A mathematical theory of plasticity for compressible powder metallurgy materials — Part III

Journal of Materials Processing Technology 100 (2000) 262±265

A mathematical theory of plasticity for compressible powder metallurgy materials Ð Part III R. Narayanasamya,*, R. Ponalagusamyb b

a Department of Production Engineering, Regional Engineering College, Tiruchirappalli 620 015, Tamilnadu, India Department of Mathematics and Computer Applications, Regional Engineering College, Tiruchirappalli 620 015, Tamilnadu, India

Accepted 17 December 1999

Abstract A new mathematical equation taking into account of all new and relevant parameters to represent the ¯ow behaviour of partially dense P/ M materials during the case of simple upsetting-compression test is proposed in this paper. The ratio of the yield stress of the P/M-porous material to the yield stress of 100% dense material is found to be increasing with increase in deformation strain. Further, a theory for simple plastic instability and work hardening rate of porous P/M-material has been developed with strain hardening index and strain rate sensitivity parameters. # 2000 Elsevier Science S.A. All rights reserved. Keywords: Upsetting-compression test; Strain hardening index; Strain rate sensitivity

1. Introduction

2. Plasticity theory for porous materials ¯ow

In P/M fabrication process, parts from metal powders are produced by compaction and sintering operations. Moreover, the porous sintered parts can be deformed again to a ®nal size within reasonable dimensional tolerance. Since the apparent density of sintered compacts changes during deformation due to the presence of pores inside, the conventional plasticity theory cannot be applied directly to this process. The mathematical theory of plasticity for porous materials has been proposed by many research workers [1±4]. However, these workers were not successful in predicting the ¯ow behaviour of partially dense materials. Narayanasamy and Ponalagusamy [5] proposed a mathematical equation for the determination of yield stress in the case of simple upsetting-compression test for P/M porous materials. However, this work involves complex mathematical processing and based on simple material law taking into the account of strain hardening index parameter. The present work involves the determination of ¯ow behaviour based on modi®ed material law taking into the account of strain rate sensitivity parameter and strain hardening index value.

The variation of ¯ow stress with densi®cation during uniaxial compression test or upsetting operation as proposed in [6] is as follows: YR ˆ Ym …2R2 ÿ 1†1=2

(2.1)

where Ym ˆ Y0 ‰1 ‡ K2 Em ‡ K3 E_ m Š K2 ˆ

1 dsm sm dEm

(2.3)

1 dsm sm d_Em  Z Eagg  1 1=2 2ÿ 2 dEagg Em ˆ R 0 K3 ˆ

Z E_ m ˆ

E_ agg 

0

2ÿ

(2.2)

1 R2

1=2

d_Eagg

(2.4) (2.5) (2.6)

On putting Rˆ1 in Eqs. (2.5) and (2.6) this yields * Corresponding author. Tel.: ‡91-431-582281; fax: ‡91-431-55213333. E-mail address: [email protected] (R. Narayanasamy).

dEm ˆ 1:0 dEagg

0924-0136/00/$ ± see front matter # 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 4 9 0 - 2

(2.7)

R. Narayanasamy, R. Ponalagusamy / Journal of Materials Processing Technology 100 (2000) 262±265

d_Em ˆ 1:0 d_Eagg

(2.8)

On putting Rˆ0.71 in the above mentioned Eqs. (2.7) and (2.8), the value of (dEm/dEagg) and (d_Em =d_Eagg ) is zero. As explained elsewhere [5], the aggregate is geometrically unstable and crumbles during deformation when the relative density is below 0.71. In the range between 0.71 and 1.00 of R, the strain transferred to the matrix increases continuously till it asymptotically approaches the strain applied to the aggregate. Differentiating Eq. (2.1) with respect to Em, the following expression is obtained: dYR dYm 1 dR ˆ …2R2 ÿ 1†1=2 ‡ Ym …2R2 ÿ 1†ÿ1=2 dEm dEm 2 dEm

(2.9)

Differentiating Eq. (2.2) with respect to Em, the following equation yields:    dYm 1 dsm Em d2 sm Em dsm 2 1 dsm d_Em ˆ Y0 ‡ ÿ ‡ sm dEm sm dE2m sm d_Em dEm dEm s2m dEm  2 E_ m d sm E_ m dsm ‡ ÿ (2.10) sm dEm d_Em s2m d_Em From Eqs. (2.9) and (2.10), the following equation is obtained: dYR dYm 1 dR ˆ …2R2 ÿ 1†1=2 ‡ Ym …2R2 ÿ 1†ÿ1=2 R dEm dEm 2 dEm

(2.11)

As proposed elsewhere [7], the material law relating sm and Em can be written as follows: mR sm ˆ K1 RnA EnR m E_ m

(2.12)

Eq. (2.12) is applicable for porous P/M materials when it is subjected to upsetting operation. When putting mˆ0 in Eq. (2.12), this becomes sm ˆ K1 RnA EnR

(2.12a)

This equation which is applicable for materials exhibiting strain hardening behaviour at room temperature is proposed in [5]. When substituting Rˆ1.0 in Eq. (2.12), this takes the form sm ˆ

K1 Enm

(2.12b)

This equation is Ludwik expression which is applicable for 100% dense metals. K1 and n are constants in Eq. (2.12). At room temperature the value of m is very low. However, it cannot be ignored when developing complex but accurate expression for the ¯ow of porous materials. Differentiating Eq. (2.12), the following equation yields:   dsm nA dR dR nR dR mR ˆ sm ‡ n ln Em ‡ ‡ m ln E_ m ‡ dEm R dEm dEm Em d_Em E_ m (2.13)

263

In Eq. (2.12), the superscript m in the term E_ mR m represents strain rate sensitivity parameter of a material and subscript m denotes the matrix of the P/M materials. Differentiating Eq. (2.13), the following equation is obtained:   d2 s m nA dR dR nR dR mR _ ‡ m ln E ˆ s ‡ n ln E ‡ ‡ m m m dE2m R dEm dEm E d_Em E_ m   2 2 nA dR nA d R n dR d2 R ‡ ‡ ‡ n ln E ‡ sm ÿ 2 m R dEm R dE2m Em dEm dE2m n dR nR m dR d2 R ÿ 2 ‡ ‡ m ln E_ m 2 Em dEm Em E_ m d_Em d_Em  m dR mR ‡ ÿ E_ m d_Em E_ 2m ‡

(2.14)

From Eqs. (2.11), (2.13) and (2.15), the following equation is obtained:  dYR nA dR dR nR dR mR ˆ Y0 ‡ n ln Em ‡ ‡ m ln E_ m ‡ dEm R dEm dEm Em d_Em E_ m   nA dR dR nR dR mR 2 ‡ Em ‡n ln Em ‡ ‡ m ln E_ m ‡ R dEm dEm Em d_Em E_ m (   nA dR 2 nA d2 R n dR d2 R ‡ Em ÿ 2 ‡ ‡ ‡ n ln E m R dEm R dE2m Em dEm dE2m  n dR nR m dR d2 R m dR mR ÿ 2 ‡ ‡ m ln E_ m 2 ‡ ÿ 2 ‡ Em dEm Em E_ m d_Em d_Em E_ m d_Em E_ m   nA dR dR nR dR mR 2 _ ‡ n ln Em ‡ ‡ m ln Em ‡ ÿ Em R dEm dEm Em d_Em E_ m    dR mR d_Em nA dR dR ‡ m ln E_ m ‡ ‡ E_ m ‡ n ln Em d_Em E_ m dEm R dEm dEm   nR dR mR dR mR ‡ ‡ m…ln E_ m † ‡ ‡m…ln E_ m † Em d_Em E_ m d_Em E_ m   m d_Em dR d2 R m dR mR d_Em ‡ m…ln E_ m † ‡ ÿ 2 ‡ E_ m E_ m dEm d_Em dEm d_Em E_ m dEm E_ m dEm   E_ m dR mR m…ln E_ m † ‡ …2R2 ÿ 1†1=2 ÿ sm d_Em E_ m 1 dR ‡ Ym R…2R2 ÿ 1†1=2 (2.15) 2 dEm

The relationship between R and Em is proposed as follows from [5]. R ˆ BECm E_ D m

(2.16)

where B, C and D are constants. These can be obtained from the experiments. Differentiating Eq. (2.16), the following equation is obtained:   dR D Cÿ1 C Dÿ1 d_Em ˆ B E_ m CEm ‡ Em D_Em (2.17) dEm dEm Differentiating Eq. (2.17), the following equation is obtained:

264

R. Narayanasamy, R. Ponalagusamy / Journal of Materials Processing Technology 100 (2000) 262±265

   d2 R D Cÿ2 Cÿ1 Dÿ1 d_Em ˆ B C E_ m …C ÿ 1†Em ‡ Em D_Em dE2m dEm  2  d_Em Dÿ1 d_Em ‡ EC…Dÿ1† E_ Cÿ2 ‡ D CECÿ1 m E_ m m m dEm dEm  2 d E_ m ‡ECm E_ Dÿ1 (2.18) m dE2m

   d2 R Cÿ1 Dÿ1 C Dÿ2 d_Em ˆ B D CEm E_ m ‡ Em …D ÿ 1†_Em d_Em dEm dEm  D dEm Dÿ1 d_Em dEm ‡ ECÿ1 ‡ C …C ÿ 1†ECÿ2 m E_ m m D_Em d_Em dEm d_Em  2 D d Em ‡ECÿ1 (2.21) m E_ m d_Em dEm

Differentiating Eq. (2.16) with respect to E_ m , the following equation is obtained:   dR C Dÿ1 Cÿ1 D dEm ˆ B Em D_Em ‡ CEm E_ m (2.19) d_Em d_Em

Substituting Eqs. (2.16)±(2.21) in Eq. (2.15), the expression for …1=Y0 †…dYR =dEm † can be obtained. From [7], the relationship between E_ m and Em is proposed as follows:

Differentiating Eq. (2.19) with respect to E_ m , the following equation is obtained:    d2 R C Dÿ2 Dÿ1 Cÿ1 dEm ˆ B D Em E_ m …D ÿ 1† ‡ E_ m CEm d_Em d_E2m  2  dEm D Cÿ1 dEm E_ ‡ D_EDÿ1 ‡ C …C ÿ 1†ECÿ2 m m Em d_Em m d_Em 2  D d Em ‡ECÿ1 (2.20) m E_ m d_E2m

Differentiating Eq. (2.22) with respect to Em, the following expression results:

Differentiating Eq. (2.19) with respect to Em, the following equation is obtained:

E_ m ˆ ln…1 ‡ Em †

dEm ˆ …1 ‡ Em † d_Em

(2.22)

(2.23)

Numerical integration of Eq. (2.15) after substituting Eqs. (2.16)±(2.21), and constants, is performed to compute the value of the yield stress YR for the P/M porous material. It is observed from Fig. 1 that the ratio of YR/Y0 is found to be increasing with increasing amount of deformation or densi®cation. The value of YR/Y0 ratio reaches unity when the matrix strain (Em) value reaches 0.60. This is in close proximity with experimental results. When the percent strain

Fig. 1. Relationship between stress ratio and Em.

R. Narayanasamy, R. Ponalagusamy / Journal of Materials Processing Technology 100 (2000) 262±265

value reaches 30±40, the 100% density is found to be developed, according to [7], for the same Al-alloy. 3. Plasticity theory for plastic instability Eq. (2.13) can be rearranged as follows:   1 dsm nA dR dR nR dR mR ˆ ‡ n ln Em ‡ ‡ m ln E_ m ‡ sm dEm R dEm dEm Em d_Em E_ m (3.1) When the material approaches 100% density, the value of dR/dEm tends to zero. Therefore, Eq. (3.1) can be written as follows: 1 dsm nR mR dR ˆ ‡ ‡ m ln E_ m sm dEm Em d_Em E_ m

(3.2)

When the density is 100% of theoretical value and dR=d_Em is zero, Eq. (3.2) becomes   1 dsm nR mR ˆ ‡ (3.3) sm dEm Em E_ m At room temperature the value of m is very low. When the deformation is carried out at high strain rate (say E_ m ˆ 1:0), Eq. (3.3) yields: 1 dsm nR ˆ sm dEm Em

(3.4)

If …1=sm †…dsm =dEm † is unity, Eq. (3.4) can be written as follows: nˆ

Em R

(3.5)

From Eq. (3.5), it is noticed that the value of n increases as R value decreases. However, the value of m and its effect on plastic instability cannot be ignored. 4. Conclusion The modi®ed plasticity theory for the determination of yield stress in the upsetting-compression test for the porous materials such as sintered P/M materials whose density

265

changes during plastic deformation is developed and proposed in this paper. The theoretical value of yield stress is in close proximity with experimental results. Further, a theory for work hardening rate and plastic instability for porous P/ M materials has been developed. 5. Nomenclature R YR Ym sm Em Y0 K1 n m A E_ m B C D

relative density of porous P/M material yield stress of partially dense material yield stress of matrix of partially dense material equivalent flow stress of matrix equivalent strain of matrix yield stress of 100% dense material strength coefficient of material constant strain hardening index value strain rate sensitivity value power law factor for strength coefficient equivalent strain rate of matrix density constant density power law constant due to strain density power law constant due to strain rate

References [1] R.G. Green, A plasticity theory for porous solids, Int. J. Mech. Sci. 14 (1972) 215. [2] M. Oyane, S. Shima, Y. Kono, Theory of plasticity for porous materials, Bull. JSME 16 (1973) 1254. [3] H.A. Kuhn, C.L. Downey, Deformation characteristics and plasticity theory of sintered powder materials, Int. J. Powder Metall. 7 (1971) 15. [4] S.M. Doraivelu et al., A new yield function for compressible P/M materials, Int. J. Mech. Sci. 26 (1984) 527. [5] R. Narayanasamy, R. Ponalagusamy, A mathematical theory of plasticity for compressible P/M materials Ð Parts I and II, J. Mater. Process. Technol., in press. [6] R. Narayanasamy, Plastic deformation behaviour of compressible solids, Unpublished Report, Regional Engineering College, Tiruchirappalli 620 015, India, 1998. [7] R. Narayanasamy, K.S. Pandey, Work hardening behaviour of sintered aluminium±alumina composite preforms during cold upset forming, Unpublished Report, Regional Engineering College, Tiruchirappalli 620 015, India.