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Elastoplastic finite element analysis for porous metals H. N. Han, H. S. Kim, K. H. Oh, and D. N. Lee
The elastoplastic fìnite element method for the deformation of porous metals has been newly formulated using the yield condition advanced by Lee and Kim. Changes in geometries and densities of porous metals, and upsetting loads with upsetting strain have been calculated. The Brinell hardnesses of porous metals with various densities dependent op indenting geometries have been analysed. The calculated results were in very good agreement with the measured data. PMj0632
(Q 1994 The Institute of Materials. Manuscript received 7 December 1993; in fìnal form 14 April 1994. Mr Han, Professor Oh, and Professor Lee are in the Department of Metallurgical Engineering, SeouI National University, Korea and Dr Kim is at the Institute for Advanced Engineering, SeouI, Korea.
Sintered products are often subjected to sizing or forged to fìnal products. Therefore, understanding the deformation behaviour of porous metals during forming is very important in achieving good quality powder metallurgical parts. Deformation behaviour of porous metals has been analysed by many investìgators using slip line fìeld theory ,1 limit analysis ,2 and rigid- plastic fìnite element analysis.3,4 The elastic- plastic deformation behaviour of porous metals is influenced by the internal pores. Analysis of deformation behaviour of porous metals requires an appropriate yield criterion which should take the pore effect into account. Lee and Kim have reviewed5 yield criteria for porous metals and found that most of them6- 10 unreasonably suggest zero yield stress only at zero relative density.
Lee and Kim5 modifìed a yield equation suggested by Doraivelu et al.6 so that it could incorporate one parameter which could be estimated from the yield stress versus inìtìal density data. The modifìed yìeld condition for porous metals may be expressed as
(2 + R2 )J2 + (1 - R2 )Jîl3 = 1'/ Yõ ..... (1)
where
끼 = YVYÕ = [(R - Rc)j(1 - Rc)]2
J2 = à[(Ull - U22? + (U22 - U33? + (U33 - Ull)2J + UI2
+ U~3 + σ31 J 1 = Uu + U22 + U33
In the above equations, Jí and J 1 are the quadratic stress deviator invariant and the linear stress invariant, respectively. YR ìs the yield stress of porous metal with relative density R, and YO is the yield stress at R = 1. For work hardening metals, YR and YO are the flow stresses of the porous and non-porous metals. The parameter Rc is an experimental parameter and may be interpreted as a critical relatìve densìty in whìch the yield stress of porous metal becomes zero: that is, 갈 = 0 at R = Rc. For non-porous metals, R = 1 and 때 =Yo, equatìon (1) becomes tbe Von Mises yield criterion, 3J2 = YÕ. Equation (1) could
140 Powder Metallurgy 1994 Vol. 37 NO.2
successfully simulate the densification process of porous iron specimens with various initial porosities under hydrostatic pressure. ll The purpose of the present work is to examine how the yield equation, equation (1), predicts various plastic deformations of porous metals and to formula te the elastoplastic fìnite element analysis of the deformation of porous metals using trus equation.
ELASTOPLASTIC FINITE ELEMENT FORMULATION For the yield condition of equation (1), under the assumptions of isothermal conditions and isotropic hardening, the yield function F at time t can be expressed
tp = (2 + tR2 rJí j3 + (1 - tR2 )tJU9-껴 t Yõ j3. . (2)
where t %, tR and tη are state variables dependent on the plastic strains t쩍 . The yield function t F is used to calculate the plastic strain increments as follows
a , 6tF tqTCEd§ 6tF de~ = t), '1 t = t T t .... t T;:' P t '1 t .... ( 3 ) ðtUij t 12. Tt q + t q T CEt q ðtuij
I /"'E tqT CECqT CE)T d tZ = CE(d[i - d양) = CEP d[i = I CE - t ~LT ’ ’ T ~F/t I dli
’ 112. 11q + ‘ql Cb1q
. (4)
where t },_ is a positive scalar, CE is an elastic stress- strain matrix, CEP is the relative density dependent elastoplastic stress- strain matrix, and trl and tqT are transposes of tl2. and tq, which are defìned by equations (5-7).
lE = [tPl1tP22tP33tPl2tP23tP3l]
tq = [tq1ltq22tq332tq122tq232‘q31]
tp니 = - ðtPjð댐 and tqij = 8lF/ 8t6i]
The elastic stress-strain matrix CE is given by
CE = E(I - v) -(1 + v)(I- 2v)
v v
1 - v 1 - v v
l - v
× symmetnc
0
O
O
1 - 2v
2(1 - v)
0
O
O
0
1 - 2v
2( 1 - v)
0
Q
Q
0
0
(5)
(6)
(7)
1 - 2v
2(1 - v)
. (8)
Equations (3- 8) are also valid ìn the case of non-porous materials.12 The relationship between the apparent plastic deformation energy increment per unit volume of porous metal dtW and that of the non-porous base metal d tWo is
-
。
• ....... 1.0
{Tengion) ) CXXXP CCuu Compression -。
0.8 ~
• ‘ 1
0
•
F 。
。
N 。0 6
Q)
>
형/: >
Rn=O.869
\、 F
.--•-
----- 。--0 745 /ι •
““‘ Ro--o -620
-:'. φ
••••• Ro=O.
0.4
a:::: 。α. 0.2 E-
O. 'ò.
-
Relative Density
1 Square of relative yield stresses of copper specimens as function of relative density:9 for curve ,fitting values of Rç = 0.442 and YO = 54 MN m - 2 were used
"--
given by
dtW = tR dtWo or 꾀 d댐 =tRtYod혀 . . . (9)
where dëo is the effective plastic strain increment of nonporous base metal. Since the yield stress is a function of the plastic work per unit volume, we can evaluate tPij and tqij using the following equations
t_ 2 tη I EET l t =:;t~1 ~ ~ I 닝 . . . . . . ( 10)
1) 3 t R I E _ ET I V 1)
l 2 + tR2 2 ( qlj = - 1 - t허 + a (1- tF) tι ðij
、‘ ---/ 4 l i
’ -/
t ··, 、
where E, ET' t허， and ðij are the elastic modulus and tangential modulus of non-porous base metal, deviatoric stresses, and the Kronecker delta respectively. For nonporous metals, t R = 1 and 써 = 1, and equations (10) and (11) become those of the Von Mises yield condition. The elastoplastic stress- strain matrix CEP and the positive scalar t). can be formulated using the equations (3), (4), (8), (10), and (11)
“
UPDATING RELATIVE DENSITY The constancy of mass during deformation of porous metals gives the following relationship
R=RoVo/ V . . . . . . . . . . (12)
where Ro and R are the relative density before and after deformation, and Vo and V are the volume before and after deformation. The volume of an axisymmetric element is given by
‘
V= 2πx dA (13) A
where x is the distance between the centroid of the element and the axisymmetric line, and A is the area of the element. The first step to calculate volume V is to relate the actual global coordinates x and y to a natural coordinate system (-1 ~ ç ~ 1, - 1 ~ η ~ 1).
x = x(ç, η)， y = y(ζ 1]) •... . . . (14)
dx and dy are given by
ðx ðx
dx
dy
“야 떠
「|
’’」
”q
”y-
”7
jU
(꺼U -(AU
%
이-%
• (15)
H an. et al. Elastoplastic fìnite element analysis for porous metals 141
‘。명 .0 0 .4 0.8
Height Strain (ε)
2 Relative density as function of heigbt strain in frictionless upsetting of sintered copper specimens: Shima’s experimental results9 (symbols) are compared with values calculated by finite element metbod (stars) and values of equation (20) witb n = 2 (lines)
1.2
The integration of a scalar function f(ζ 1]) in the natural coordinate can be obtained by applying the integration formula successively, namely
f(ç, 1]) dç dη = L L WiWJ(Çi, 끼j) (16) - 1 J-1 ) 1
where Wi and Wj are the weight fac tors, and m and n are the order of in tegration. For a function f(x , y) = 2πx defined over an isoparametric element, integration can be evaluated as
n 7
d 값
-r ! u 써
V
γ r 」-
7 | 、
/ j
’ l i
n l !
「·l i
--V
= L L WiWJ(Çi’ 1]j) IJ(çj, 깨j) 1 (17) J 1
where I J(ζ， ηj) 1 is the determinant of a Jacobian matrix and is given by
ðx ðy ðx ðy IJ(Çi, ηJ I = ~ ~-' - -;- ~-'~ ...... ( 18)
J " 8ç ð1] 81] 8ç
The updated relative density R can be obtained using equations (12), (17), and (18).
APPLICATIONS Simple upsetting Shima and Oyane9 performed frictionless upsetting of 20 mm diameter, 25 mm high sintered cylindrical copper specimens with various relative densities: Fig. 1 shows the variation of 야 with R for the specimens. The mechanical properties of the non-porous copper specimens (v=O'3 , E=117GNm- 2, 0'0=54MNm - 2) could be represented by9
Oeq = 946(6eq + 0·OO3 )05833 for O < 6eq < O·034
O'eq = 482(8eq - 0'0105)0'3333 for εeq > 0'034, where O'eq is the equivalent stress in MN m - 2, 8eq is the equivalent strain, v is Poisson’s ratio, and E is Young’s modulus. The best fitting of the data in Fig. 1 to the YR- R relationship in equation (1) gives Rç = 0-442.
Figure 2 shows Shima’s measured relative densities of sintered copper specimens with different initial relative densities as a function of height strain in frictionless upsetting, along with those obtained by analytical and
Powder Metallurgy 1994 Vo l. 37 NO. 2
a calculated grid distortion; b calculated relative density; c calculated hydrostatic pressure distribution (MPa); d measured density distribution4
5 Calculated (assuming a friction coefficient of 0'25) and measured properties of 20% reduced siotered copper
implying that the numerical method is very accurate. T he good correlation between the measured and calculated results indicates that the finite element method based on equation (1) is reliable.
Figure 4 shows the radial strain of the porous copper specimens as a function of height strain during frictionless upsetting. The values calculated by the fìni te element method are in very good agreement with the data of Shima and Oya ne.9
Mori et al.4 performed upsetting experiments on sintered cylindrical copper specimens under sticking and Coulomb frictions. The initial height and diameter of the specimens were both 20 mm. The mechanical properties (ûeq in
142 Han er al. Elastoplastìc fì nite element analysis for porous melals
10mm 」 i
，..、
‘UOUOUUUU U C l
12.5mm
(a )
c c
3 Initial mesh for finite element analysis of frictionless upsetting of siotered copper specimens
numerical methods. Before discussing the results, the analytical method is first explained. Assuming homogeneous and rigid plas tic deformation in sîmple upsetting and the following relationshîp (equatîon (19)) between P oisson’s ratio v and relative density R, suggested by Zhdanovich,13
Kim and Ch014 derived a relationship between rela tive densities before and after deforrnation (equation (20))
(b )
v = O'SRn (19) • •
l /n p (ne) R = I
exp(ne) + (1 - Rô)/Rô
where ε is the height strain, defined as - ln(hjho), with ho and h being the height of the specimen before and after deformation respectîvely. Kuhn15 obtained n = 2 in equation (19) from specimens on porous metals. 1n fact, the relationshîp \1 = 0'SR2 was also used to deri ve the yield condition of equation (1). The numerical method used is the elastoplastic fìnite element method described above. Figure 3 shows the ini tial mesh for the finite element calculation. The numerical method and equatîon (2이 with n = 2 are seen to give almost identîcal results,
(20)
(c )
85:
87!
9o:
ω
. 。
----
l-- ---- -- ---- -- - - --
* . 。
I ·0
」·•
이
••••• Ro=O.869
““‘ Ro= O.745 ••••• Ro=O.620
(d )
- speclmens . 。
0
·깅 。ι
I O 。0 .4 0.8
Height Strain (ê)
4 Relationship between radial and height strains in frictiooless upsetting of sintered copper specimens with various initial relative densities: Shima’s experimental results9 (symbols) are compared with calculated \'alues ( lines)
1.2
Powder Metallurgy 1994 Vol. 37 NO. 2
A=0.860E+00
8 = 0 .872E+00
C=0.884E+00
0=0.896E+00
E=0.908E+00
F=0.920E+00
A=O.700E+02
8=0.920E+02
C=0.114E+03
D=0.136E +03
E=0.158E+03
F=0.180E+03
87 85
'-
'-.....
‘
(a)
A=0.700E +02
8=0 .920E+02
C=0.114E+03
0=0.136E+03
E=0.158E+03
F=0.180E+03
(c)
Han et al. Elastoplastic finite element analysis for porous metals 143
으
•
87i .κ‘ •
•
•
•
90} -“
•
•
•
( b)
87
/ /
/
A=0.860E+00
8=0.872E+00
C=0.884E+00
0=0.896E+00
E=0.908E+00
F=0.920E+00
"\85
182
. _._._.~ . _ . _._.-‘_._._.-‘._._‘-
(d)
a calculated grid distortion; b calculated relative density; c calculated hydrostatic pressure distribution (MPa); d measured density distribution
6 Calculated and measured properties of 20% reduced sintered copper specimens under sticking friction
MN m - 2) of the non-porous copper specimens (v = 0'3, E = 117 GN m- 2, (Jo = 94 MN m - 2
) were as follows 4
(Jeq = 431(εeq + 0.00851 )0'32
Since Rc of this system is not known, Rc for the previous example, Rc = 0 , 442, was used in the calculation.
Calculations have been performed for various friction conditions on material with an initial relative density of 0.8 1. Figure 5a- d shows the calculated grid distortion, relative density, hydrostatic pressure, and the measured relative density (based on hardness) distributions4 in a specimen reduced by 20% assuming a friction coefficient of 0.25. Figure 6a- d shows the results for sticking friction. lt can be seen that the calculated relative density distributions are comparable with the measured data. It is seen from Figs. 5 and 6 that for 20% reduction, the relative density is low half way down the side surface and the
F
centre of the diejmetal interface. The grid distortion and relative density distributions in 40 and 60% reduced specimens calculated assuming a friction coefficient of 0.25 are shown in Fig. 7. The highest density is observed at the centre of the workpiece and near the edge region of the diejmetal interface. It can be seen that fo lding occurs at the outer edge region of the 40% reduced specimen. Most regions reach relative densities above 0.99 and the density is lowest at a point half way down the side surface where the specimen becomes barrelled at a reduction of 60%. Figure 8 shows the measured load- displacement data4
along with values calculated for various friction coefficients. It can be seen that the forming loads are insensitive to friction and the calculated values are in good agreement with the measured data.
Figure 9 shows the average relative density of a sintered porous alloy steel with an initial relative density of 0.825
A=0.860E+00
8=0.882E+00
C=0.904E+00
0=0.926E+00
E=0.948E+00
F=0.970E+00
A=0.850E +00
8=0.878E +00
C=0.906E+00
0=0.934E +00
E=0.962E+00
F=0.990E+00
(a) (b)
a. grid distortion; b density distribution
7 Calculated grid distortion and density distributions in 40% reduced (top) and 60% reduced (bottom) sintered copper specimens assuming a friction coefficient of 0.25
Powder Metallurgy 1994 Vo l. 37 No. 2
144 H an et al. Elastoplastic fìnite element analysis for porolls metals
200
150
••••• Measured 웅융융용e μ=0.0
g • 11 • 11 μ=0.05 ••••• μ=0.25 •••• • Sticklng
Data, μ=0.25
z i
Tf1oo o 。_.
50
0 O. 0.2 0.4
Height Str미n (ç )
8 Upsetting loads as function of height strflin for various friction conditions: Mori’s experimental results4 (symbols) are compared with calculated values for given coefficient of friction
O.
as a function of height strain measured by Kim and ChO.14
They analysed the experimental data based on equation (20). The best fitting of the data yielded n = 2.37 rather than n = 2 in this equation which was derived assuming frictionless homogeneous dεformation. However, finite element analysis of the deformation assuming a friction coefficient of 0.1 gives a better result as shown in Fig. 9. The calculated effect of coefficient of friction on average relative density versus height strain is shown in Fig. 10. The average relative density increases slowly with increasing friction. lt follows from the results in Figs. 9 and 10 that it is not necessary to take values other than 2 as the η value.
Indenting Tabata et α1. 16 measured the BrineU hardnesses of sin tered copper specimens as a function of djD, where d and D are the respective diameters of the indented mark and the indenter, and the results are shown in Fig. 11. Hardness increased monotonically with increasing djD, except for
。
• 〉、’
•-.- m
.。
。
.。
이gm디 φ〉-←o
。」。
φ
>l{) <
c:6 .0
Equation (20), n=2.0 - - - Equ이ion (20), n=2.37
• • • • FEM, μ=0.0 • i •• FEM , μ=0.1 口口디口디 Measured Dat。
0.2 0.4 0.6 0.8 Heiaht Strain
(1':)
9 Average relative density of porous alloy steel specimens with initial relative density of 0.825 as function of height strain for various frictional conditions: Kim’s experimental results14 (symbols) are compared with values calculated assuming coefficient of friction of 0.1
1.0
Powder Metallllrgy 1994 Vol. 37 NO. 2
。
• 〉、..
-•-Equ이ion (20), n=2 .. … FEM, μ=0.0
• •••• FEM , μ=0.05 ““‘ FEM , μ=0.25 • !' ••••• FEM , Sticking • ~.~ . _.~ .‘ _ • .: . _.;: ----’ ‘ !
.-m
.。
。
.。
m「』@디 @〉:0-@ι。。
」
@ > 。
<
c:6. 0.2 0.4 0.6 Height Strain
(1':)
10 Calculated average relative densities of specimen of Fig. 9 as function of height strain for various friction conditions
0.8
specimens of higher initial densities. It is to be expected that the hardness increases with increasing initial relative density. There had, however, previousJy been no theoretical explanation of the hardness data. The authors of the present paper have made a finite element analysis of the two-dimensional axisymmetric frictio nless indentation of specimens 24.5 mm in diameter and 30 mm in height by a 5 mm diameter hemispherical iron indenter. The dimensions of the specimens and indenter are identical to those used in Tabata’s experiment. The mechanical properties (O'ea in MN m - 2) of the non-porous copper specimens (v = 0, 3, E = 117 GN m- 2, 0'0 = 54 MN m- 2
) and Rc show the following relationships16,17
Û eq = 5586획~7， Rc= 0.375
The initial mesh for indentation is shown in Fig. 12. The Brinell hardness (HB) is load F divided by indented surface area 18
HB = Fj{(πDj2)[D - (D2 - d2)]1/2} = FjπDt . . (21 )
where F, D, d, and t are applied load, diameter of indenter, diameter of indentation, and depth of indentation respectively. The calculated Brinell hardnesses as a function of djD are shown in Fig. 11 along with the Brinell hardnesses
70
ID 60 工
cñ 50
” @
동 40 」
。
:x: 30
••••• Ro =0.96 ••••• Ro =0.90 ““‘ Ro =0.85 디口口口口 Ro =0.80 。0000 Ro =0.75
-g 20 」
ID 10
0 0.0
11
0.4 0.6 d/O
Relationship between the Brinell hardness and diameter ratio dlD: symbols indicate experimental results16 and curves indicate calculated values
0.2 0.8 1.0
、~
‘ \__
‘
30mm
12.25mm
12 Initial mesh for indenting
measured by Tabata et α[.16 A good correlation between the measured and calculated values can be seen. The Brinell hardness is usually measured at constant F and D. For an indenting load of 2450 N, the usual Brinell hardness value is shown in Fig. 13 as a function of initial relative density. A linear relationshìp between the ìnìtìal relatìve densìty and the hardness is seen within the experimental range of Tabata et al.
Calculated grid distortions and relatíve density distributìons in the specimens with initial relative densities of 0.8 and 0.9 during indentation are shown in Fig. 14. The shape of the densificatìon zone under the ìndenter is similar to that of the plas tic zone of non-porous metal indented by a spherical indenter.18 1t is interesting to note that the region next to the indenter sinks rather than bulges.
CONCLUSIONS The calculated radial strain versus height strain relationships and relative densities of porous metals subjected to frictionless upsetting are in good agreement with the experimental values. The calculated density distributions in porous specimens upset under various friction conditions are comparable with the measured distributions. The average relative density in upset porous metals increases slowly with increasing friction. The Brinell hardness calculated for a given load and indenter diameter increases linearly with the initial relative density of the specimen and
50
0140 工
m 없 30 I
1
‘O :1: 20 --@ I
꼽 10
0 O. O. O.
Initial Relative Density
13 Relationship between 8rinell hardness (load = 2450 N ) and initial relative density: symbols indicate experimental results16 and trace indicates calculated values
Han et al. Elastoplastic fìnite element analysis for porous metals 145
(a)
( b)
a Ro= 0'8; b Ro= O'9
A=0.820E+OO 8=0.844E+00 C-O.8SBE+OO D=0.892E +00 E톨0.916E+00
F"-0.940E+OO
A-0.920E+OO 8=0.932E+00 C-0.944E+00 D=0.95SE+00 E=0.968E+00 F"-0.980E+00
14 Calculated grid distortions (above) and relative density distributions (below) in specimens with initial relative densities of 0.8 and 0'9, indented to dl D = 0.63
correlates well with the measured data, although the measured hardness tends to increase with increasing ratio of indentation to indenter diameter.
ACKNOWLEDGEMENT This work has been supported by the Korea Science and Engineering Foundation through the Researcb Center for Thin F ilm Fabrication and Crystal Growing of Advanced Materials, Seoul National University, Seoul, Korea.
REFERENCES 1. M. OYANE and T. TABAT: J. ‘φn Soc. Technol. Plast., 1974, 15, 43. 2. H.-K. OH and J.-K. LEE: J. Mech. Work. Technol. , 1985, 11, 53. 3. Y. T. 1M and s. KOBAYASHI: in ‘Finite element analysis of plastic
deformation of porous materials’, (ed. S. R. Reid), 103; 1985, Oxford, Pergamon Press.
4. K. MORI, S. SHIMA, and K. OSAKADA: Bull. Jpn Soc. Mech. Eng., 1980, 23, 516.
5. D. N. LEE and H. S. KIM: Powder Meta.ll., 1992, 35, 275. 6. s. M. DORAIVELU, H. L. GEGEL, J. S. GUNASEKERA, J. C. MALAS,
J. T. MORGAN, and J. F. THOMAS: 1nt. J. Mech. Sci., 1984, 26, 527. 7. A. L. GURSON: J. Eng. Mater. Techno l. (Trαns. ASME) , 1977,
99, 2. 8. V. TVERGAARD: Int. J. Fract., 1982, 18, 237. 9. S. SHIMA and M. OYANE: 1m. J. Mech. Sci., 1976, 18, 285.
10. s. B. BlNER and w. A. SPITZIG: Acta Metall. Mater., 1990, 38, 603.
Powder Metallurgy 1994 Vol. 37 No. 2
146 Hαn et al. Elastoplastic fìnite element analysis for porous metals
11. H. N. HAN, H. S. KIM, and D. N. LEE: Scr. Metall. Mater., 1993, 29, 1211.
12. K.-J. BATHE: ‘Finite elemeot procedures in eogineering analysis’, 388; 1982, Englewood Cliffs, NJ, Prentice-Hall.
13. G. M. ZHDANOVICH: ‘Theory of compaction of metal powders’, 1; 1969, OH, Foreign Technology Division, Wright-Patterson AFB.
14. K. T. KIM and Y. H. CHO: Powder Metall. , 1993, 36, 129.
15. H. A. KUHN: in ‘Powder metallurgy processing', (ed. H. A. Kuhn and A. Lawley), 118; 1978, New York, Academic Press.
16. T. TABATA, S. MASAKI, and T. YABE: Trans. Jpn Soc. Mech. Eng., 1987, 53, 2029 (in Japanese) .
17. T. TABATA and s. MAS찌: J. Eng. Mater. Technol. ( Trans. ASME) , 1990, 112, 95.
18. G. E. DlliTER: ‘Mechanical metallurgy’, 3rd edn, 326; 1986, New York, McGraw-Hil l.
First Announcement
SUPERPLASTICITY - 60 YEARS AFTER PEARSON
7- 8 December 1994 Manchester
Organ;섭ed by the Sup뼈l없tic Foηning Committee of the Manufacturing Technology Division of The Institute of Materials
The meeting will be held to commemorate the 60th anniversary of the publication by the former Institute of Metals of the classic paper by C. E. Pearson entitled ‘The viscous properties of extruded eutectic alloys ’. Not only did Pearson observe very high tensile strains in metallic materials but he also understood important microstructural aspects of the phenomenon, which subsequently became known as superplasticity. He was also the first to demonstrate the feasibility of superplastic biaxial bulge forming using gas pressure, a procedure which is the basis of the forming techniques employed commercially today.
Scope The conference wiU review the current understanding of superplastic behaviour in metals and other crystalline solids, but in particular will be concerned with the current status of, and future trends in, superplastic forming of commercially useful shapes. The markets in which superplastically formed components are finding applications will be reviewed as will the progress in forming machinery, associated equipment, and manufacturing techniques.
The conference is intended to include both invited and contributed papers dealing with mechanisms of superplastic flow , cavitation, diffusion bondingl superplastic forming, superplasticity in ceramics and intermetallics, and high strain rate superplasticity. In addition contributed papers are intended to deal with Al alloys, Ti alloys, stainless steels, and commercial aspects of diffusion bonding and/or superplastic forming.
Further information For further information please contact:
Conference Department (C441 ) The Institute of Materials 1 CarIton H ouse Terrace London S\VIY 5DB Tel: 071-8394071 (direct line: 071-235 1391); Fax: 071-823 1638.
Powder Metallurgy 1994 Vo l. 37 No. 2
-
146 Han et al. Elastoplastic finite element analysis [0 1' porous metals
11. H. N. fV\N, H. S. KlM, and D. N. LEE: Scr. Metall. Mαter .， 1993, 29, 1211.
12. K.-J . ßATHE: ‘Finite element procedures in en밍neering analysis', 388; 1982, Englewood Cliffs, NJ, Prentice-Hall.
13. G. M. ZHDANOVICH: ‘Theory of compaction of metal powders’, 1; 1969, OH, Foreign Technology Division, Wright-Patterson AFB
14. K. T. KlM and Y. H. CHO: Powder Metall., 1993, 36, 129.
15. H. A. KUHN: in ‘Powder metallurgy processing’, (ed. H. A. Kuhn and A. Lawley), 118; 1978, New York, Academic Press.
16. T . TABATA, S. MASAKI, and T. YABE: Trans. Jpn Soc. Mech. Eng., 1987, 53, 2029 (in Japanese).
17. T . TABATA and s. MASAKl: J. Eng. Mater. Techno l. ( Trans ASME) , 1990, 112, 95.
18. G. E. Dæ TER: ‘Mechanical metallurgy’, 3rd edn, 326; 1986, New York, McGraw-Hill.
First Announcement
SUPERPLASTICITY - 60 YEARS AFTER PEARSON
7 - 8 D ecember 1994 Manchester
Organ:상ed by the Sup뼈l없tic Foπning Committee of the Mam따âcturing Technology Division of 1he Institute of A1a뼈iaμ
The meeting will be held to commemorate the 60th anniversary of the publication by the former Institute of Metals of the classic paper by C. E. Pearson entitled ‘The viscous properties of extruded eutectic alloys’. Not only did Pearson observe very high tensile strains in metallic materials but he also understood important microstructural aspects of the phenomenon, which subsequently became known as superplasticity. He was also the first to demonstrate the feasibility of superplastic biaxial bulge forming using gas pressure, a procedure which is the basis ofthe forming techniques employed commercially today.
Scope The conference wiU review the current understanding of superplastic behaviour in metals and other crystalline solids, but in particular wiJl be concerned with the current status of, and future trends in, superplastic forming of commercially useful shapes. The markets in which superplastically formed components are finding applications will be reviewed as will the progress in forming machinery, associated equipment, and manufacturing techniques.
The conference is intended to include both invited and contributed papers dealing with mechanisms of superplastic flow, cavitation , diffusion bondingl superplastic forming, superplasticity in ceramics and intermetallics, and high strain rate superplasticity. In addition contributed papers are intended to deal with Al alloys, Ti alloys, stainless steels, and commercial aspects of diffusion bonding and/or superplastic forming.
Further information For further information please contact:
Conference Department (C441 ) The Institute of Materials 1 Carlton House Terrace London SWIY 5DB Tel: 071-8394071 (direct line: 071-235 1391 ); Fax: 071-823 1638.
Powder Metallurgy 1994 Vol. 37 NO.2
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