- Email: [email protected]
M materials
Journal of Materials Processing Technology 86 (1999) 159 – 162
A mathematical theory of plasticity for compressible P/M materials R. Narayanasamy a,*, R. Ponalagusamy b a
b
Department of Production Engineering, Regional Engineering College, Tiruchirappalli 620 015, Tamil Nadu, India Department of Mathematics and Computer Applications, Regional Engineering College, Tiruchirappalli 620 015, Tamil Nadu, India Received 25 August 1997
Abstract A mathematical equation for the calculation of the yield stress in the case of the simple upsetting-compression test is proposed for P/M sintered preforms of materials. The new yield function developed by Doraivelu and his co-workers, taking into account the hydrostatic stress, is considered for the development of the above equation. Numerical integration is carried out in order to compute the yield stress, using the above-mentioned mathematical equation. The ratio of the yield stress of the P/M porous material to the yield stress of the fully dense material is found to increase with increase in densification due to deformation. Further, a theory for the calculation of stresses such as the hoop and hydrostatic has been developed using the plasticity equations. © 1999 Published by Elsevier Science S.A. All rights reserved. Keywords: Plasticity; P/M materials; Compression test
1. Introduction P/M fabrication has recently drawn much attention because this method is used widely for the manufacturing of various components in industry. In this process, mechanical parts from metal powders are produced by compaction and sintering operations. Moreover, the porous sintered parts can be deformed again to a final size within quite reasonable dimensional tolerance. Since the apparent density of sintered compacts changes during deformation due to the presence of internal pores, conventional plasticity theory cannot be applied directly to this process. Hence, it is necessary to establish a plasticity theory for porous materials. Many researchers [1–3] have proposed plasticity theory for porous materials. However, they are not successful for predicting the flow behaviour of partially dense materials. Thus, a special yield function which takes into account the effect of hydrostatic stress was developed for the plasticity analysis of compressible solids by * Corresponding author. [email protected]
Fax:
+91
431
552133;
e-mail:
Doraivelu and his co-workers [4], considering the distortion energy due to the total stress tensor. The present work describes the variation of the flow stress with increasing densification during the uniaxial compression test. Plasticity equations for the calculation of stresses are also developed.
2. Plasticity equations
2.1. Plastic flow The variation of the flow stress with densification during uniaxial compression testing (or the upsetting operation) resulting from both geometric and matrixstrain-hardening effects as given in [5], is as follows: YR = Ym(2R 2 − 1)1/2
(1)
Where: Ym = Yo(1+Ko¯m)
(2)
K= (1/s¯ m)(ds¯ m/do¯m)
(3)
and:
0924-0136/99/$ - see front matter © 1999 Published by Elsevier Science S.A. All rights reserved. PII S0924-0136(98)00305-7
160
o¯m =
R. Narayanasamy, R. Ponalagusamy / Journal of Materials Processing Technology 86 (1999) 159–162
&
o¯agg
[2−(1/R 2)]1/2 do¯agg
(4)
0
Putting R= 1 into Eq. (4) yields: (do¯m/do¯agg)=1.0
(5)
whilst putting R=0.71 into the above-mentioned Eq. (4) yields: (do¯m/do¯agg)=0.0
dR dYR dYm = (2R 2 − 1)1/2 +2YmR(2R 2 −1) do¯m do¯m do¯m
(7)
Differentiating Eq. (2) with reference to o¯m, yields the following equation:
d2s¯ dYm Yo ds¯ m = + o¯m 2m do¯m s¯ m do¯m do¯ m
n
(8)
From Eqs. (7) and (8), the following equation is obtained:
n
+2YmR(2R 2 −1) − 1/2
+ 2YmR(2R 2 − 1) − 1/2
dR do¯m
s¯ m =K(oo +o¯m)n
where n is the strain-hardening index value, K is the material constant or strength coefficient and oo is a constant. Differentiating Eq. (10), the following equation is obtained: ds¯ m = Kn[oo +o¯m]n − 1 do¯m
(11)
Differentiating Eq. (11), the following equation is obtained: d2s¯ m =Kn(n− 1)[oo + o¯m]n − 2 do¯ 2m
R= A[oo1 + o¯m]B
(13)
Similarly, Eq. (12) can be rearranged and written as follows:
(16)
dR = AB[oo1 + o¯m]B − 1 do¯m
(17)
From Eqs. (16) and (17), the following is obtained after rearrangement: R
dR = BA 2[oo1 + o¯m]2B − 1 do¯m
(18)
Substituting Eqs. (16)–(18) into Eq. (15), the following equation is obtained: dYR = Yo[n(oo + no¯m)/(oo + o¯m)2][2A 2(oo1 + o¯m)2B −1]1/2 do¯m
n
no¯m BA 2(oo1 + o¯m)2B − 1 (oo + o¯m)
[2A 2(oo1 + o¯m)2B − 1]
(19)
Numerical integration is carried out after substituting the values for the constants in order to compute the value of the yield stress YR for the porous material. It is observed from Fig. 1 that the ratio of YR /Yo is found to increase with increasing level of deformation or densification.
2.2. Calculation of stresses As given in [5], the relationship between the strain increment and the deviatoric stress component according to the Levy–Mises equation is as follows: do1 =
do¯ [2s1 − R 2(s2 + s3)] 2YR
do2 =
do¯ [2s2 − R 2(s3 + s1)] 2YR
do3 =
do¯ [2s3 − R 2(s1 + s2)] 2YR
(12)
Eq. (11) can be rearranged and written as follows:
(15)
where A, B and oo1 are constants that can be obtained by experiment. Differentiating Eq. (16), the following equation is obtained:
(9)
(10)
dR do¯m
The relationship between R and o¯m is as follows:
+ 2Yo 1+
However, the material law relating s¯ m and o¯m is written as follows:
1 ds¯ m =n/(oo + o¯m) s¯ m o¯m
dYR = Yo[n(oo + no¯m)/(oo + o¯m)2](2R 2 − 1)1/2 do¯m
d s¯ dYR Yo ds¯ m = +o¯m 2m (2R 2 +1) do¯m s¯ m do¯m do¯ m 2
(14)
Substituting Eqs. (13) and (14) into Eq. (9), the following equation is found:
(6)
This shows that at relative densities of below 0.71, the aggregate is geometrically unstable and crumbles during deformation. In the range of R between 0.71 and 1.0, the strain transferred to the matrix increases continuously until it asymptotically approaches the strain applied to the aggregate. Differentiating Eq. (1) with reference to o¯m, the following equation is obtained:
1 d2s¯ m = n(n− 1)/[oo + o¯m]2 s¯ m do¯ 2m
(20)
The ratio do1/do2 is obtained from Eq. (20). This is written as follows:
R. Narayanasamy, R. Ponalagusamy / Journal of Materials Processing Technology 86 (1999) 159–162
161
Fig. 1. Relationship between the stress ratio (YR /Yo) and the strain of the matrix (oagg), where YR is the yield stress of porous material and Yo is the yield stress of fully dense material (input data: aluminium compact; Yo =100 MPa; K= 135 MPa; n =0.26; oo =0.02; A= 0.97; B= 0.24; oo1 = 0.02).
do1 2s1 −R 2(s2 + s3) = do2 2s2 −R 2(s3 + s1)
(21)
Since s3 is negligible in Eq. (21) for the upsetting operation or compression test, this equation becomes: do1 2s1 −R 2s2 = do2 2s2 −R 2s1
sm = 1/3(su + sz )
2a − R 2 2 − R 2a
(23)
2a −R 2 =b 2− R 2a
(24)
or:
Since the ratio of strain increments do1/do2 (or) dou /doz (the ratio of the diametral strain increment to the axial strain increment or Poisson’s ratio) is less than 0.50 for the porous metals, Eq. (24) becomes: [2a −R 2]/[2 − R 2a]B 0.50
Plasticity equations for the calculation of the yield stress in the upsetting-compression test for porous materials such as sintered P/M materials, for which the density changes during plastic deformation, have been developed and proposed here. Further, the hoop stress and the hydrostatic stress can be calculated from the equations developed using the Levy–Mises rule. It is also found that the derived plasticity equations are adequate for describing the deformation behaviour of sintered porous metals. 4. Nomenclature
(26)
(b = 0.50 and R= 1.0, is the condition for fully dense material). Therefore, for porous P/M material the ratio of stresses su and sz is as follows: su s1 = B 0.80 sz s1
3. Conclusions
(25)
Eq. (25) can be rearranged and written as follows: su s 1 = = 0.80 sz s2
(28)
(22)
Let s1/s2 =a and do1/do2 =b. The Eq. (22) becomes: b=
From Eq. (27), the hoop stress can be calculated by substituting the value for sz for a given level of deformation. The equation for the hydrostatic stress is as follows:
(27)
R YR Ym s¯ m o¯m s1, s2 and s3
relative density of aggregate (or) porous P/M material yield stress of aggregate or partially dense material yield stress of the matrix equivalent flow stress of the matrix equivalent strain of the matrix principal stresses in directions 1, 2 and 3, respectively
162
R. Narayanasamy, R. Ponalagusamy / Journal of Materials Processing Technology 86 (1999) 159–162
do1, do2 and principal strain increments in direcdo3 tions 1, 2 and 3, respectively sr, su and sz principal stresses in the radial, hoop and axial direction, respectively Yo yield stress of fully dense material References [1] R.J. 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 Met. 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] V. Seetharaman, et al., Plastic Deformation Behaviour of Compressible Solids, unpublished report, Universal Energy Systems Inc., Dayton, OH, 45432.
.
.
A mathematical theory of plasticity for compressible P/M materials R. Narayanasamy a,*, R. Ponalagusamy b a
b
Department of Production Engineering, Regional Engineering College, Tiruchirappalli 620 015, Tamil Nadu, India Department of Mathematics and Computer Applications, Regional Engineering College, Tiruchirappalli 620 015, Tamil Nadu, India Received 25 August 1997
Abstract A mathematical equation for the calculation of the yield stress in the case of the simple upsetting-compression test is proposed for P/M sintered preforms of materials. The new yield function developed by Doraivelu and his co-workers, taking into account the hydrostatic stress, is considered for the development of the above equation. Numerical integration is carried out in order to compute the yield stress, using the above-mentioned mathematical equation. The ratio of the yield stress of the P/M porous material to the yield stress of the fully dense material is found to increase with increase in densification due to deformation. Further, a theory for the calculation of stresses such as the hoop and hydrostatic has been developed using the plasticity equations. © 1999 Published by Elsevier Science S.A. All rights reserved. Keywords: Plasticity; P/M materials; Compression test
1. Introduction P/M fabrication has recently drawn much attention because this method is used widely for the manufacturing of various components in industry. In this process, mechanical parts from metal powders are produced by compaction and sintering operations. Moreover, the porous sintered parts can be deformed again to a final size within quite reasonable dimensional tolerance. Since the apparent density of sintered compacts changes during deformation due to the presence of internal pores, conventional plasticity theory cannot be applied directly to this process. Hence, it is necessary to establish a plasticity theory for porous materials. Many researchers [1–3] have proposed plasticity theory for porous materials. However, they are not successful for predicting the flow behaviour of partially dense materials. Thus, a special yield function which takes into account the effect of hydrostatic stress was developed for the plasticity analysis of compressible solids by * Corresponding author. [email protected]
Fax:
+91
431
552133;
e-mail:
Doraivelu and his co-workers [4], considering the distortion energy due to the total stress tensor. The present work describes the variation of the flow stress with increasing densification during the uniaxial compression test. Plasticity equations for the calculation of stresses are also developed.
2. Plasticity equations
2.1. Plastic flow The variation of the flow stress with densification during uniaxial compression testing (or the upsetting operation) resulting from both geometric and matrixstrain-hardening effects as given in [5], is as follows: YR = Ym(2R 2 − 1)1/2
(1)
Where: Ym = Yo(1+Ko¯m)
(2)
K= (1/s¯ m)(ds¯ m/do¯m)
(3)
and:
0924-0136/99/$ - see front matter © 1999 Published by Elsevier Science S.A. All rights reserved. PII S0924-0136(98)00305-7
160
o¯m =
R. Narayanasamy, R. Ponalagusamy / Journal of Materials Processing Technology 86 (1999) 159–162
&
o¯agg
[2−(1/R 2)]1/2 do¯agg
(4)
0
Putting R= 1 into Eq. (4) yields: (do¯m/do¯agg)=1.0
(5)
whilst putting R=0.71 into the above-mentioned Eq. (4) yields: (do¯m/do¯agg)=0.0
dR dYR dYm = (2R 2 − 1)1/2 +2YmR(2R 2 −1) do¯m do¯m do¯m
(7)
Differentiating Eq. (2) with reference to o¯m, yields the following equation:
d2s¯ dYm Yo ds¯ m = + o¯m 2m do¯m s¯ m do¯m do¯ m
n
(8)
From Eqs. (7) and (8), the following equation is obtained:
n
+2YmR(2R 2 −1) − 1/2
+ 2YmR(2R 2 − 1) − 1/2
dR do¯m
s¯ m =K(oo +o¯m)n
where n is the strain-hardening index value, K is the material constant or strength coefficient and oo is a constant. Differentiating Eq. (10), the following equation is obtained: ds¯ m = Kn[oo +o¯m]n − 1 do¯m
(11)
Differentiating Eq. (11), the following equation is obtained: d2s¯ m =Kn(n− 1)[oo + o¯m]n − 2 do¯ 2m
R= A[oo1 + o¯m]B
(13)
Similarly, Eq. (12) can be rearranged and written as follows:
(16)
dR = AB[oo1 + o¯m]B − 1 do¯m
(17)
From Eqs. (16) and (17), the following is obtained after rearrangement: R
dR = BA 2[oo1 + o¯m]2B − 1 do¯m
(18)
Substituting Eqs. (16)–(18) into Eq. (15), the following equation is obtained: dYR = Yo[n(oo + no¯m)/(oo + o¯m)2][2A 2(oo1 + o¯m)2B −1]1/2 do¯m
n
no¯m BA 2(oo1 + o¯m)2B − 1 (oo + o¯m)
[2A 2(oo1 + o¯m)2B − 1]
(19)
Numerical integration is carried out after substituting the values for the constants in order to compute the value of the yield stress YR for the porous material. It is observed from Fig. 1 that the ratio of YR /Yo is found to increase with increasing level of deformation or densification.
2.2. Calculation of stresses As given in [5], the relationship between the strain increment and the deviatoric stress component according to the Levy–Mises equation is as follows: do1 =
do¯ [2s1 − R 2(s2 + s3)] 2YR
do2 =
do¯ [2s2 − R 2(s3 + s1)] 2YR
do3 =
do¯ [2s3 − R 2(s1 + s2)] 2YR
(12)
Eq. (11) can be rearranged and written as follows:
(15)
where A, B and oo1 are constants that can be obtained by experiment. Differentiating Eq. (16), the following equation is obtained:
(9)
(10)
dR do¯m
The relationship between R and o¯m is as follows:
+ 2Yo 1+
However, the material law relating s¯ m and o¯m is written as follows:
1 ds¯ m =n/(oo + o¯m) s¯ m o¯m
dYR = Yo[n(oo + no¯m)/(oo + o¯m)2](2R 2 − 1)1/2 do¯m
d s¯ dYR Yo ds¯ m = +o¯m 2m (2R 2 +1) do¯m s¯ m do¯m do¯ m 2
(14)
Substituting Eqs. (13) and (14) into Eq. (9), the following equation is found:
(6)
This shows that at relative densities of below 0.71, the aggregate is geometrically unstable and crumbles during deformation. In the range of R between 0.71 and 1.0, the strain transferred to the matrix increases continuously until it asymptotically approaches the strain applied to the aggregate. Differentiating Eq. (1) with reference to o¯m, the following equation is obtained:
1 d2s¯ m = n(n− 1)/[oo + o¯m]2 s¯ m do¯ 2m
(20)
The ratio do1/do2 is obtained from Eq. (20). This is written as follows:
R. Narayanasamy, R. Ponalagusamy / Journal of Materials Processing Technology 86 (1999) 159–162
161
Fig. 1. Relationship between the stress ratio (YR /Yo) and the strain of the matrix (oagg), where YR is the yield stress of porous material and Yo is the yield stress of fully dense material (input data: aluminium compact; Yo =100 MPa; K= 135 MPa; n =0.26; oo =0.02; A= 0.97; B= 0.24; oo1 = 0.02).
do1 2s1 −R 2(s2 + s3) = do2 2s2 −R 2(s3 + s1)
(21)
Since s3 is negligible in Eq. (21) for the upsetting operation or compression test, this equation becomes: do1 2s1 −R 2s2 = do2 2s2 −R 2s1
sm = 1/3(su + sz )
2a − R 2 2 − R 2a
(23)
2a −R 2 =b 2− R 2a
(24)
or:
Since the ratio of strain increments do1/do2 (or) dou /doz (the ratio of the diametral strain increment to the axial strain increment or Poisson’s ratio) is less than 0.50 for the porous metals, Eq. (24) becomes: [2a −R 2]/[2 − R 2a]B 0.50
Plasticity equations for the calculation of the yield stress in the upsetting-compression test for porous materials such as sintered P/M materials, for which the density changes during plastic deformation, have been developed and proposed here. Further, the hoop stress and the hydrostatic stress can be calculated from the equations developed using the Levy–Mises rule. It is also found that the derived plasticity equations are adequate for describing the deformation behaviour of sintered porous metals. 4. Nomenclature
(26)
(b = 0.50 and R= 1.0, is the condition for fully dense material). Therefore, for porous P/M material the ratio of stresses su and sz is as follows: su s1 = B 0.80 sz s1
3. Conclusions
(25)
Eq. (25) can be rearranged and written as follows: su s 1 = = 0.80 sz s2
(28)
(22)
Let s1/s2 =a and do1/do2 =b. The Eq. (22) becomes: b=
From Eq. (27), the hoop stress can be calculated by substituting the value for sz for a given level of deformation. The equation for the hydrostatic stress is as follows:
(27)
R YR Ym s¯ m o¯m s1, s2 and s3
relative density of aggregate (or) porous P/M material yield stress of aggregate or partially dense material yield stress of the matrix equivalent flow stress of the matrix equivalent strain of the matrix principal stresses in directions 1, 2 and 3, respectively
162
R. Narayanasamy, R. Ponalagusamy / Journal of Materials Processing Technology 86 (1999) 159–162
do1, do2 and principal strain increments in direcdo3 tions 1, 2 and 3, respectively sr, su and sz principal stresses in the radial, hoop and axial direction, respectively Yo yield stress of fully dense material References [1] R.J. 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 Met. 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] V. Seetharaman, et al., Plastic Deformation Behaviour of Compressible Solids, unpublished report, Universal Energy Systems Inc., Dayton, OH, 45432.
.
.