- Email: [email protected]
Development of constitutive equations of copper-silicon alloys
Journalof
Materials Processing Technology ELSEVIER
Journal of Materials ProcessingTechnology60 (1996) 621--627
Development of constitutive equations of copper-silicon alloys Z.J. Gronostajski
Technical University of Wroclaw, ul. Lukasiewicza 3/5, 50-I11 Wroctaw, Poland
Abstract
The paper shows the development of constitutive equations describing the flow stress of silicon bronzes. The fn'st model can be used only for description of characteristic values of flow stress, the further models make possible to follow the course of stress-strain curves. The second model bases on structural processes, the third one bases on the internal state of materials and the last one on the internal state of materials and structural processes.
Keywords: constitutive equation, silicon bronze, strain rate, temperature, strain rate history.
1. Introduction
During the last years a finite element analysis has been intensively developed for simulating 3-1) deformation of a materials whose elastic strain is negligibly small compared to the plastic strain. Such conditions exist in a large number of the press-forming operations. The procedures for the analysing of such problems are elaborated by Kobayashi et all [1,2], Wang [3], Ouate and Zienkiewicz [4], and Saran and Wagoner [5]. The analysis of the metal working operations needs accurate knowledge of the forming limit curves [6-8] and of the constitutive equations describing the flow behaviour of materials under different operating conditions. CAD of forming operations is no longer limited by the memory capacity of computers or data processing capabilities but by the imperfections of the available constitutive models that describe the behaviour of materials under different deformation conditions. The investigations for creation of better models have been conducted in the last years in the different scientific centres [9-13]. The works are more connected with steels than with other non ferrous materials. The constitutive equations worked out for steels can not be directly used for CAD of forming processes of copper and its alloys [14]. Only in the last year a few constitutive equations describing the flow stress of coppersilicon alloys were developed. The constitutive equations can be divided into fourth main types. First type can be used only for determination of characteristic levels of flow stress of copper and silicon bronzes, e.g. maximum flow stress (¢rpm) and flow stress of steady-state flow (crp,) [15]. The model can not be
the third, pure mathematical model, new parameters it means internal state of materials and internal state evolution rate were introduced [16]. The model with good accuracy describes the flow stress as a function of deformation parameters only in lower range of temperature. The last model contains internal state parameters as well as structural processes parameters. The model can be used for description of flow stress in wide range of temperature and strain rate with good accuracy [17]. The main aim of the work is to give short presentation of the above mentioned models.
2. Model describing characteristic values of flow stress and their positions
For description of characteristic values of flow stress was used Arrhenius equation k = B[sinh(atrp)]" exp(-~----T)
(i)
where: B, a, u - material constants, Q - activation energy, R - gas constant (8,314 J/mole K), trp -flow stress equal to trpm or trps. -strain rate
applied to follow the course of ¢ r p - ~ curves. The second
To correlate the date at different temperatures and strain rates, temperature compensated strain rate parameter Z ( Zener-
model is used for description of the course of a p - 6 curves.
Hollomon parameter) given by relation Z = i r I ' % T )
The model takes into account the structural process parameters, but it cannot be used for analysing of the strain rate history. In
introduced into equation (1). In this way the following equation was obtained
0924-0136/96/$15.00 © 1996 Elsevier Seienc~ S.A. All rights reserved PI10924-0136 (96) 02396-5
was
622
ZJ.
Gronostajski / Journal of Materials Processing Technology 60 (1996) 621-627
.1=,=] k ~
, ,
Calculation of flow stress according to equation (1) needs the values of constants B, a; u and activation energy Q. The method of calculation of these parameters are given el~where [15]. For calculation of activation energy of dynamic recrystallization Q = Qm, and activation energy of steady state flow Q = Q, the equation (1) can be rearrangement to the following form
=f. L
==>
I
) =.~=,r= k ~
& )=,~,r ~] where: ()c~r[
(3)
) =,~,r
(7)
,,
k t,~ g
Q=
-f==,zl
-the flow stress change rate at e,& and T, c,~,T
&rpB)
- the base flow stress change rate in
conditions where the other terms of the fight side of equation (2) are equal to zero,
where: Q = Qm for nrp = apm and Q = Q, for trp = trps Finally the dependence of trpm and trps from deformation
(c~rpzl - the flow stress change rate produced by c~ ) =,~,r
conditions was obtained by transformation of equation (2) as follows
dynamic recovery, %
&r pR |
- the flow stress change rate produced by
(4) dynamic recrystallization, For calculation of
trpm
have to be used parameters
Zm,,Bra,um and a m , and for calculation of trps parameters Zs,Bs,u= and a s . The method of computation of above mentioned parameters and their values are given elsewhere [15]. Zener-Hoiomon parameter was also used for description of
&rr°) - the flow stress change rate produced by d~ - ~,~,r strain ageing. It was assumed that at lowest temperature and highest strain rate used in experiment there is no recovery processes and such conditions were taken as the base conditions of experiments. The base flow stress curve of materials was def'med by following function
maximum flow stress position named critical strain 8or and strain of beginning of steady state flow e,. Dependence of critical strain as a function of deformation conditions according to Sellars [18] can be written as follows
=~
= M m Z . t- = k.,DPZ,~ ~-
(5)
where: Mm , p, lm ,kin - material constants, D - initial grain size. Usually values of p = 0,5 and !m is altered in the range from 0,12 to 0,17 [18]. Similar relation was assumed for calculation of strain at beginning of the steady state flow 6,.
apB=A(e+¢o)"
(8)
splaned at strain e = esp with function crpB=4=
(9)
Base rate of flow stress change of materials is determined by following relations: tT~O'pB = A I (10) = > 6=p =:~ 69°'~= = A 4 = + =o) n-I
6 s = MsZs t" = ksDPZst"
(6)
where: M=, l, and k, -material constants. Values of the material constants of investigated bronzes are given in paper [15].
The flow stress change rate produced by dynamic recovery is postulated in the form
cgtrpz & )=,~,r
3. Model based on rate changes of structural processes
Change of flow stress of materials caused by structural processes is described by following differential equation
= (b _ a6 )( cg°'pB] k &
(11)
)=,~=,r=
where: a and b - temperature-dependent and strain ratedependent material constants. After integration of equation (11) one can obtain the relation describing the flow stress change caused by dynamic recovery
Z,£ Gronostajski/ Journal of Materials Processing Technology 60 (1996) 621-627 (12)
O'pZ = (b-aEX~rpB)~,~ ,rn +a~(°'pB)~,~,rn dE
623
trp = (1 - b + at/trpB)E,~n,Tn
(17)
0 The flow stress change rate due to strain ageing is directly proportional to current change of flow stress with strain and it can be expressed as follows
For strain greater than critical strain ~ > ~cr substituting equation (15) to (7) the following relation was obtained (oq9"__~)
= 2(trpm - trps)
1
(13)
kE~-e~ k
t:~ . s,k,T where: x - material constant dependent on strain rate, temperature and type of materials. The integration of (13) leads to the following relation that det'mes the flow stress change produced by strain ageing
(14)
trpo = xf(trpB)s,~ n,ra dE
2
which after integration leads to
2
IEs-Ecr ~,
z
Integration constant was calculated from boundary condition, t h a t f o r z = ¢ c r , tr w = Crw,n
l'0460"pm
For description of flow stress change rate caused by dynamic recrystallization the following function was assumed
C = 0'966trps
(80"pR ]
and finally the flow stress for ¢ > 6 can be written as follows
= 2(~rpm--Crps)
¢7~ J o,•. T
1
cosh2f
Ea--Eor
4
+
+
(Ear+E s
kcs-E¢~ k
2 (19)
'~ ) ,,~,,r,
0,96trps
k & ) ,,~,r k & ) ,,~,r (15)
The equation (15) is present in relation (7) only when strain is bigger than critical strain 6 , because it was assumed that dynamic recrysatallization fraction in the range of deformation from s = 0 , 7 ¢ to e = 6 is rather small and in elaborated model can be neglected. The final model has two forms dependent on the strain range. For strains less than critical strain ¢ ~ ¢ ~ the model was obtained by substituting equations (11) and (13) into equation (7)
+
1,04tTpm
2
4. Model taking into account internal state of material
The change of flow stress during deformation is given by the following relation
o'1, = M(T)P(T, dr)o"w (6, k w, T)
(20)
Where: tr w (E, k w, T) - internal state of materials
M(T) - a function that describes the change of the (--~--]ri~r=(1-b+ag)(&:rpB] ,,
k
~
+x(CrpB), kn ra (16) )~,~n,r~
'
'
Integration of above equation gives the function for the flow stress
°'p =(1-b+ag]{°'~B].~ e I,,ka,Ta +(tc-a)f (°'PB)r
,1"~dE+C1
The boundary conditions, that for 8 = O, ~rp = 0 and ~rpB = O, and equality of t¢ and a, lead to integration constant Ct = O. Finally in the range of deformation 6 -< % flow stress is given
by
elastic properties of a materials with temperature, -a function describing the effect of sudden changes of strain rate on the flow stress, k w - evolution rate of internal state of material. introduction of function M(T) into equation (20) arises from the fact, that material's response on the stress is more affected by elastic strain caused by the stress, than by absolute value of stress. Function P(T, ir) allows to obtain correct material's response produced by a sudden change of strain rate. The last function together with the change of internal state of materials can be used for calculation of strain history effect. Each material is described by base internal state-strain curve, from which using elaborated model the ~rw - 6 curves at different temperatures and strain rates can be computed.
P(T,k)
ZJ. Gronostajskt /Journal o f Matertals Processing Technology 60 (1996) 621-627
624
For description of t r ~ ( e , k ~ , T ) courses the model was created, which has two forms dependent on the strain range. First form concerns strain range e < ecr, where flow stress increases monotoniealy with strains, and the second one includes the strain range e > ecr, where flow stress decreases and passes into steady state flow.
( tSO'wB] & ),,k,,,rn = f ' t r w ' ~" k " r (
(25)
one can obtain the following relation
(--~] 6,k,,,,r=[a6+b]f(°'w)'~'~"'r
(26)
4.1. Model f o r strain 6 < 6cr
For orw < trwsp assuming that the base internal state curves of Rate of change of internal state of material, according to Fig. 1 in the range of strain e < $cr is described by the following relation:
,
,
\
o~
(21)
)E,~.a,TB
materials is described by following spline function J Ale,
e < e,p
(27)
.B(4 =[ A(e+eo)',e;,
where: esp and awsp spline strain and spline stress respectively where: ~dtrw) ,.~,.r
- the internal state
of functions WwB(e)= Ale and o-wB(6) = A(e + eo)n.
ehange rate in
the integration of equation (26) at boundary condition ,that ~rw(0 ) = 0 gives the following relation
conditions determined by ~, kw and T, \
tr
W,B
1
- the base internal state change rate in
conditions determined by base evolution rate of internal state kwe and base temperature Tn, A - function describing the response of material with def'med base internal state on the change of strain rate and deformation temperature.
For ~w -> o'wsp the integration of equation (26) at the boundary condition, that for e - -- esp ' , O"w = O'wsp leads to
(~%)~'k-'r=A[ ae2-e~'22 ÷ b ( x - e ; ~ v ) + ( - ~ - ~ ) ~ ] "
~6.,w1
(29)
6'wB = A [g+E°)u
. . . . . . .
where
j~W
e~,
strain computed from the following equality
trw = trwn, (Fig.l).
4. 2. Model for strain 6)6cr
For description of internal state change rate in the range of strain e > ecr the following relation was used
VI I i I Esp EB l~'sp E "~ Fig.1.Cb~teristic valuesof strainfor specifiedvaluesof intmmalstateof materials.
2(tYwm-tr~s)
(t~trw~
L)
~,kw,T
1
es - ecr
\ 6 s - 6cr ~
The function X is described by the following relation
2 (30)
A=ae+b
(22) where:
Materials constants a and b are given by equations: a = aa(kw) + a I (kw)T+...+a4(kw)T 4
(23)
b = ba(kw) + bl(kw)T+...+b4(kw)T 4 From equations (21), (22) and
(24)
irwin-internal state of materials at critical strain,
trws -internal state of materials in the range of steady state flow. The method of Crwra and trws determination is similar to the calculation method of characteristic values of flow stress presented in chapter 2.
ZJ. Gronostajsk#I Journal of Materials Processing Technology 60 (1996) 621-627 P=k m
Integration of equation (30) leads to following relation
(~w)',i- .r=
2
'g"L
t
-
+c
(31)
which after taking into account the boundary condition, that for 6 = Scr, trw is determined by equation (28), leads to the following intemul state equations: -for O'wm(O'wsp
------{,
-.)]-,,=}
+AiIa - +b.cr] and for aw~ ~ crw~ (
2
+"
r
/
- L[s , - e c r t -'°"-
2
/
r,o. ,~v,]"
,2
="-=', +"(=o,-=,,)+t---r-J J
n -
(36)
where: m-coefficient of strain rate sensitivity. The values of P(T,~:), should be calculated using the sudden change of strain rate test. Because the differences of values of coefficient m caused by change of strain rate within the range of 0.01-10s "l were small, for f u ~ e r calculations was assumed the average values of the coefficient at each temperature of investigations. The effect of temperature on the coefficient of strain rate sensitivity was described by following polynomial m = K o + KIT + K2 T2 + K3T 3 + K4T 4
(37)
Coefficient of polynomial is given elsewhere [16]. (32)
2
625
-
), ~
/
'wsp /
/
(33) In order to calculate the flow stress according to elaborated model the functions M ( T ) and P(T,~) have to be know. 4.3. Calculation o f function
M(T)
4. 5. Strain rate history Strain rate sensitivity materials like copper-silicon alloys and copper are characterised by such behaviour, that the sudden change of strain rate causes increase of the flow stress to the level lower than the flow stress obtained during continuous deformation of material from the beginning with higher final strain rate. To render properly that phenomena it was assumed that the material's response produced by a change of the viscous properties described by function P(T,k)occurs simultaneously with the change of the strain rate, whereas the change of the material's internal state is somewhat delayed relative to the change of this rate and is detel"mined by material internal state evolution rate. To describe the material internal state evolution rate caused by a change in the strain rate, the following differential equation was used
Function M ( T ) is determined by ratio of Young modules Er at particular temperature T to the Young modules E re at ambient temperature To (To--293K) at the same, constant strain rate. M = Er
(34)
dc
=
(38)
where: L - a temperature dependent coefficient. The internal state evolution rate is proportional to temperature, which is expressed by coefficient L in the form L = lexp(-QIRT)
The Young modules were investigated within the temperature range of 293-1073K and strain rate of 0.01-10S -I. The effect of strain rate on the function M ( T ) is very small and without making bigger errors can be neglected. Then function M(T) can be written in the following form
1 k 2 -~
(39)
where: 1 - a material constant, Q - activation energy.
4. 6. Final form o f model describing the flow stress
Values of coefficients kl and k2 are given elsewhere [16].
Final form of model describing the flow stress-strain curves of copper-silicon alloys and copper at different conditions of deformation is obtained by combination of equations (20), (35) and (36)
4.4. Calculation o ffunction P(T,~;)
o,=(k, +k2 , o.(,.,.,)
M = kI
+
(35)
1
It has been assumed that function P(T,~) can be described by the following equation
"m
°
,40>
Internal state of material Crw(S,kw,T) is determined by two different equations depending on strain range.
Z.J. Gronosiajski /Journal of Malerds
626
Ifs9
cr and
by (2% If E > E cr and o,(trWSP,
Technology 60 (1996) 621-627
is described by
equations (1 1), (13) and (15) respectively, where instead of the
is determined
flow stress
CT,,,< rrWSP then a,(.s,i,,T)
equation (28), and if u, 2 D,,,$~ , a,(~,i,,T)
Processing
up
the internal
state of material
cW was
introduced, the following model for strains less than critical then ~J~(E,.?,,,,T) is determined by
equation (32), and if u,,.,,, 2 u,,,,~, c~(E,~,,,,T) is described by equation (33). Internal state evolution rate is given by relation (38).
strain E S &,was obtained
(%),i T =(I-b+a&)(%) . +K(u~B)e,k,,~B * .ww,.Ts
W’
(42) 5. Model taking into account the internal structural processes
state and the Integration (42) at the boundary conditions, that
The flow stress changes during deformation was defined by the equation (20) used in third model, the internal state of
UW
=
0
and uwB = 0, and for equality of
K
for
E = 0,
and a, gives the
function for the internal state of materials
materials was described by relation similar to (7) used in second model.
uw= (l -b +aE)(uw~),,iwB,TB (41)
(43)
For strain greater than critical strain the following relation was obtained
-
the
internal
state
change
rate
.c& ,r (44) determined by E,.?,
i T,
- the internal
state change rate in
which afk &=&,,
integration
CT, = 6,
at the boundary
condition,
that for
kadsto
conditions where the other terms of the right side of equation (4 1) are equal to zero, CT,=
“-;“-tg#---(“2”-E)]+ (45)
- the internal state change rate produced ~,h,,,r OQ6u,
by dynamic recovery,
+ 1,04a, 2
- the internal state change rate produced &ix ,r
Final model describing changes of flow stress of investigated
by dynamic recrystallization,
materials for E s E, was obtained by substituting equation (43)
- the internal state change rate produced 6,~W.T
into (20)
by strain ageing.
(46)
Taking into account that for description of the internal state change rate produced by dynamic recovery, the internal state change rate caused by strain ageing, and the internal change rate produced by dynamic recrystallization
state
were used
and for E z E, by rearrangement of following equations (45) and (20)
ZJ. Gronostajsla/ Journal of MaterialsProcessing Technology60 (1996) 621-627
o., =.,,,,,,<,.. =.,,.,,,, <"---=,:,hr 2
0,96trws + 1,04trwm
2
Los - c~ k
[2] (47)
}
6. Conclusion
The above described models have been applied to calculation of silicon bronzes flow stress as a function of strain in the temperature ranging from 293 to 1073K and at strain rate of 0.01 - 6.7 s -1 . It has been found, that the first model used to calculation of the values and positions of maximum and steady state flow stresses determines these parameters with good accuracy. The second model based on structural processes can be applied with good accuracy to follow the course of flow stress strain curves in the whole range of deformation conditions without taking into account the strain rate history. The third model based on the internal slate of materials and the internal state evolution rate can be used to calculation of the course of flow stress with strain and the effect of strain rate history, but the properly results have been obtained only in the lower range of temperature (293 - 500 K). The last model taking into account the structural processes and the internal state of materials does not has shortcoming which exist in the third model and can be used to calculation of flow stress - strain curves as well as to design the metal forming processes. The further development of constitutive equations decribing the flow stress of copper-silicon alloys and other copper alloys should created the possibility of calculation of the effect of deformation interruption on the flow stress.
References
[1]
Metal Forming (Edit. by Koistinen), Plenum Press, New
_+:.,. :)]+
2
S. Kohayashi and J.H. Kim, Deformation analysis of axisymetric sheet metal forming processes by the rigidplastic finite element method, in Mechanics of Sheet
627
[3]
York 1978. C.H. Toh and S. Kobayashi, Int. J. Mach. Tool Des. Res.,25 (1985) 15. N.M. Wang, A rigid-plastic rate-sensitive finite element
procedure for sheet metal forming calculations, in Numerical Methods in Industrial Forming Processes [4] [5] [6] [7] [8] [9]
(Edited by Pittman J.F.T., and all.) Pineridge Press, Swansea, 1982. E. Onate and O.C. Zienkiewicz, Int. J. Mech. Sci., 25 (1983) 305. M.J. Saran and R.H. Wagoner, J. of Appl. Mech.,S& (1991)499. J. Gronostajski and Z. Zimniak, Proc. Int. Conf. on Sheet Metal, Bermingham (1995) 77. J. Gronostajski and Z. Zimniak, J. Mater. Proc. Technol., 32 (1992) 263. W. Choi, P.P. Gillis and S.E. Jones, Met. Trans.,4, 20A, (1975) 989. K.P. Rao, Y.K.D.V. Prasad and E.B. Hawbolt, Proc. of
Int. Conf. on Advances in Mater. and Process. Technol., AMPT'93, Dublin, 3 (1993) 1529. [10] K. Nakanishi, Research Report of the Faculty of Engineering, Kagoshima University, No 30, November, [ll]
1988. J.R. Kiepaezko, T. Sasaki and T. Kurokawa, Trans. of
Japan Soc. for Aeronautical and Space Sciences, 36 (1993) 170. Z. Marciniak and A. Konieczay, J. Mech. Work. Technol., 15 (1987) 15. [13] A.K. Miller, Unified Constitutive Equations for Creep and Plasticity, Elsevier, London, 1987. [14] Z.J. Gronostajski, Proc. of XIV Int. Conf. on Advanced Mater. and Technol., AMT'95, Zakopane, (1995) 141. [15] Z.J. Gronostajski, Rudy i Metale Nie~elazne, 40 (1995) 40. [16] Z.J. Gronostajski, Proc. of Int. Conf. on Advances in Mat. and Proc. Technol. AMPT'95, Dublin, 2 (1995) 765? [17] Z.J. Crronostajski i Z. Misiotek, , Archiv of Metallurgy, 40 (1995) 399. [18] C.M. Sellars, Proc of Int. Conf. on Hot Worla'ng and Forming Processes, The Metal Society, London 1980. [12]
Materials Processing Technology ELSEVIER
Journal of Materials ProcessingTechnology60 (1996) 621--627
Development of constitutive equations of copper-silicon alloys Z.J. Gronostajski
Technical University of Wroclaw, ul. Lukasiewicza 3/5, 50-I11 Wroctaw, Poland
Abstract
The paper shows the development of constitutive equations describing the flow stress of silicon bronzes. The fn'st model can be used only for description of characteristic values of flow stress, the further models make possible to follow the course of stress-strain curves. The second model bases on structural processes, the third one bases on the internal state of materials and the last one on the internal state of materials and structural processes.
Keywords: constitutive equation, silicon bronze, strain rate, temperature, strain rate history.
1. Introduction
During the last years a finite element analysis has been intensively developed for simulating 3-1) deformation of a materials whose elastic strain is negligibly small compared to the plastic strain. Such conditions exist in a large number of the press-forming operations. The procedures for the analysing of such problems are elaborated by Kobayashi et all [1,2], Wang [3], Ouate and Zienkiewicz [4], and Saran and Wagoner [5]. The analysis of the metal working operations needs accurate knowledge of the forming limit curves [6-8] and of the constitutive equations describing the flow behaviour of materials under different operating conditions. CAD of forming operations is no longer limited by the memory capacity of computers or data processing capabilities but by the imperfections of the available constitutive models that describe the behaviour of materials under different deformation conditions. The investigations for creation of better models have been conducted in the last years in the different scientific centres [9-13]. The works are more connected with steels than with other non ferrous materials. The constitutive equations worked out for steels can not be directly used for CAD of forming processes of copper and its alloys [14]. Only in the last year a few constitutive equations describing the flow stress of coppersilicon alloys were developed. The constitutive equations can be divided into fourth main types. First type can be used only for determination of characteristic levels of flow stress of copper and silicon bronzes, e.g. maximum flow stress (¢rpm) and flow stress of steady-state flow (crp,) [15]. The model can not be
the third, pure mathematical model, new parameters it means internal state of materials and internal state evolution rate were introduced [16]. The model with good accuracy describes the flow stress as a function of deformation parameters only in lower range of temperature. The last model contains internal state parameters as well as structural processes parameters. The model can be used for description of flow stress in wide range of temperature and strain rate with good accuracy [17]. The main aim of the work is to give short presentation of the above mentioned models.
2. Model describing characteristic values of flow stress and their positions
For description of characteristic values of flow stress was used Arrhenius equation k = B[sinh(atrp)]" exp(-~----T)
(i)
where: B, a, u - material constants, Q - activation energy, R - gas constant (8,314 J/mole K), trp -flow stress equal to trpm or trps. -strain rate
applied to follow the course of ¢ r p - ~ curves. The second
To correlate the date at different temperatures and strain rates, temperature compensated strain rate parameter Z ( Zener-
model is used for description of the course of a p - 6 curves.
Hollomon parameter) given by relation Z = i r I ' % T )
The model takes into account the structural process parameters, but it cannot be used for analysing of the strain rate history. In
introduced into equation (1). In this way the following equation was obtained
0924-0136/96/$15.00 © 1996 Elsevier Seienc~ S.A. All rights reserved PI10924-0136 (96) 02396-5
was
622
ZJ.
Gronostajski / Journal of Materials Processing Technology 60 (1996) 621-627
.1=,=] k ~
, ,
Calculation of flow stress according to equation (1) needs the values of constants B, a; u and activation energy Q. The method of calculation of these parameters are given el~where [15]. For calculation of activation energy of dynamic recrystallization Q = Qm, and activation energy of steady state flow Q = Q, the equation (1) can be rearrangement to the following form
=f. L
==>
I
) =.~=,r= k ~
& )=,~,r ~] where: ()c~r[
(3)
) =,~,r
(7)
,,
k t,~ g
Q=
-f==,zl
-the flow stress change rate at e,& and T, c,~,T
&rpB)
- the base flow stress change rate in
conditions where the other terms of the fight side of equation (2) are equal to zero,
where: Q = Qm for nrp = apm and Q = Q, for trp = trps Finally the dependence of trpm and trps from deformation
(c~rpzl - the flow stress change rate produced by c~ ) =,~,r
conditions was obtained by transformation of equation (2) as follows
dynamic recovery, %
&r pR |
- the flow stress change rate produced by
(4) dynamic recrystallization, For calculation of
trpm
have to be used parameters
Zm,,Bra,um and a m , and for calculation of trps parameters Zs,Bs,u= and a s . The method of computation of above mentioned parameters and their values are given elsewhere [15]. Zener-Hoiomon parameter was also used for description of
&rr°) - the flow stress change rate produced by d~ - ~,~,r strain ageing. It was assumed that at lowest temperature and highest strain rate used in experiment there is no recovery processes and such conditions were taken as the base conditions of experiments. The base flow stress curve of materials was def'med by following function
maximum flow stress position named critical strain 8or and strain of beginning of steady state flow e,. Dependence of critical strain as a function of deformation conditions according to Sellars [18] can be written as follows
=~
= M m Z . t- = k.,DPZ,~ ~-
(5)
where: Mm , p, lm ,kin - material constants, D - initial grain size. Usually values of p = 0,5 and !m is altered in the range from 0,12 to 0,17 [18]. Similar relation was assumed for calculation of strain at beginning of the steady state flow 6,.
apB=A(e+¢o)"
(8)
splaned at strain e = esp with function crpB=4=
(9)
Base rate of flow stress change of materials is determined by following relations: tT~O'pB = A I (10) = > 6=p =:~ 69°'~= = A 4 = + =o) n-I
6 s = MsZs t" = ksDPZst"
(6)
where: M=, l, and k, -material constants. Values of the material constants of investigated bronzes are given in paper [15].
The flow stress change rate produced by dynamic recovery is postulated in the form
cgtrpz & )=,~,r
3. Model based on rate changes of structural processes
Change of flow stress of materials caused by structural processes is described by following differential equation
= (b _ a6 )( cg°'pB] k &
(11)
)=,~=,r=
where: a and b - temperature-dependent and strain ratedependent material constants. After integration of equation (11) one can obtain the relation describing the flow stress change caused by dynamic recovery
Z,£ Gronostajski/ Journal of Materials Processing Technology 60 (1996) 621-627 (12)
O'pZ = (b-aEX~rpB)~,~ ,rn +a~(°'pB)~,~,rn dE
623
trp = (1 - b + at/trpB)E,~n,Tn
(17)
0 The flow stress change rate due to strain ageing is directly proportional to current change of flow stress with strain and it can be expressed as follows
For strain greater than critical strain ~ > ~cr substituting equation (15) to (7) the following relation was obtained (oq9"__~)
= 2(trpm - trps)
1
(13)
kE~-e~ k
t:~ . s,k,T where: x - material constant dependent on strain rate, temperature and type of materials. The integration of (13) leads to the following relation that det'mes the flow stress change produced by strain ageing
(14)
trpo = xf(trpB)s,~ n,ra dE
2
which after integration leads to
2
IEs-Ecr ~,
z
Integration constant was calculated from boundary condition, t h a t f o r z = ¢ c r , tr w = Crw,n
l'0460"pm
For description of flow stress change rate caused by dynamic recrystallization the following function was assumed
C = 0'966trps
(80"pR ]
and finally the flow stress for ¢ > 6 can be written as follows
= 2(~rpm--Crps)
¢7~ J o,•. T
1
cosh2f
Ea--Eor
4
+
+
(Ear+E s
kcs-E¢~ k
2 (19)
'~ ) ,,~,,r,
0,96trps
k & ) ,,~,r k & ) ,,~,r (15)
The equation (15) is present in relation (7) only when strain is bigger than critical strain 6 , because it was assumed that dynamic recrysatallization fraction in the range of deformation from s = 0 , 7 ¢ to e = 6 is rather small and in elaborated model can be neglected. The final model has two forms dependent on the strain range. For strains less than critical strain ¢ ~ ¢ ~ the model was obtained by substituting equations (11) and (13) into equation (7)
+
1,04tTpm
2
4. Model taking into account internal state of material
The change of flow stress during deformation is given by the following relation
o'1, = M(T)P(T, dr)o"w (6, k w, T)
(20)
Where: tr w (E, k w, T) - internal state of materials
M(T) - a function that describes the change of the (--~--]ri~r=(1-b+ag)(&:rpB] ,,
k
~
+x(CrpB), kn ra (16) )~,~n,r~
'
'
Integration of above equation gives the function for the flow stress
°'p =(1-b+ag]{°'~B].~ e I,,ka,Ta +(tc-a)f (°'PB)r
,1"~dE+C1
The boundary conditions, that for 8 = O, ~rp = 0 and ~rpB = O, and equality of t¢ and a, lead to integration constant Ct = O. Finally in the range of deformation 6 -< % flow stress is given
by
elastic properties of a materials with temperature, -a function describing the effect of sudden changes of strain rate on the flow stress, k w - evolution rate of internal state of material. introduction of function M(T) into equation (20) arises from the fact, that material's response on the stress is more affected by elastic strain caused by the stress, than by absolute value of stress. Function P(T, ir) allows to obtain correct material's response produced by a sudden change of strain rate. The last function together with the change of internal state of materials can be used for calculation of strain history effect. Each material is described by base internal state-strain curve, from which using elaborated model the ~rw - 6 curves at different temperatures and strain rates can be computed.
P(T,k)
ZJ. Gronostajskt /Journal o f Matertals Processing Technology 60 (1996) 621-627
624
For description of t r ~ ( e , k ~ , T ) courses the model was created, which has two forms dependent on the strain range. First form concerns strain range e < ecr, where flow stress increases monotoniealy with strains, and the second one includes the strain range e > ecr, where flow stress decreases and passes into steady state flow.
( tSO'wB] & ),,k,,,rn = f ' t r w ' ~" k " r (
(25)
one can obtain the following relation
(--~] 6,k,,,,r=[a6+b]f(°'w)'~'~"'r
(26)
4.1. Model f o r strain 6 < 6cr
For orw < trwsp assuming that the base internal state curves of Rate of change of internal state of material, according to Fig. 1 in the range of strain e < $cr is described by the following relation:
,
,
\
o~
(21)
)E,~.a,TB
materials is described by following spline function J Ale,
e < e,p
(27)
.B(4 =[ A(e+eo)',e;,
where: esp and awsp spline strain and spline stress respectively where: ~dtrw) ,.~,.r
- the internal state
of functions WwB(e)= Ale and o-wB(6) = A(e + eo)n.
ehange rate in
the integration of equation (26) at boundary condition ,that ~rw(0 ) = 0 gives the following relation
conditions determined by ~, kw and T, \
tr
W,B
1
- the base internal state change rate in
conditions determined by base evolution rate of internal state kwe and base temperature Tn, A - function describing the response of material with def'med base internal state on the change of strain rate and deformation temperature.
For ~w -> o'wsp the integration of equation (26) at the boundary condition, that for e - -- esp ' , O"w = O'wsp leads to
(~%)~'k-'r=A[ ae2-e~'22 ÷ b ( x - e ; ~ v ) + ( - ~ - ~ ) ~ ] "
~6.,w1
(29)
6'wB = A [g+E°)u
. . . . . . .
where
j~W
e~,
strain computed from the following equality
trw = trwn, (Fig.l).
4. 2. Model for strain 6)6cr
For description of internal state change rate in the range of strain e > ecr the following relation was used
VI I i I Esp EB l~'sp E "~ Fig.1.Cb~teristic valuesof strainfor specifiedvaluesof intmmalstateof materials.
2(tYwm-tr~s)
(t~trw~
L)
~,kw,T
1
es - ecr
\ 6 s - 6cr ~
The function X is described by the following relation
2 (30)
A=ae+b
(22) where:
Materials constants a and b are given by equations: a = aa(kw) + a I (kw)T+...+a4(kw)T 4
(23)
b = ba(kw) + bl(kw)T+...+b4(kw)T 4 From equations (21), (22) and
(24)
irwin-internal state of materials at critical strain,
trws -internal state of materials in the range of steady state flow. The method of Crwra and trws determination is similar to the calculation method of characteristic values of flow stress presented in chapter 2.
ZJ. Gronostajsk#I Journal of Materials Processing Technology 60 (1996) 621-627 P=k m
Integration of equation (30) leads to following relation
(~w)',i- .r=
2
'g"L
t
-
+c
(31)
which after taking into account the boundary condition, that for 6 = Scr, trw is determined by equation (28), leads to the following intemul state equations: -for O'wm(O'wsp
------{,
-.)]-,,=}
+AiIa - +b.cr] and for aw~ ~ crw~ (
2
+"
r
/
- L[s , - e c r t -'°"-
2
/
r,o. ,~v,]"
,2
="-=', +"(=o,-=,,)+t---r-J J
n -
(36)
where: m-coefficient of strain rate sensitivity. The values of P(T,~:), should be calculated using the sudden change of strain rate test. Because the differences of values of coefficient m caused by change of strain rate within the range of 0.01-10s "l were small, for f u ~ e r calculations was assumed the average values of the coefficient at each temperature of investigations. The effect of temperature on the coefficient of strain rate sensitivity was described by following polynomial m = K o + KIT + K2 T2 + K3T 3 + K4T 4
(37)
Coefficient of polynomial is given elsewhere [16]. (32)
2
625
-
), ~
/
'wsp /
/
(33) In order to calculate the flow stress according to elaborated model the functions M ( T ) and P(T,~) have to be know. 4.3. Calculation o f function
M(T)
4. 5. Strain rate history Strain rate sensitivity materials like copper-silicon alloys and copper are characterised by such behaviour, that the sudden change of strain rate causes increase of the flow stress to the level lower than the flow stress obtained during continuous deformation of material from the beginning with higher final strain rate. To render properly that phenomena it was assumed that the material's response produced by a change of the viscous properties described by function P(T,k)occurs simultaneously with the change of the strain rate, whereas the change of the material's internal state is somewhat delayed relative to the change of this rate and is detel"mined by material internal state evolution rate. To describe the material internal state evolution rate caused by a change in the strain rate, the following differential equation was used
Function M ( T ) is determined by ratio of Young modules Er at particular temperature T to the Young modules E re at ambient temperature To (To--293K) at the same, constant strain rate. M = Er
(34)
dc
=
(38)
where: L - a temperature dependent coefficient. The internal state evolution rate is proportional to temperature, which is expressed by coefficient L in the form L = lexp(-QIRT)
The Young modules were investigated within the temperature range of 293-1073K and strain rate of 0.01-10S -I. The effect of strain rate on the function M ( T ) is very small and without making bigger errors can be neglected. Then function M(T) can be written in the following form
1 k 2 -~
(39)
where: 1 - a material constant, Q - activation energy.
4. 6. Final form o f model describing the flow stress
Values of coefficients kl and k2 are given elsewhere [16].
Final form of model describing the flow stress-strain curves of copper-silicon alloys and copper at different conditions of deformation is obtained by combination of equations (20), (35) and (36)
4.4. Calculation o ffunction P(T,~;)
o,=(k, +k2 , o.(,.,.,)
M = kI
+
(35)
1
It has been assumed that function P(T,~) can be described by the following equation
"m
°
,40>
Internal state of material Crw(S,kw,T) is determined by two different equations depending on strain range.
Z.J. Gronosiajski /Journal of Malerds
626
Ifs9
cr and
by (2% If E > E cr and o,(trWSP,
Technology 60 (1996) 621-627
is described by
equations (1 1), (13) and (15) respectively, where instead of the
is determined
flow stress
CT,,,< rrWSP then a,(.s,i,,T)
equation (28), and if u, 2 D,,,$~ , a,(~,i,,T)
Processing
up
the internal
state of material
cW was
introduced, the following model for strains less than critical then ~J~(E,.?,,,,T) is determined by
equation (32), and if u,,.,,, 2 u,,,,~, c~(E,~,,,,T) is described by equation (33). Internal state evolution rate is given by relation (38).
strain E S &,was obtained
(%),i T =(I-b+a&)(%) . +K(u~B)e,k,,~B * .ww,.Ts
W’
(42) 5. Model taking into account the internal structural processes
state and the Integration (42) at the boundary conditions, that
The flow stress changes during deformation was defined by the equation (20) used in third model, the internal state of
UW
=
0
and uwB = 0, and for equality of
K
for
E = 0,
and a, gives the
function for the internal state of materials
materials was described by relation similar to (7) used in second model.
uw= (l -b +aE)(uw~),,iwB,TB (41)
(43)
For strain greater than critical strain the following relation was obtained
-
the
internal
state
change
rate
.c& ,r (44) determined by E,.?,
i T,
- the internal
state change rate in
which afk &=&,,
integration
CT, = 6,
at the boundary
condition,
that for
kadsto
conditions where the other terms of the right side of equation (4 1) are equal to zero, CT,=
“-;“-tg#---(“2”-E)]+ (45)
- the internal state change rate produced ~,h,,,r OQ6u,
by dynamic recovery,
+ 1,04a, 2
- the internal state change rate produced &ix ,r
Final model describing changes of flow stress of investigated
by dynamic recrystallization,
materials for E s E, was obtained by substituting equation (43)
- the internal state change rate produced 6,~W.T
into (20)
by strain ageing.
(46)
Taking into account that for description of the internal state change rate produced by dynamic recovery, the internal state change rate caused by strain ageing, and the internal change rate produced by dynamic recrystallization
state
were used
and for E z E, by rearrangement of following equations (45) and (20)
ZJ. Gronostajsla/ Journal of MaterialsProcessing Technology60 (1996) 621-627
o., =.,,,,,,<,.. =.,,.,,,, <"---=,:,hr 2
0,96trws + 1,04trwm
2
Los - c~ k
[2] (47)
}
6. Conclusion
The above described models have been applied to calculation of silicon bronzes flow stress as a function of strain in the temperature ranging from 293 to 1073K and at strain rate of 0.01 - 6.7 s -1 . It has been found, that the first model used to calculation of the values and positions of maximum and steady state flow stresses determines these parameters with good accuracy. The second model based on structural processes can be applied with good accuracy to follow the course of flow stress strain curves in the whole range of deformation conditions without taking into account the strain rate history. The third model based on the internal slate of materials and the internal state evolution rate can be used to calculation of the course of flow stress with strain and the effect of strain rate history, but the properly results have been obtained only in the lower range of temperature (293 - 500 K). The last model taking into account the structural processes and the internal state of materials does not has shortcoming which exist in the third model and can be used to calculation of flow stress - strain curves as well as to design the metal forming processes. The further development of constitutive equations decribing the flow stress of copper-silicon alloys and other copper alloys should created the possibility of calculation of the effect of deformation interruption on the flow stress.
References
[1]
Metal Forming (Edit. by Koistinen), Plenum Press, New
_+:.,. :)]+
2
S. Kohayashi and J.H. Kim, Deformation analysis of axisymetric sheet metal forming processes by the rigidplastic finite element method, in Mechanics of Sheet
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[3]
York 1978. C.H. Toh and S. Kobayashi, Int. J. Mach. Tool Des. Res.,25 (1985) 15. N.M. Wang, A rigid-plastic rate-sensitive finite element
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1988. J.R. Kiepaezko, T. Sasaki and T. Kurokawa, Trans. of
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