Download Semiviscosity flow equation with variable parameters under

SCIENCE IN CHINA (Series E)
VO~.41 NO. 3
June 1998
Semiviscosity flow equation with variable parameters
under superplastic tension conditions *
SONG Yuquan
(Xi%&),
GAO Bai'en
(&a,!%)
and WANG Xiwen (E43C)
(Superplasticity and Plasticity Institute, Jilin University of Technology, Changchun 130025, China)
Received October 8, 1997
Abstract
The accurate multinomial expression of the relation of lga-lgi of the three typical superplastic alloys is
given. Then by means of the analytic expression of the constitutive parameters m ( i ) and k (i ) , the semiviscosity
flow equation in which m and k are both variable can be obtained. And the equation includes the three parameters
m,, mk and 7, which are related to the curve of m-lgi Finally, it is pointed out that the bigger the values of m, I )
and m , / m k , the better the superplasticity of alloys. Furthermore, the Backofen's equation is only one of the special
cases of semiviscosity flow equations with variable parameters.
.
Keywords:
superplastic tension, semiviscosity flow equation.
As early as in 1964, Backofen et al. believed that under superplastic conditions hardly any
strain-hardening takes place during the deformation of material but there is a strong strain-ratehardeningr" . And they also established the semiviscosity equation
d = kErn,
(1)
which indicates the relation between real stress a and strain rate i , where m is the strain rate
sensitivity index, k, the material parameter, and additionally, m and k are both regarded as
constants. This equation has played a basic part in the history of superplasticity mechanics research. But except ceramics and intermetallic compound, most materials' m and k will have great
change with the variation of E , so it is inaccurate to regard m and k as constants. In this paper,
we attempt to establish the semiviscosity flow equation with m and k being variable by means of
the analytic expression of rn ( ;) and k ( ;) and to describe the materials' superplasticity with the
geometric shape of the curve of rn .
1 Accurate expression of semiviscosity equation
As long as the experimental curve of lgo-lg&of material is drawn, the simulative multinomial
expression of this curve can be accurately worked out with a computer. Let z = lg;, that is
11
lga(E) = C a p i .
(2)
i=O
The respective coefficients of eq. (1) for experimental curves of the three materials in fig. 1 are
as follows:
For superplastic alloy Zn-A122 % :
+ Project supported in part by the National Natural Science Foundation of China.
No. 3
SEMIVISCOSITY FLOW EQUATION WITH VARIABLE PARAMETERS
243
Fig. 1. The experimental curve and the function simulated curve of lgo-lgi. (a) Superplastic alloy Zn-Al22%, experi-
mental temperature being 250C
;
(b) superplastic alloy Al-Zn-Mg, experimental temperature being 550C ['I ; (c) su-
perplastic alloy Al-Zn-Mg-Cu-Cr, experimental temperature being 520°C 14]. 1, Experimental curve; 2, cosine function
simulated symmetrical curve; 3, negative-power function simulated symmetrical curve; 4, cosine function step by step
simulated non-symmetrical curve; 5, negativepower function step by step simulated non-symmetrical curve.
a . = 4.221 910
x lo-*, as
=
1.893 360 x lo-'.
For superplastic alloy Al-Zn-Mg :
a. = 0.624 526, a l = - 2.364 580,
I
= - 6.283 880, a3 = - 6.586 200,
a4 = - 3.390 580, a 5 = - 0.640 606, a6 = 0.115 561, a7 = 7.329 930 x
a s = 1.189 960
a2
a9 = 6.582 760
X
X
(4)
lop4.
For superplastic alloy Al-Zn-Mg-Cu-Cr :
a . = 69.691 841, a l = 383.463 387, a2 = 930.656 207, a3 = 1286.555 192,
a4 = 1 123.039 777, a s = 647.704 110, a6 = 250.157 339, a , = 63.908 002,
(5)
as = 10.333 920, a9 = 0.954 312, ale = 3.815 326 x
As eq. ( 2 ) (also considering eqs. ( 3 )-( 5) at the same time) can accurately describe the
corresponding relations of a and E of the three alloys, it can help to solve the quantitative mechanical problem. But multinomial expression (2) does not include the constitutive parameters k
and m , so it is of little use to the understanding of the mechanical essence of superplastic tension.
2 Flow equation with variable parameters
2.1 Curve of m-lg& is symmetrical
The analytic expression of m ( E ) and k ( E ) is given by simulating the curve of m-lgh with
cosine function in reference [ 5 ] .
m(E) = mm2
+ mk
+
mm 2
- mkcos(lg~mglgEz) Y
(6)
SCIENCE IN CHINA (Series E)
244
Vol. 41
To simulate the curve with the negative-power function, the analysis expression of m (E ) and
k ( € ) is as follows:
m ( € )=
mm
lgd- l g ~ ,
Introducing eqs. ( 6 ) and (7) into eq. ( I ) , we obtain the superplastic semiviscosity flow
equation simulated with cosine function:
Introducing eqs. (8) and (9) into eq. ( I ) , we obtain the semiviscosity flow equation simulated with negative-power function:
2.2
Curve of m -lg; is non-symmetrical
As the curve of m-lg; is non-symmetrical, it must be simulated step by step. The flow equation simulated with cosine function step by step is as follows:
< E < Em),
(12)
(hm < i< i B ) .
(EA
I')
m -m
Igi -1gi
(*)10-[2(+~)~(
The flow equation simulated with negative-power function step by step is as follows:
The implication of parameters in eqs. (6)-(13)
If the curve of m-lg; is symmetrical, then
is indicated in figure 2 .
If the curve of m -lg€ is non-symmetrical, then
IgEm - IgEA, VB = igEB - lgE,.
For the above-mentioned three materials, if
the curve of m-lg€ is approximately regarded as
symmetrical, introducing m ,, m k, lg;,, lgk and 1
of eq. ( 1 3 ) from ref. [ 5 ) into eqs. (10) and
VA=
Fig. 2. Schematic diagram of the curve of rn-lg;.
No. 3
SEMIVISCOSITY FLOW EQUATION WITH VARIABLE PARAMETERS
245
( l l ) , then similarly the corresponding simulated curve can be plotted with a computer (see
curves 2 and 3 in figure 1) .
If the curve of m-lge is dealt with as the practical non-symmetrical case, introducing m,,
m ~ m, ~ ? l ~ a ~ IgEB,
, l ~ T]A
e ~and
,
IB
of eqs. ( 1 3 ) and ( 2 4 ) from ref. [ 6 ] into eqs. ( 1 2 ) and
( 1 3 ) , then the corresponding simulated curve can be plotted with a computer (see curves 4 and 5
in figure 1) .
Figure 1 shows that, in the range of superplastic deforming strain rate ( ( a ) E = 1 0 - ~ . ~ 10-0.3 , ( b ) ;= 10-3.18- 10-0.8 1 ( C ) ;= 10-3.01-10-1.39 ) , theoretical curves all coincide with
experimental curve 1 very well. But, as the curves of m-lgE of the above three materials are not
completely symmetrical, the non-symmetrical simulated theoretical curves coincide with the experimental curve better.
3
Discussion
1 ) From the above, we can see that for the known typical superplastic material, we only need
to determine the following through experiment:
{
(mm?l g ~ m )(, m u 1 IgEA), ( m k ~IgEB),
,
71.4
= IgE, - l g ; ~ , VB =
k,
= a,eGrnm.
I&
- lgiml
(14)
For the material with symmetrical curve of m , that is, V A = I B = 1, m u = m k =
~ mkl introducing eq. ( 14) into eq. ( 1 0 ) or ( 11) , we obtain the symmetrical semiviscosity constitutive
# VB, m u
equation with variable parameters. For the non-symmetrical curve of m , that is,
f m kB, introducing eq. ( 14) into eq. ( 10) or ( 11) , we can obtain the non-symmetrical semiviscosity constitutive equation with variable parameters.
For the newly developed superplastic material, the curve of m-lga should be determined
through experiment first, and only if the curve is arched, similarly to the above, the corresponding semiviscosity constitutive equation with variable parameters can be established.
2 ) In special cases, supposing m is a constant, namely, mk = m, = m , introducing it into
eq. ( l o ) , we have
a = k , ~ ~ .
(15)
Introducing m k = m, = m into eq. ( l l ) , we have
lim
Hence, o = k,Emm.
From the above, we know that Backofen's semiviscosity flow equation is only one of the very
special cases of semiviscosity with variable parameters, and in fact the parameter k in Backofen's
the average of m with the cosine function simequation is the parameter k, and its rn equals
ulated method and m , the maximum of m with the negative-power function simulated method.
In common cases, it only mirrors the sensitivity of the flow stress a to the strain rate ; or the re-
m,
lation between a and ;at any point. Only to those materials whose m and k are both constant,
such as the vast majority of ceramics and most of the superplastic intermetallic compound, can
Backofen's equation be applied.
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Vol. 41
3) In the superplasticity research field, it is commonly recognized that rn is the mechanical
index of judging the superplasticity of materials. It has been proved by a large quantity of facts
that for the same complete semiviscosity materials, the bigger the value of & or rn,, the better its
superplasticity. But for different superplastic materials, there are often materials found that alor rn, is bigger, their superplasticity is not better correspondingly, while the mathough their ;
or m, have better superplasticity. This uncommon phenomenon has long
terials with smaller ;
puzzled researchers during the development of superplasticity research. Although a further solution to this problem was given in ref. [ 5 ] by taking the geometric shape of the curve of rn-lg; into account, it is not well satisfactory. For the sake of simplicity in description, take the idealistic
case, namely, the curve of rn-lg; is symmetrical, as an example, and discuss it by considering
eqs. (10) and (11). The theory does not lose its generality. But it must be pointed out that during the superplastic deformation, the smaller the flow resistance a ( E. ) , the better its superplasticity . The superplasticity strain rates are very small and all in the range of 10-5-10- s- , namely, ;is always smaller than 1 . Considering the two conditions, from eqs. ( 10) and ( 11) , we can
see that, the bigger the values of rn , rn,/ m k, and 7, the better the superplasticity of the material. Thus, we cannot judge the superplasticity of the material only by the single average of m
or the maximum of rn but by the curve of m-lg€, namely, by the three parameters m,, mk and
7, which are related with the geometric shape of the curve. From this, we understand why we
usually get the uncommon puzzling answer when we use the value of rn to compare the superplasticity of dif'ferent materials.
' '
References
1 Backofen, W . A . , Turner, I . R . , Avery, D. H . , Superplasticity in an Al-Zn alloy, Trans. ASM Quart., 1964, 57:
980.
2
Takaai, T. , Katoh. K. , Nakayarna, Y. , Superplastic deformation behaviours of Zn-22% A1 of eutectoid, Light Metal. (in
Japanese), 1991, 41(3): 152.
3
Matuki,
448.
4
Kojima, Y . , Furukawa, T. , Kobayashi, M. et al. , Grain refining and superplasticity of Al-Zn-Mg-Cu-Cr alloys, Light
Metal. (in Japanese), 1986, 36(7) :. 421.
K.,Yamada,
M . , Superplastic beehavior of Al-Zn-Mg alloys, J . JapanJnst. Metals. (in Japanese), 1973, 37:
5
Song Yuquan, Cheng Yongchun, Wang Xiwen, A mechanical analysis of the constitutive parameters in the semiviscosity flow
equations under the superplastic tension conditions, Science in China (in Chinese), Ser. E, 1997, 27(5) : 385.
6 Song Yu-quan, Zhao Jun, A mechanical analysis of the superplastic free bulging of metal sheet, Mater. Sci. Eng. 1986,
84: 122.
.