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Deformation and strengthening of sintered ferrous material
Journal of Materials Processing Technology 187–188 (2007) 694–697
Deformation and strengthening of sintered ferrous material X.P. Qin, L. Hua ∗ School of Materials Science and Engineering, Wuhan University of Technology, Wuhan 430070, PR China
Abstract Plastic deformation is an important process to improve performance and obtain final product for sintered powder materials. Sintered ferrous material is widely used in many engineering departments and its deformation characters are typical for most of sintered powder materials. The influential factors, such as deformation strengthening, instantaneous and initial relative densities etc., on following yield strength is studied; an approach to determine the plastic stress coefficient and hardening exponent is derived, based on yield criterion and strengthening laws of plastic deformation of sintered powder materials and uni-axial compression experiment of sintered ferrous cylinder specimens. A unified form of yield criterion function is obtained, in which initial and following yield of sintered ferrous material are expressed. © 2006 Elsevier B.V. All rights reserved. Keywords: Sintered ferrous material; Plastic deformation; Strengthening; Yield criterion
1. Introduction Sintered ferrous materials are made by process of compacting and sintering ferrous powder and non-metal powder. The plastic deformation is a main way to improve performance of sintered ferrous materials and obtain the final product. As most sintered powder materials, sintered ferrous material is of complex characters in plastic deformation. For example, radical changes of both volume and density of preform take place with the deformation, and densification and strengthening are resulted in from the plastic deformation. Hence, the initial yield and following yield of sintered ferrous material are related to many factors. In this paper, the yield criterion of sintered ferrous material is researched, and the parameters in the yield criterion for sintered ferrous material are defined based on the theoretical and experimental analysis. Thus, a unified form of function which can expresses both initial yield and following yield of sintered ferrous material is obtained. 2. Deformation analysis for sintered powder material 2.1. Yield criterion From 1970s, many researchers have introduced yield criteria for sintered powder materials (or compressible materials), ∗
Corresponding author. Tel.: +86 27 87168391; fax: +86 27 87168391. E-mail address: [email protected] (L. Hua).
0924-0136/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2006.11.060
which are based on experimental and theoretical analysis [1–9]. A typical theorem is that the plastic deformation occurs when the elasticity strain energy reaches a critical value. The formulation can be written as AJ2 + BJ12 = Y 2 = δY02
(1)
where A, B, δ are the parameters related to property of sintered material and the parameters A and B are the functions of ρ, material transient density [1–9], and Y, Y0 are the instantaneous yield strength of sintered powder material and yield strength of its matrix material respectively, and J1 is the first invariant of stress tensor, i.e., the linear stress invariant, J2 is the second invariant of deviatoric stress tensor, i.e., the quadratic stress deviator invariant. J2 = σij σij /2 and J1 = σ ii where σij is the stress deviator and σ ij is the stress tensor and ρ is the relative density of working material during deformation. On condition of uni-axial compression deformation, the stress components are σ 1 = σ 2 = 0, σ 3 = −Y. Many researchers suggest the expressions to determine A and B, as shown by Table 1 In Table 1 when sintered powder materials are isotropy, namely plastic Poisson’s ratio μ = 0.5ρ2 , the expressions to determine both A and B are only correlative with ρ, i.e., ⎫ A = 2(1 + μ) = 2 + ρ2 ⎬ (1 − 2μ) (1 − ρ2 ) ⎭ B= = 3 3
(2)
X.P. Qin, L. Hua / Journal of Materials Processing Technology 187–188 (2007) 694–697 Table 1 The parameters A and B and δ from different scholars Scholars
A
B
δ
Doraivelu [4] Park [8] R. Narayanasamy [9]
2 + ρ2
(1 − ρ2 )/3
2ρ2 − 1 1.44ρ2 /(2.44 − ρ) [(ρ − ρ0 )/(1 − ρ0 )]2
2(1 + μ) 2 + ρ2
(1 − 2μ)/3 (1 − ρ2 )/3
The expressions to define δ from different researchers are shown in Table 1. In Table 1, 0 < ρ0 ≤ ρ ≤ 1 and ρ0 is the initial relative density. Based on Table 1 and assuming ρ0 = 0.3, the values of δ are shown in Table 2 for different ρ. As shown in Table 2, when ρ is equal to 1.0, namely sintered powder material is changed into fully densified material, the values of parameter δ from different investigators are approximately the same. On the other condition the values of the δ may be quite different. When ρ is below 0.7, δ of Doraivelu are negative. When ρ is equal to ρ0 , δ of R. Narayanasamy is zero, namely yield strength of sintered powder materials is zero. Obviously, it is difficult to say that the values of δ are in agreement. 2.2. Parameter δ According to plastic potential theory [10] and by partially differentiating Eq. (1), the constitutive equations of sintered powder materials in plastic deformation can be written as B dεi = dλ σi − 1 − 6 (3) σm A where σ i and dεi are the principal stresses and the strain increments respectively, and σ m is the hydrostatic stress, and dλ is a non-negative constant. Initial yield strength of sintered powder materials may be considered to be correlative with relative density and yield strength of its matrix material. Following yield strength of sintered powder materials during deformation is correlative with relative density and yield strength of its matrix material as well as workhardening capacity. For most of sintered powder materials, on condition of uni-axial compression deformation stress–strain relationship [5] is σ = Kεn
(4)
where σ, ε are the real stress and strain respectively, and K, n are the plastic stress coefficient and the hardening exponent Table 2 The relationship between δ and ρ ρ
δ of Doraivelu [4]
δ of Park [8]
␦ of R. Narayanasamy, et al. [9]
0.3 (ρ0 ) 0.5 0.7 0.8 0.9 1.0
−0.82 −0.5 −0.02 0.28 0.62 1
0.06 0.19 0.41 0.56 0.76 1
0 0.08 0.33 0.51 0.73 1
695
respectively based on uni-axial compression test for sintered powder materials. For a uni-axial compression test with cylinder specimens of sintered powder materials, there are σ 1 = σ 2 = 0, σ 3 = −Y, dε1 = dε2 and μ = 0.5ρ2 . Substituting these expressions and Eq. (4) into Eqs. (1) and (3), we have ⎛ ⎞ 2 ρ 1 − ρ0 ⎠ ε = ln ⎝ (5) ρ0 1 − ρ 2 ⎡ ⎛ Y = Kεn = K⎣ln ⎝
ρ ρ0
⎞⎤n 2 1 − ρ 0 ⎠⎦ 1 − ρ2
(6)
And parameter δ can be written as ⎡ ⎛ ⎞⎤2n 2 2 2 2 Y ρ 1 − ρ0 ⎠⎦ σ K δ = 2 = 2 = 2 ⎣ln ⎝ ρ0 1 − ρ 2 Y0 Y0 Y0
(7)
From Eqs. (5)–(7), the value of δ in the yield criterion is correlative with relative density and yield strength of its matrix material as well as plastic stress coefficient and hardening exponent, which represents that yield criterion of sintered powder materials is correlative with yield strength of matrix material and following yield strength during deformation, and following yield strength is correlative with initial and instantaneous relative density as well as plastic stress coefficient and hardening exponent namely work-hardening capacity. In the other word, Y or δ is also correlative with the type of the material, deformation and densified degree, initial density and deformation condition etc. Thus, the parameter δ or yield criterion of sintered powder materials can be only derived by means of the experiment. 3. Yield and strengthening for sintered ferrous material 3.1. Uni-axial compression experiment for sintered ferrous material Uni-axial compression of sintered ferrous powder cylinder specimen in the room temperature is carried out in order to define parameters in yield criterion. Specimens are made of deoxidized ferrous powder, compositions of which are listed in Table 3. Sintering is conducted at temperature 1100 ◦ C for 2 h in an environment of coal gas. Experimental equipment and facilities used are Model WI-60 material test machine, Model Dvcc-51-ch microcomputer measure system and Model YD-21 dynamic resistance strain instrument. Initial parameters of the specimens are shown in Table 4. The two end surfaces of specimens are ground, polished and daubed with zinc stearate lube, diluted by alcohol, for reducing friction. Every Table 3 The ingredient of sintered ferrous powder cylinder specimen Ingredient (%)
Fe
Cu
Mo
C
Muriatic acid infusible
Content
Surplus
1.0
0.5
0.35–0.45
<0.3
696
X.P. Qin, L. Hua / Journal of Materials Processing Technology 187–188 (2007) 694–697
Table 4 The initial parameters of sintered ferrous powder cylinder specimens and experiment regressive equations Specimen no.
Weight (g)
Height (mm)
Diameter (mm)
Initial relative density (%)
Regressive equation (σ/MPa)
1 2 3 4 5 6 7 8 9 10
54.00 53.30 55.50 53.10 51.60 50.20 53.60 54.30 54.00 48.00
25.00 25.26 25.44 24.60 24.82 24.84 25.10 25.18 25.04 24.72
20.06 20.30 20.30 20.28 20.30 20.04 20.20 20.20 20.16 20.16
0.8590 0.8359 0.8642 0.8567 0.8235 0.8214 0.8542 0.8627 0.8627 0.7799
σ = 782.34ε0.2652 σ = 742.12ε0.2989 σ = 784.89ε0.2741 σ = 789.91ε0.2743 σ = 693.55ε0.2797 σ = 699.79ε0.2890 σ = 759.24ε0.2632 σ = 776.01ε0.2653 σ = 739.71ε0.2470 σ = 652.43ε0.2586
specimen is compressed many times, and pressure forces and dimensions of the specimens for each time are measured and recorded. By means of regressive analysis for data shown in Table 4 from the experiments, instantaneous yield strength (real stress) and real strain ε are obtained. σ–ε curves are shown by Fig. 1. By matching and analyzing σ–ε curve of two kinds of specimens with different initial relative density, relationships between real stress and real strain are derived, which are in agreement with power function as shown by Eq. (4). Experimental regressive equations are shown by Table 4. From Table 4, K and n are different for specimens with different initial relative densities. The main effect factor to them is the initial relative density. If ρ0 − K, ρ0 − n are used as reference frame (coordinates system) respectively, relation curves among them can be derived as shown in Fig. 2. By matching and drafting curves in Fig. 2 and analyzing regressive equations in Table 4, the relationships between n, K and ρ0 are defined as that follows: K = −0590.92 + 1583.12ρ0 (MPa) n = 0.3513 − 0.0947ρ0
.
(8)
Fig. 2. A − ρ0 and n − ρ0 .
3.2. Discussion for plastic deformation of sintered ferrous material (1) Based on Eqs. (7) and (8), when ρ0 = 0.3733, K = 0, δ = 0 and Y 2 = δY02 = 0, which represents that when initial relative density of sintered ferrous material is close to relative density of loose packed ferrous powder [11], namely ρ0 = 0.4 the yield strength vanishes, which are in accordance with the result of the experiment and Ref. [11]. (2) When ρ0 = 1, namely sintered ferrous material is changed into fully densified material, from Eqs. (4) and (8) if ε = 0.2% is assumed as yield point of material [10] Y0.2 = 201 MPa which is close to the yield strength of matrix iron (or homogeneous ingredient melted iron) [12]. In the same way from Eq. (2) when ρ = 1, A = 3, B = 0, the yield criterion of sintered ferrous material, namely Eq. (1) is changed into Von Mises criterion of the fully densified material [10], which represents that yield criterion of the fully densified materials can be researched as special case of sintered powder (compressible) materials, and on the other hand that the yield criterion of sintered powder materials proposed in the paper is applicable for the fully densified materials. 4. Conclusions
Fig. 1. Real stress–real strain curve for two kinds of initial relative density sintered ferrous powder specimen.
(1) The yield criterion of sintered powder materials may be applied to not only the powder materials but also the fully densified material, and yield criterion for the fully densified material may be investigated as special case of sintered
X.P. Qin, L. Hua / Journal of Materials Processing Technology 187–188 (2007) 694–697
powder (compressible) materials. Sintered ferrous material is a typical representative sintered powder material, plastic deformation and yield criterion of which present many common characters of most sintered powder materials. (2) Following yield strength is correlative with not only initial and instantaneous relative density but also plastic stress coefficient and hardening exponent, which are different for different sintered powder materials. Plastic stress coefficient and hardening exponent can be determined by uni-axial compression experiment of sintered ferrous material. (3) Plastic stress coefficient and hardening exponent are derived by means of uni-axial compression experiment of sintered ferrous powder cylinder specimen so that the effect of deformation strengthening and instantaneous and initial relative density on following yield strength is revealed and a unified form of yield criterion function is obtained, in which initial and following yield criterion of sintered ferrous material are expressed. Acknowledgement This work was supported by the National Natural Science Foundation of China under grant Nos. 50175086 and 50175085. The supports are gratefully acknowledged.
697
References [1] H.A. Kuhn, C.L. Downey, Deformation characteristics and plasticity theory of sintered materials, Int. J. Powder Metall. 7 (1) (1971) 15–25. [2] R.J. Green, A plasticity theory for porous solids, Int. J. Mech. Sci. 18 (6) (1976) 215–224. [3] S. Shima, M. Oyane, Plasticity theory for porous metals, Int. J. Mech. Sci. 18 (6) (1976) 285–291. [4] S.M. Doraivelu, H.L. Gegel, J.S. Gunasekera, et al., A new yield function for compressible P/M materials, Int. J. Mech. Sci. 26 (9/10) (1984) 527–535. [5] Z.Z. Zhao, L. Hua, H.C. Wang, The deformation and densification of sintered powder materials, Int. J. Sci. Sintering 19 (2) (1987) 65– 80. [6] L. Hua, Z.Z. Zhao, et al., Plastic theory and application of sintered materials, Chin. J. Mech. Eng. 28 (4) (1992) 94–102. [7] X.P. Ren, Yield criterion of P/M, Powder Metall. Technol. 10 (1) (1992) 8–12. [8] J.J. Park, Constitutive relations to predict plastic deformations of porous metals in compaction, Int. J. Mech. Sci. 37 (7) (1995) 709–719. [9] R. Narayanasamy, R. Ponalagusamy, K.R. Subranmanian, Generalized yield criteria of porous sintered powder metallurgy metals, J. Mat. Pro. Tech. 110 (2001) 182–185. [10] R. Wang, Foundation of Plasticity Mechanics, Science press, Beijing, 1982. [11] C.Y. Wu, Principles of Powder Deformation Mechanics, Metallurgy Industry Press, Beijing, 2003. [12] Machine Design Handbook Compiling Group/Machine Design Handbook, Machine Industry Press, Beijing, 1989.
Deformation and strengthening of sintered ferrous material X.P. Qin, L. Hua ∗ School of Materials Science and Engineering, Wuhan University of Technology, Wuhan 430070, PR China
Abstract Plastic deformation is an important process to improve performance and obtain final product for sintered powder materials. Sintered ferrous material is widely used in many engineering departments and its deformation characters are typical for most of sintered powder materials. The influential factors, such as deformation strengthening, instantaneous and initial relative densities etc., on following yield strength is studied; an approach to determine the plastic stress coefficient and hardening exponent is derived, based on yield criterion and strengthening laws of plastic deformation of sintered powder materials and uni-axial compression experiment of sintered ferrous cylinder specimens. A unified form of yield criterion function is obtained, in which initial and following yield of sintered ferrous material are expressed. © 2006 Elsevier B.V. All rights reserved. Keywords: Sintered ferrous material; Plastic deformation; Strengthening; Yield criterion
1. Introduction Sintered ferrous materials are made by process of compacting and sintering ferrous powder and non-metal powder. The plastic deformation is a main way to improve performance of sintered ferrous materials and obtain the final product. As most sintered powder materials, sintered ferrous material is of complex characters in plastic deformation. For example, radical changes of both volume and density of preform take place with the deformation, and densification and strengthening are resulted in from the plastic deformation. Hence, the initial yield and following yield of sintered ferrous material are related to many factors. In this paper, the yield criterion of sintered ferrous material is researched, and the parameters in the yield criterion for sintered ferrous material are defined based on the theoretical and experimental analysis. Thus, a unified form of function which can expresses both initial yield and following yield of sintered ferrous material is obtained. 2. Deformation analysis for sintered powder material 2.1. Yield criterion From 1970s, many researchers have introduced yield criteria for sintered powder materials (or compressible materials), ∗
Corresponding author. Tel.: +86 27 87168391; fax: +86 27 87168391. E-mail address: [email protected] (L. Hua).
0924-0136/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2006.11.060
which are based on experimental and theoretical analysis [1–9]. A typical theorem is that the plastic deformation occurs when the elasticity strain energy reaches a critical value. The formulation can be written as AJ2 + BJ12 = Y 2 = δY02
(1)
where A, B, δ are the parameters related to property of sintered material and the parameters A and B are the functions of ρ, material transient density [1–9], and Y, Y0 are the instantaneous yield strength of sintered powder material and yield strength of its matrix material respectively, and J1 is the first invariant of stress tensor, i.e., the linear stress invariant, J2 is the second invariant of deviatoric stress tensor, i.e., the quadratic stress deviator invariant. J2 = σij σij /2 and J1 = σ ii where σij is the stress deviator and σ ij is the stress tensor and ρ is the relative density of working material during deformation. On condition of uni-axial compression deformation, the stress components are σ 1 = σ 2 = 0, σ 3 = −Y. Many researchers suggest the expressions to determine A and B, as shown by Table 1 In Table 1 when sintered powder materials are isotropy, namely plastic Poisson’s ratio μ = 0.5ρ2 , the expressions to determine both A and B are only correlative with ρ, i.e., ⎫ A = 2(1 + μ) = 2 + ρ2 ⎬ (1 − 2μ) (1 − ρ2 ) ⎭ B= = 3 3
(2)
X.P. Qin, L. Hua / Journal of Materials Processing Technology 187–188 (2007) 694–697 Table 1 The parameters A and B and δ from different scholars Scholars
A
B
δ
Doraivelu [4] Park [8] R. Narayanasamy [9]
2 + ρ2
(1 − ρ2 )/3
2ρ2 − 1 1.44ρ2 /(2.44 − ρ) [(ρ − ρ0 )/(1 − ρ0 )]2
2(1 + μ) 2 + ρ2
(1 − 2μ)/3 (1 − ρ2 )/3
The expressions to define δ from different researchers are shown in Table 1. In Table 1, 0 < ρ0 ≤ ρ ≤ 1 and ρ0 is the initial relative density. Based on Table 1 and assuming ρ0 = 0.3, the values of δ are shown in Table 2 for different ρ. As shown in Table 2, when ρ is equal to 1.0, namely sintered powder material is changed into fully densified material, the values of parameter δ from different investigators are approximately the same. On the other condition the values of the δ may be quite different. When ρ is below 0.7, δ of Doraivelu are negative. When ρ is equal to ρ0 , δ of R. Narayanasamy is zero, namely yield strength of sintered powder materials is zero. Obviously, it is difficult to say that the values of δ are in agreement. 2.2. Parameter δ According to plastic potential theory [10] and by partially differentiating Eq. (1), the constitutive equations of sintered powder materials in plastic deformation can be written as B dεi = dλ σi − 1 − 6 (3) σm A where σ i and dεi are the principal stresses and the strain increments respectively, and σ m is the hydrostatic stress, and dλ is a non-negative constant. Initial yield strength of sintered powder materials may be considered to be correlative with relative density and yield strength of its matrix material. Following yield strength of sintered powder materials during deformation is correlative with relative density and yield strength of its matrix material as well as workhardening capacity. For most of sintered powder materials, on condition of uni-axial compression deformation stress–strain relationship [5] is σ = Kεn
(4)
where σ, ε are the real stress and strain respectively, and K, n are the plastic stress coefficient and the hardening exponent Table 2 The relationship between δ and ρ ρ
δ of Doraivelu [4]
δ of Park [8]
␦ of R. Narayanasamy, et al. [9]
0.3 (ρ0 ) 0.5 0.7 0.8 0.9 1.0
−0.82 −0.5 −0.02 0.28 0.62 1
0.06 0.19 0.41 0.56 0.76 1
0 0.08 0.33 0.51 0.73 1
695
respectively based on uni-axial compression test for sintered powder materials. For a uni-axial compression test with cylinder specimens of sintered powder materials, there are σ 1 = σ 2 = 0, σ 3 = −Y, dε1 = dε2 and μ = 0.5ρ2 . Substituting these expressions and Eq. (4) into Eqs. (1) and (3), we have ⎛ ⎞ 2 ρ 1 − ρ0 ⎠ ε = ln ⎝ (5) ρ0 1 − ρ 2 ⎡ ⎛ Y = Kεn = K⎣ln ⎝
ρ ρ0
⎞⎤n 2 1 − ρ 0 ⎠⎦ 1 − ρ2
(6)
And parameter δ can be written as ⎡ ⎛ ⎞⎤2n 2 2 2 2 Y ρ 1 − ρ0 ⎠⎦ σ K δ = 2 = 2 = 2 ⎣ln ⎝ ρ0 1 − ρ 2 Y0 Y0 Y0
(7)
From Eqs. (5)–(7), the value of δ in the yield criterion is correlative with relative density and yield strength of its matrix material as well as plastic stress coefficient and hardening exponent, which represents that yield criterion of sintered powder materials is correlative with yield strength of matrix material and following yield strength during deformation, and following yield strength is correlative with initial and instantaneous relative density as well as plastic stress coefficient and hardening exponent namely work-hardening capacity. In the other word, Y or δ is also correlative with the type of the material, deformation and densified degree, initial density and deformation condition etc. Thus, the parameter δ or yield criterion of sintered powder materials can be only derived by means of the experiment. 3. Yield and strengthening for sintered ferrous material 3.1. Uni-axial compression experiment for sintered ferrous material Uni-axial compression of sintered ferrous powder cylinder specimen in the room temperature is carried out in order to define parameters in yield criterion. Specimens are made of deoxidized ferrous powder, compositions of which are listed in Table 3. Sintering is conducted at temperature 1100 ◦ C for 2 h in an environment of coal gas. Experimental equipment and facilities used are Model WI-60 material test machine, Model Dvcc-51-ch microcomputer measure system and Model YD-21 dynamic resistance strain instrument. Initial parameters of the specimens are shown in Table 4. The two end surfaces of specimens are ground, polished and daubed with zinc stearate lube, diluted by alcohol, for reducing friction. Every Table 3 The ingredient of sintered ferrous powder cylinder specimen Ingredient (%)
Fe
Cu
Mo
C
Muriatic acid infusible
Content
Surplus
1.0
0.5
0.35–0.45
<0.3
696
X.P. Qin, L. Hua / Journal of Materials Processing Technology 187–188 (2007) 694–697
Table 4 The initial parameters of sintered ferrous powder cylinder specimens and experiment regressive equations Specimen no.
Weight (g)
Height (mm)
Diameter (mm)
Initial relative density (%)
Regressive equation (σ/MPa)
1 2 3 4 5 6 7 8 9 10
54.00 53.30 55.50 53.10 51.60 50.20 53.60 54.30 54.00 48.00
25.00 25.26 25.44 24.60 24.82 24.84 25.10 25.18 25.04 24.72
20.06 20.30 20.30 20.28 20.30 20.04 20.20 20.20 20.16 20.16
0.8590 0.8359 0.8642 0.8567 0.8235 0.8214 0.8542 0.8627 0.8627 0.7799
σ = 782.34ε0.2652 σ = 742.12ε0.2989 σ = 784.89ε0.2741 σ = 789.91ε0.2743 σ = 693.55ε0.2797 σ = 699.79ε0.2890 σ = 759.24ε0.2632 σ = 776.01ε0.2653 σ = 739.71ε0.2470 σ = 652.43ε0.2586
specimen is compressed many times, and pressure forces and dimensions of the specimens for each time are measured and recorded. By means of regressive analysis for data shown in Table 4 from the experiments, instantaneous yield strength (real stress) and real strain ε are obtained. σ–ε curves are shown by Fig. 1. By matching and analyzing σ–ε curve of two kinds of specimens with different initial relative density, relationships between real stress and real strain are derived, which are in agreement with power function as shown by Eq. (4). Experimental regressive equations are shown by Table 4. From Table 4, K and n are different for specimens with different initial relative densities. The main effect factor to them is the initial relative density. If ρ0 − K, ρ0 − n are used as reference frame (coordinates system) respectively, relation curves among them can be derived as shown in Fig. 2. By matching and drafting curves in Fig. 2 and analyzing regressive equations in Table 4, the relationships between n, K and ρ0 are defined as that follows: K = −0590.92 + 1583.12ρ0 (MPa) n = 0.3513 − 0.0947ρ0
.
(8)
Fig. 2. A − ρ0 and n − ρ0 .
3.2. Discussion for plastic deformation of sintered ferrous material (1) Based on Eqs. (7) and (8), when ρ0 = 0.3733, K = 0, δ = 0 and Y 2 = δY02 = 0, which represents that when initial relative density of sintered ferrous material is close to relative density of loose packed ferrous powder [11], namely ρ0 = 0.4 the yield strength vanishes, which are in accordance with the result of the experiment and Ref. [11]. (2) When ρ0 = 1, namely sintered ferrous material is changed into fully densified material, from Eqs. (4) and (8) if ε = 0.2% is assumed as yield point of material [10] Y0.2 = 201 MPa which is close to the yield strength of matrix iron (or homogeneous ingredient melted iron) [12]. In the same way from Eq. (2) when ρ = 1, A = 3, B = 0, the yield criterion of sintered ferrous material, namely Eq. (1) is changed into Von Mises criterion of the fully densified material [10], which represents that yield criterion of the fully densified materials can be researched as special case of sintered powder (compressible) materials, and on the other hand that the yield criterion of sintered powder materials proposed in the paper is applicable for the fully densified materials. 4. Conclusions
Fig. 1. Real stress–real strain curve for two kinds of initial relative density sintered ferrous powder specimen.
(1) The yield criterion of sintered powder materials may be applied to not only the powder materials but also the fully densified material, and yield criterion for the fully densified material may be investigated as special case of sintered
X.P. Qin, L. Hua / Journal of Materials Processing Technology 187–188 (2007) 694–697
powder (compressible) materials. Sintered ferrous material is a typical representative sintered powder material, plastic deformation and yield criterion of which present many common characters of most sintered powder materials. (2) Following yield strength is correlative with not only initial and instantaneous relative density but also plastic stress coefficient and hardening exponent, which are different for different sintered powder materials. Plastic stress coefficient and hardening exponent can be determined by uni-axial compression experiment of sintered ferrous material. (3) Plastic stress coefficient and hardening exponent are derived by means of uni-axial compression experiment of sintered ferrous powder cylinder specimen so that the effect of deformation strengthening and instantaneous and initial relative density on following yield strength is revealed and a unified form of yield criterion function is obtained, in which initial and following yield criterion of sintered ferrous material are expressed. Acknowledgement This work was supported by the National Natural Science Foundation of China under grant Nos. 50175086 and 50175085. The supports are gratefully acknowledged.
697
References [1] H.A. Kuhn, C.L. Downey, Deformation characteristics and plasticity theory of sintered materials, Int. J. Powder Metall. 7 (1) (1971) 15–25. [2] R.J. Green, A plasticity theory for porous solids, Int. J. Mech. Sci. 18 (6) (1976) 215–224. [3] S. Shima, M. Oyane, Plasticity theory for porous metals, Int. J. Mech. Sci. 18 (6) (1976) 285–291. [4] S.M. Doraivelu, H.L. Gegel, J.S. Gunasekera, et al., A new yield function for compressible P/M materials, Int. J. Mech. Sci. 26 (9/10) (1984) 527–535. [5] Z.Z. Zhao, L. Hua, H.C. Wang, The deformation and densification of sintered powder materials, Int. J. Sci. Sintering 19 (2) (1987) 65– 80. [6] L. Hua, Z.Z. Zhao, et al., Plastic theory and application of sintered materials, Chin. J. Mech. Eng. 28 (4) (1992) 94–102. [7] X.P. Ren, Yield criterion of P/M, Powder Metall. Technol. 10 (1) (1992) 8–12. [8] J.J. Park, Constitutive relations to predict plastic deformations of porous metals in compaction, Int. J. Mech. Sci. 37 (7) (1995) 709–719. [9] R. Narayanasamy, R. Ponalagusamy, K.R. Subranmanian, Generalized yield criteria of porous sintered powder metallurgy metals, J. Mat. Pro. Tech. 110 (2001) 182–185. [10] R. Wang, Foundation of Plasticity Mechanics, Science press, Beijing, 1982. [11] C.Y. Wu, Principles of Powder Deformation Mechanics, Metallurgy Industry Press, Beijing, 2003. [12] Machine Design Handbook Compiling Group/Machine Design Handbook, Machine Industry Press, Beijing, 1989.