6, and D (initial relative density) = 0.8.
196
D.-C. Chen / Journal of Materials Processing Technology 180 (2006) 193–200
Fig. 7. Relative density distributions at various friction factors, (a) m = 0.3 and (b) m = 0.9, during rolling. R = 100 mm, V = 5 mm/s, d0 /t0 = 0.2, r = 20%, and D (initial relative density) = 0.8.
equations for the plastic deformation of rigid-plastic and rigid-viscoplastic materials involve equilibrium equations, yield criteria, constitutive equations and compatibility conditions. The duality of the boundary value problem and the variation problem can be seen clearly by considering the construction of the function [15]:
π=
σ¯ ε¯˙ dV − v
Fi ui dS
(6)
SF
where π is a function of the total energy and work, σ¯ the effective stress, ε¯˙ the effective strain rate, Fi represents the surface tractions and ui is the velocity component. The incompressibility constraint on admissible velocity fields in Eq. (6) can be removed by using the penalized form of the incompressibility. Hence, the variational form used as the basis for finite element discretization is given by:
␦π =
v
ε˙ v ␦˙εv dV −
σ¯ ␦ε¯˙ dV + K v
Fi ␦ui dS = 0
(7)
Fig. 8. Relative density distributions at various roll diameters and speed, (a) R = 50 mm, V = 2.5 mm/s and (b) R = 200 mm, V = 10 mm/s, during rolling. d0 /t0 = 0.2, m = 0.6, r = 20%, and D (initial relative density) = 0.8. the flow formulation approach using an updated Lagrange procedure. In the finite element discretization procedure, Eq. (7) is converted into a set of nonlinear algebraic equations. The Newton–Raphson method and direct iteration methods are adopted to solve the non-linear equations (see previous question) in DEFORM-2D and DEFORM-3D. The direct iteration method is used to generate a suitable initial estimate for the Newton–Raphson method, which is then employed to obtain a rapid final convergence. The convergence criteria for the iteration are the velocity error norm ||v||/||v|| 0.005, and the force 1/2 error norm ||F||/||F|| 0.05, where ||v|| is defined as (vT v) . For a detailed description of the finite element theory and the modeling formulations, readers are referred to Ref. [15]. Fig. 1 depicts the rolling process for a porous metal containing an internal void. In this figure, R and V are the radius and peripheral velocity of the roll, respectively, L is the contact length between the sheet and the roll, X is the distance from the internal void to the exit of the roll gap, and t0 and t1 are the
Table 1 The stress–strain relationship of the aluminum A6062 Strain
SF
where ␦ε¯˙ and ␦˙εv are the variations in strain rate and volumetric strain rate, respectively, and K is a penalty constant having a very large positive value. Eq. (6) or (7) represents the basic equation for the finite element formulation. This study applies the commercial finite element codes DEFORM-2D and DEFORM-3D to simulate the closure behavior of voids inside porous metal sheets during the sheet rolling process. The finite element code is based on
0 0.2 1 2
Strain rate 1
100
138 168.312 204.828 208
139 170.312 206.828 210
Flow stress = f (temperature, strain rate, strain) (MPa); temperature = 20 ◦ C.
D.-C. Chen / Journal of Materials Processing Technology 180 (2006) 193–200
197
Fig. 9. Relative density distributions at various void length and initial relative density, (a) Lv0 /w0 = 1, D = 0.8, (b) Lv0 /w0 = 0.5, D = 0.8, (c) Lv0 /w0 = 0.25, D = 0.8, and (d) Lv0 /w0 = 1, D = 0.9, during rolling. R = 100 mm, V = 5 mm/s, t0 = 10 mm, w0 = 40 mm, d0 /t0 = 0.2, m = 0.6. initial and final thickness of the porous metal sheet, respectively. The internal void is assumed to have an initial circular cross-section with a diameter of d0 . The final height and length of the internal void are given by h1 and l1 , respectively. The present simulations apply the following assumptions: (1) the rolls are rigid, (2) the sheet is a porous material, (3) the deformation of the porous metal sheet occurs as the result of plane strain, (4) the properties of the porous metal sheet are isotropic, and (5) a constant friction factor exists at the interface between the rolls and the porous metal sheet.
4. Mesh configurations of porous metal sheet As shown in Fig. 2, the present two-dimensional model consists of two objects: the roll and the porous metal sheet.
Isoparametric four-node elements are used. The porous metal sheet is divided into approximately 1000 elements. In the threedimensional model shown in Fig. 3, the porous metal sheet is divided into 6060 nodes, 25,500 elements and 6850 surface polygons. Although the setting of the mesh density in the deformation zones can be varied according to the strain or strain rate distributions, a uniform mesh density is chosen here. The friction factor, m, between the sheet and the rolls is specified as 0.6 to reflect dry friction conditions. The material of the sheet used in the present simulation is porous aluminum A6062. The stress–strain relationship of the aluminum A6062 is presented in Table 1. The material con-
198
D.-C. Chen / Journal of Materials Processing Technology 180 (2006) 193–200
stants are A = 2.27, B = 0.49, and C = 2.30 for porous metal sheet. Fig. 4(a and b) shows the relative density distributions of the porous metal at the roll gap during rolling with sheet reductions of r = 10 and 20%, respectively. In both cases, R = 100 mm, d0 /t0 = 0.2, m = 0.6, and D (initial relative density) = 0.8. It is clear that the relative density is greatest at the front and rear regions of the internal void in both cases. Furthermore, it is observed that the increased rolling force associated with a greater reduction percentage increases the magnitude of the relative density. Figs. 5 and 6 show the effective stress and effective strain distributions, respectively, at the roll gap during rolling of the porous metal sheet under reductions of r = 10 and 20%. The rolling conditions are as presented above for Fig. 3. The results indicate that the larger the reduction and front–rear of voids are, the larger the effective stress and the effective strain. It is apparent that a larger reduction yields a greater relative density in porous metals. Fig. 7(a and b) shows the relative density distributions of the porous metal at the roll gap during rolling with friction factors of m = 0.3 and 0.9, respectively. The rolling conditions are specified as R = 100 mm, d0 /t0 = 0.2, r = 20%, and D = 0.8 in both cases. The relative density is clearer greater at the front and rear regions of the internal void. Furthermore, it is apparent that a higher friction factor yields a slight increase in the relative density. Fig. 8(a and b) illustrates the relative density distributions of the porous metal at the roll gap during rolling with roll diameters of R = 50 mm (V = 2.5 mm/s) and 200 mm (V = 10 mm/s), respectively. The rolling conditions are given by d0 /t0 = 0.2, m = 0.6, r = 20%, D = 0.8 in both figures. As in the case above, the relative density increases at the front and rear regions of the internal voids. Additionally, a larger roll diameter and speed is seen to increase the relative density slightly since the larger diameter and speed generates a higher rolling force. Fig. 9 presents the relative density distributions of the porous metal at the roll gap during rolling for various void lengths, Lv0 , and initial relative densities, D. The rolling conditions are R = 100 mm, V = 5 mm/s, t0 = 10 mm, w0 (width of sheet) = 40 mm, d0 /t0 = 0.2, m = 0.6, and D = 0.8–0.9. It can be seen that a longer void length, Lv0 /w0 , results in a slightly higher final relative density. This result is to be expected since when the value of Lv0 /w0 is high, then necessarily a reduced area of porous metal sheet remains. It is noted that when the initial relative density has a value of 0.9, the final density is 1.0.
Fig. 10. Effects of roll diameter upon final void aspect ratio.
Fig. 11 shows the critical reduction, i.e. the reduction which achieves complete closure of the voids, for various roll radii. In this figure, the region above each curve represents complete void closure. It can be seen that the critical reduction percentage decreases with increasing roll radius, but increases with increasing d0 /t0 . The critical reductions obtained for 0.04 d0 /t0 0.20 and 5 R/t0 20 are found to be 15% rc (%) 30%.
5. Results and discussion The rolling conditions used in the current numerical simulations are presented in Table 2. Fig. 10 illustrates the effects of ratio of the roll radius to the sheet thickness upon the aspect ratio of the final void at the exit for rolling conditions of d0 = 2.0 mm and D = 0.8. It can be seen that as the roll diameter increases, the final aspect ratio of the internal void decreases. This result is reasonable since a larger roll diameter generates larger flow stresses in porous metals, and these stresses suppress the complete closure of the voids.
Fig. 11. Effects of roll diameter on critical reductions.
D.-C. Chen / Journal of Materials Processing Technology 180 (2006) 193–200
199
Table 2 Rolling conditions for porous metal simulations No.
Void diameter, d0 /t0
Friction factor, m
Initial relative density
Roll radius, R/t0
Reduction, r (%)
Case 1 Case 2 Case 3
0.2 0.1 0.04
0.3, 0.6, 0.9 0.3, 0.6, 0.9 0.3, 0.6, 0.9
0.8, 0.9 0.8 0.8
5, 10, 20 5, 10, 20 5, 10, 20
5–25 2.5–20 2.5–10
6. Experiments on porous metal sheet rolling with internal voids In sheet rolling processes, the lubricity at the interface between the porous metal sheet and the rolls plays a key role in determining various rolling results, including void closure. The lubricity is expressed by the constant shear friction parameter τ s = mk, where k is the yield shear stress of the deforming material and m is the friction factor (0 m 1). In the present study, the friction factor between the sheet and the rolls was determined by the so-called ring compression test. The aspect ratio of the test piece was 6:3:2. The surface of the test piece was polished slightly with sandpaper and then cleaned using a cloth with an acetone solution to ensure the presence of dry friction. The experimental measurements of the internal ring diameter
reductions were then compared with the analytically obtained results. It was found that the two sets of values correlated most closely for a friction factor of 0.6. Fig. 12 shows the appearance of the rolled product and the internal voids for different reductions under rolling conditions of R = 100 mm, V = 5 mm/s, t0 = 10 mm, w0 = 40 mm, m = 0.6, and d0 /t0 = 0.15. It can be seen that at a reduction of 27%, the final void height decreases to zero, hence indicating that the void is completely closed. 7. Conclusions This paper has used the finite element method of continuum mechanics to investigate the void closure behavior inside a porous metal sheet during the sheet rolling process. The effects of various rolling conditions on the final dimensions of the void defect have been discussed systematically. These rolling conditions include the ratio of the roll radius to the sheet thickness, the diameter of the defective void, the thickness reduction, the friction factor between the porous metals and rolls, the initial relative density of the porous metal sheet, etc. From the present simulation results, it can be concluded that: (1) the relative density is greatest at the front and rear regions of the internal void. Furthermore, the larger the values of the reduction percentage, the friction factor and the roll diameter, the higher the relative density; (2) the larger the reduction and front–rear of voids are, the larger the effective stress and the effective strain; (3) a longer void length, Lv0 /w0 , results in a slightly higher relative density; (4) the critical reduction decreases with increasing roll radius, but increases with increasing d0 /t0 . The current simulation results provide a valuable source of reference for the rolling of porous metal sheets with internal void defects. Acknowledgement The authors gratefully acknowledge the support provided to this study by the National Science Council of the Republic of China under Grant No. NSC 92-2212-E-274-005. References
Fig. 12. Appearance of rolled product and void: (a) r = 8%, (b) r = 18%, and (c) r = 27%. R = 100 mm, V = 5 mm/s, t0 = 10 mm, w0 = 40 mm, m = 0.6, d0 /t0 = 0.15.
[1] O. Moriya, T. Tsuyoshi, Ductile fracture of porous material, J. Jpn. Soc. Technol. Plast. 14 (149) (1973) 439–445. [2] O. Moriya, T. Tsuyoshi, Slip-line field theory and upper-bound theory for porous materials, J. Jpn. Soc. Technol. Plast. 15 (156) (1974) 43–51. [3] O. Moriya, S. Susumu, O. Kunio, Ductile fracture of sintered iron compacts, J. Jpn. Soc. Technol. Plast. 20 (226) (1979) 1037–1044. [4] B.B. Hwang, S. Kobayashi, Deformation characterization of powdered metals in compaction, Int. J. Mach. Tools Manuf. 30 (2) (1990) 309–323.
200
D.-C. Chen / Journal of Materials Processing Technology 180 (2006) 193–200
[5] H.N. Han, H.S. Kim, K.H. Oh, D.N. Lee, Elastoplastic finite element analysis for porous metals, Powder Metall. 37 (2) (1994) 140–146. [6] H.N. Han, H.S. Kim, K.H. Oh, D.N. Lee, Analysis of coefficient of friction in compression of porous metal rings, Powder Metall. 37 (4) (1994) 259–264. [7] S. Nagaki, M. Goya, R. Sowerby, The influence of void distribution on the yielding of an elastic–plastic porous solid, Int. J. Plast. 9 (4) (1993) 199–211. [8] H.K. Cho, J. Suh, K.T. Kim, Densification of porous alloy steel preforms at high temperature, Int. J. Mech. Sci. 36 (4) (1994) 317–328. [9] P.T. Wang, M.E. Karabin, Evolution of porosity during thin plate rolling of powder-based porous aluminum, Powder Technol. 78 (1994) 67–76. [10] Q. Shi, S. Zhang, Y. Sun, D. Zhang, The characteristics of the deformation and flow process in the rolling of metal sheets and metal pow-
[11]
[12]
[13] [14] [15]
ders of composite materials, J. Mater. Process. Technol. 63 (1997) 421– 425. T. Hirohata, S. Masaki, S. Shima, Experiment on metal powder compaction by differential speed rolling, J. Mater. Process. Technol. 111 (2001) 113–117. H.A. Kuhn, C.L. Downey Jr., Deformation characteristics and plasticity theory of sintered powdered materials, Int. J. Powder Met. 7 (1971) 15– 25. R.J. Green, A plasticity theory for porous solids, Int. J. Mech. Sci. 14 (1972) 215–224. S. Shima, M. Oyane, Plasticity theory for porous metals, Int. J. Mech. Sci. 18 (1976) 285–291. S. Kobayashi, S.I. Oh, T. Altan, Metal Forming and the Finite Element Method, Oxford University Press, New York, 1989.