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An implicit functional representation of stress-strain behavior
MECH. RES. COMM.
Vol. 5(4), 185-190, 1978.
Pergamon Press.
Printed in USA.
AN IMPLICIT FUNCTIONAL REPRESENTATION OF STRESS-STRAIN BEHAVIOR
K.P. Walker* and E. Krempl Department of Mechanical Engineering, Aeronautical Engineering & Mechanics, Rensselaer Polytechnic Institute, Troy, New York 12181
(Received 23 March 1978; accepted for print 6 July 1978)
Introduction
In a previous note, "An Implicit Functional Theory of Viscoplasticity"
[i],
an integral representation of metallic material deformation behavior was developed.
We now proceed to specialize this theory to the infinitesimal iso-
thermal deformation of isotropic materials.
For the uniaxial case elastic
perfectly plastic and work-hardening behavior is shown to be part of this functional theory.
Analysis The constitutive equations appropriate to infinitesimal isothermal deformation may be obtained from equations
(5) of [i] by substituting de for ~ and ~ for E, respectively and by letting ~ = 0. The kernel J (Co, t - t')
in equation
(3) of
(i), (2), (3) and
[i] is assumed to be the constant Jo' this form being deemed
suitable at low homologous temperatures
[2].
We note that our theory has
separate repositories for plasticity phenomena, and for rate dependence
(the microstructure parameter R)
(the time dependence of the kernels).
For the purpose of demonstration we assume that the isotropic form of the fourth rank tensor G is given by
Pratt & Whitney Aircraft,
East Hartford,
Connecticut 06108.
oo95-6~15/78/o7oi-o1855o2.oo/o
Copyright
(e)
197S Pergamon P r e s s
186
K.P.
WALKER and E. KREMPL
Gijk~(R,R',R '~,~o, 2t- t' - t ~') = ~ (R,R',R 'j,8o~ 2t- t' - t '~)@ij@k£ + b (R, R'~ R '~,Oo, 2t- t' - t ~) (§ik$j£ + ~i~$jk) ,
(i)
and, as a particular form for the function ~ we choose ~ (R, R', R", ~o' 2t - t' - t '~) = le -[2g (R)-g (R')-g (R~')]e-7 (2t-tt-t H) + ~e -[2f(R)-f(R')-f(Rz')] in which ~
[i - e
-~(2t-t'-t ~)
] ,
(2)
f (R), g(R) and 7 are functions of @o~ not explicitly written and
f (R) > g(R) for 0 " R < ~
In (2), k is to be identified with the Lame
constant for the material under small elastic deformations;
f and g are func-
tions of the structure parameter R; and 7 is taken to be a constant.
A simi-
lar expression with different k, f. g and 7 might be adopted for the hereditary function ~.
For simplicity we assume ~ = I~-2 ~ <~ i.e., Poisson's
ratio ~ is constant
[3].
Although we shall make this simplifying assumption
it should be made clear that it is not necessary to do so. With these stipulations the governing equations for the homogeneous uniaxial state of stress are [4] t ~ [e-[g(R)-g(R')]e-~(t-tl ) +e-[f(R)-f(R')][l e-~(t-t' ) ] ~¢ii dt t ; ell = E j ] ~t' o (3) t (4)
~ dt' ; R = Jo ~[' a --~--~ ~t' - b !~t-TI~ t t
~O ~
= 1~ E ~! ~ ie-[2g(R)-g(R')-g(Ri')}e-~(2t-t'-tz' ) o
o
+ e-{mf (R)-f (R')-f (RH) ] [i- e -7(2t-t'-tH)]l Bell ~Cll~t"dt'dt H ;
(5)
~t' i i Ee-~(2t-t'-t~' ) + [ 1 e -~(2t-t'-tH) I ~CII ~ell po%0=~1 E ] ~t' ~t H dt'dtZ' " o
(6)
o
In these equations E is Young's modulus.
Note that equation
the equation for a three-parameter viscoelastic
solid
(3) is similar to
[5] in which the instant-
aneous and equilibrium elastic moduli have been replaced with functions which depend on the structure parameter R. tonic loading process equations
It is shown in [4] that for a slow mono-
(3) - (6) may be reduced to
IMPLICIT REPRESENTATION OF STRESS-STRAIN BEHAVIOR
i ~11 = E
eo
[f (R)-f (R') } -~ell -dt'; ~t'
187
(7)
t
=
el---
b
I
dt' ;
(S)
o t
t
Oo* = 71 E o~ e-[2f(Rl-f(R')-f(R'o ~ )} Be~t,llBell~t,,dt'dt" ; 1
(9)
2
po~ = ~ Eel1.
(i0)
For suitable values of a- b we can model typical rate-independent plasticity phenomena by proper selection of the function f(R).
First we assume that
f(R) = ~R~ where ~ is a constant. Substitution of (9) and (i0) into (8) gives~ loading
for this case and for monotonic
(delI 2 0) using ell = 0 and R = 0 for t = 0 t Bell ~j2-~o{ a e l l - b ~ e-~(R-R') dt'} o St'
R= Jo(a4~-bv~)=Jo
This is an implicit relation for the determination of R. to obtain a relation of the form ell = m(R). into
(7) yields O l l =
h (R).
(ii)
It can be solved
Substitution of this relation
The parameter R can be eliminated to yield the
desired stress-strain relation Eell
=
- -
a ~Ii
--
~ max
1
(12)
(% max
~o E where o
max
a~J
o
For the slope we obtain doll dell
aE (I -
Cll/Cmax)
(13)
a- b Oll/Oma x
Examination of (12) and (13) shows that for all values of a and b d°ll dell (Oll= 0) = E
and
d°ll dell
(Oll=~max)
= 0~
188
K.P.
W A L K E R and E. KREMPL
and that ~ii reaches Omax for large ell.
The shape of the s t r e s s - s t r a i n
d i a g r a m b e t w e e n ell = 0 and ~ii = ~max is d e t e r m i n e d by the values of a and b as shown in Fig. l. reproduce
For very small values of a - b equations
(12) and
(13)
elastic p e r f e c t l y plastic behavior.
To investigate stress-strain
the b e h a v i o r during u n l o a d i n g d i a g r a m equations
u n l o a d i n g history.
(7) - (I0) have to be solved for the loading-
Of p a r t i c u l a r
any point ~Ii on the u n l o a d i n g
from any point u °ii ~ ell o on a
interest
branch
is the slope during unloading.
[4]
doll
dR - -
dell
For Cll = 0
+ E.
(14)
GUll dell
(intercept with the e-axis)
we obtain the elastic
values of a and b. The slope at the point of u n l o a d i n g dR evaluating ~ at ~ii = ~ii" We obtain o[4] Ii d~ll de
(
O
(~ii=~ii)
For
= E
can be d e t e r m i n e d
by
(a - b)
max
1
slope for all
+
.
~o ii
Ii a+b
(15)
max
The slope at u n l o a d i n g values of
( a - b)
depends
close
therefore
to elastic
o on the ~ll ~ a and b.
unloading
is
obtained
F r o m the above it is clear that the choice of f(R) flat s t r e s s - s t r a i n
curve.
a - b m u s t be very small.
To approach
as seen
= ~ R leads to an u l t i m a t e l y
the e l a s t i c - p e r f e c t l y
In this c a s e , R would never grow
get only elastic behavior~
see e q u a t i o n
present
in the endochronic
theories
provide
for this flexibility
(12).
plastic b e h a v i o r
[3].
it
[4] and we would
Such a flexibility
Recent developments,
was not
however,
[6].
we show how this theory can be extended
To this end we select
in Fig.1.
A l t h o u g h we can make a - b as small as we please
can never be exactly zero.
For illustration
For small
to work-hardening.
[4] f (R) =
and solve the equations
(n I + n 2 R ) ~
given in Fig.2
(R+ i)
for completely
trolled condition at a strain amplitude of 1%.
(16)
reversed
Parameters
strain con-
are selected to
IMPLICIT REPRESENTATION OF STRESS-STRAIN BEHAVIOR
a-b .-a
a -b -= 0.001
189
0.4 a -b
0.95
1.0 0.9 0.8
///
0.7 O1 1
0.6
E
0.5 o.41 0.3
I .......
,~,
a-b -7"
0.2
: o.ool
0.1
0 . 5 1.0 1.5 2.0 ~ 5
3.0 3.5 4 . 0 4 . 5 5.0
Cll
Fig. 1.
STRESS-STRAIN
DIAGRAM
611 a
E with
E
Omax/E /
0"max/E = 1; E = 5; a -
1
40
20 10
zI 0
uJ Pr k-
lO
"°APL~G//
II
--20
-30 --40, -1.2
I -0.8
1 0.4
I 0
I 0.4
STRAIN Fig. 2.
~ = E 1 c + E2
/
STRAIN
HARDENING
HYSTERESIS
LOOPS
-{,(RI-, (R', I e
#, = '/2 ( E I + E 2) c 2 f(R)
offO
e
=
1
E1
=
27400
E2
-
17772600
Jo ~~ t ~ d t ' ;
, t = 72 E 1 c 2 + '/~ E 2
I 1.2
(%)
o
v
I 0.8
-t2,~I-,R')-,R")
= (n 1 + n 2 R ) l n
t a~ ae ()t'
__
C';')t" dt dt
__
r #t
;
(R+I).
a
=
.01
b
=
.005
n1
=
36
n2
=
4
190
K.P.
simulate steel.
W A L K E R and E. KREMPL
a strongly cyclic h a r d e n i n g b e h a v i o r The equations
were n u m e r i c a l l y
calculator
using a trapezoidal
Concluding
Remarks
The c o n t i n u o u s l y
rule.
integrated
The results
evolving m i c r o s t r u c t u r e
effects of p l a s t i c i t y
similar to type 304 stainless
in this theory.
on a p r o g r a m m a b l e are shown in Fig. 2.
parameter
It grows
pocket
is responsible
for the
slowly with d e f o r m a t i o n
upon
first loading and then more rapidly as loading proceeds.
Immediately
each successive
than before unload-
unloading
ing.
The d i s s i p a t i o n
i.e.,
during
initial
under fast loading equations
of plastic work into heat is small where R is small, loading and u n l o a d i n g
conditions
stress-strain
curves
for finite
the e q u i l i b r i u m loading
the stress
(3) - (6), by replacing
we assume that f(R) > g(R)
curves
the growth of R is much smaller
response
after
(instantaneous
rates of loading
It is also shown in
strain curve
f(R) with g(R)
7~ 0, these
stress-strain
[4].
is determined,
in equations
two functions
and equilibrium)~
form a limiting
Since
set of
with s t r e s s - s t r a i n
It is also possible
to be elastic by setting g(R)
from
(7) - (I0).
falling below the i n s t a n t a n e o u s
curves.
[4] that
and above
to assume
the fast
= 0.
Acknowledgement The research
reported
on in this paper was supported by the National
Science
Foundation.
References
[i] [2] [3] [4] [5] [6]
K.P. W a l k e r and E. Krempl, An Implicit Functional Theory of Viscoplasticity: preceding paper in this issue. E. Krempl, On the Interaction of Rate- and H i s t o r y - D e p e n d e n c e in Structural Metals, Acta M e c h a n i c a ~ 53 (1975). K.C. Valanis, On the Foundations of the E n d o c h r o n i c Theory of Plasticity, Archives of Mechanics 2 7 7 857 (1975). K.P. Walker~ Ph.D. Thesis, R e n s s e l a e r Polytechnic Institute, Troy, New York (1976) . R.M. Christensen, "Theory of Viscoelasticity. An Introduction," Academic Press (1971). K.C. Valanis, A New Intrinsic Time Measure for the E n d o c h r o n i c Description of Plastic Behavior, D i v i s i o n of Materials Engineering, The University of Iowa, Iowa City, Iowa 52242, Report G-123-DME-77-003, April 1977.
Vol. 5(4), 185-190, 1978.
Pergamon Press.
Printed in USA.
AN IMPLICIT FUNCTIONAL REPRESENTATION OF STRESS-STRAIN BEHAVIOR
K.P. Walker* and E. Krempl Department of Mechanical Engineering, Aeronautical Engineering & Mechanics, Rensselaer Polytechnic Institute, Troy, New York 12181
(Received 23 March 1978; accepted for print 6 July 1978)
Introduction
In a previous note, "An Implicit Functional Theory of Viscoplasticity"
[i],
an integral representation of metallic material deformation behavior was developed.
We now proceed to specialize this theory to the infinitesimal iso-
thermal deformation of isotropic materials.
For the uniaxial case elastic
perfectly plastic and work-hardening behavior is shown to be part of this functional theory.
Analysis The constitutive equations appropriate to infinitesimal isothermal deformation may be obtained from equations
(5) of [i] by substituting de for ~ and ~ for E, respectively and by letting ~ = 0. The kernel J (Co, t - t')
in equation
(3) of
(i), (2), (3) and
[i] is assumed to be the constant Jo' this form being deemed
suitable at low homologous temperatures
[2].
We note that our theory has
separate repositories for plasticity phenomena, and for rate dependence
(the microstructure parameter R)
(the time dependence of the kernels).
For the purpose of demonstration we assume that the isotropic form of the fourth rank tensor G is given by
Pratt & Whitney Aircraft,
East Hartford,
Connecticut 06108.
oo95-6~15/78/o7oi-o1855o2.oo/o
Copyright
(e)
197S Pergamon P r e s s
186
K.P.
WALKER and E. KREMPL
Gijk~(R,R',R '~,~o, 2t- t' - t ~') = ~ (R,R',R 'j,8o~ 2t- t' - t '~)@ij@k£ + b (R, R'~ R '~,Oo, 2t- t' - t ~) (§ik$j£ + ~i~$jk) ,
(i)
and, as a particular form for the function ~ we choose ~ (R, R', R", ~o' 2t - t' - t '~) = le -[2g (R)-g (R')-g (R~')]e-7 (2t-tt-t H) + ~e -[2f(R)-f(R')-f(Rz')] in which ~
[i - e
-~(2t-t'-t ~)
] ,
(2)
f (R), g(R) and 7 are functions of @o~ not explicitly written and
f (R) > g(R) for 0 " R < ~
In (2), k is to be identified with the Lame
constant for the material under small elastic deformations;
f and g are func-
tions of the structure parameter R; and 7 is taken to be a constant.
A simi-
lar expression with different k, f. g and 7 might be adopted for the hereditary function ~.
For simplicity we assume ~ = I~-2 ~ <~ i.e., Poisson's
ratio ~ is constant
[3].
Although we shall make this simplifying assumption
it should be made clear that it is not necessary to do so. With these stipulations the governing equations for the homogeneous uniaxial state of stress are [4] t ~ [e-[g(R)-g(R')]e-~(t-tl ) +e-[f(R)-f(R')][l e-~(t-t' ) ] ~¢ii dt t ; ell = E j ] ~t' o (3) t (4)
~ dt' ; R = Jo ~[' a --~--~ ~t' - b !~t-TI~ t t
~O ~
= 1~ E ~! ~ ie-[2g(R)-g(R')-g(Ri')}e-~(2t-t'-tz' ) o
o
+ e-{mf (R)-f (R')-f (RH) ] [i- e -7(2t-t'-tH)]l Bell ~Cll~t"dt'dt H ;
(5)
~t' i i Ee-~(2t-t'-t~' ) + [ 1 e -~(2t-t'-tH) I ~CII ~ell po%0=~1 E ] ~t' ~t H dt'dtZ' " o
(6)
o
In these equations E is Young's modulus.
Note that equation
the equation for a three-parameter viscoelastic
solid
(3) is similar to
[5] in which the instant-
aneous and equilibrium elastic moduli have been replaced with functions which depend on the structure parameter R. tonic loading process equations
It is shown in [4] that for a slow mono-
(3) - (6) may be reduced to
IMPLICIT REPRESENTATION OF STRESS-STRAIN BEHAVIOR
i ~11 = E
eo
[f (R)-f (R') } -~ell -dt'; ~t'
187
(7)
t
=
el---
b
I
dt' ;
(S)
o t
t
Oo* = 71 E o~ e-[2f(Rl-f(R')-f(R'o ~ )} Be~t,llBell~t,,dt'dt" ; 1
(9)
2
po~ = ~ Eel1.
(i0)
For suitable values of a- b we can model typical rate-independent plasticity phenomena by proper selection of the function f(R).
First we assume that
f(R) = ~R~ where ~ is a constant. Substitution of (9) and (i0) into (8) gives~ loading
for this case and for monotonic
(delI 2 0) using ell = 0 and R = 0 for t = 0 t Bell ~j2-~o{ a e l l - b ~ e-~(R-R') dt'} o St'
R= Jo(a4~-bv~)=Jo
This is an implicit relation for the determination of R. to obtain a relation of the form ell = m(R). into
(7) yields O l l =
h (R).
(ii)
It can be solved
Substitution of this relation
The parameter R can be eliminated to yield the
desired stress-strain relation Eell
=
- -
a ~Ii
--
~ max
1
(12)
(% max
~o E where o
max
a~J
o
For the slope we obtain doll dell
aE (I -
Cll/Cmax)
(13)
a- b Oll/Oma x
Examination of (12) and (13) shows that for all values of a and b d°ll dell (Oll= 0) = E
and
d°ll dell
(Oll=~max)
= 0~
188
K.P.
W A L K E R and E. KREMPL
and that ~ii reaches Omax for large ell.
The shape of the s t r e s s - s t r a i n
d i a g r a m b e t w e e n ell = 0 and ~ii = ~max is d e t e r m i n e d by the values of a and b as shown in Fig. l. reproduce
For very small values of a - b equations
(12) and
(13)
elastic p e r f e c t l y plastic behavior.
To investigate stress-strain
the b e h a v i o r during u n l o a d i n g d i a g r a m equations
u n l o a d i n g history.
(7) - (I0) have to be solved for the loading-
Of p a r t i c u l a r
any point ~Ii on the u n l o a d i n g
from any point u °ii ~ ell o on a
interest
branch
is the slope during unloading.
[4]
doll
dR - -
dell
For Cll = 0
+ E.
(14)
GUll dell
(intercept with the e-axis)
we obtain the elastic
values of a and b. The slope at the point of u n l o a d i n g dR evaluating ~ at ~ii = ~ii" We obtain o[4] Ii d~ll de
(
O
(~ii=~ii)
For
= E
can be d e t e r m i n e d
by
(a - b)
max
1
slope for all
+
.
~o ii
Ii a+b
(15)
max
The slope at u n l o a d i n g values of
( a - b)
depends
close
therefore
to elastic
o on the ~ll ~ a and b.
unloading
is
obtained
F r o m the above it is clear that the choice of f(R) flat s t r e s s - s t r a i n
curve.
a - b m u s t be very small.
To approach
as seen
= ~ R leads to an u l t i m a t e l y
the e l a s t i c - p e r f e c t l y
In this c a s e , R would never grow
get only elastic behavior~
see e q u a t i o n
present
in the endochronic
theories
provide
for this flexibility
(12).
plastic b e h a v i o r
[3].
it
[4] and we would
Such a flexibility
Recent developments,
was not
however,
[6].
we show how this theory can be extended
To this end we select
in Fig.1.
A l t h o u g h we can make a - b as small as we please
can never be exactly zero.
For illustration
For small
to work-hardening.
[4] f (R) =
and solve the equations
(n I + n 2 R ) ~
given in Fig.2
(R+ i)
for completely
trolled condition at a strain amplitude of 1%.
(16)
reversed
Parameters
strain con-
are selected to
IMPLICIT REPRESENTATION OF STRESS-STRAIN BEHAVIOR
a-b .-a
a -b -= 0.001
189
0.4 a -b
0.95
1.0 0.9 0.8
///
0.7 O1 1
0.6
E
0.5 o.41 0.3
I .......
,~,
a-b -7"
0.2
: o.ool
0.1
0 . 5 1.0 1.5 2.0 ~ 5
3.0 3.5 4 . 0 4 . 5 5.0
Cll
Fig. 1.
STRESS-STRAIN
DIAGRAM
611 a
E with
E
Omax/E /
0"max/E = 1; E = 5; a -
1
40
20 10
zI 0
uJ Pr k-
lO
"°APL~G//
II
--20
-30 --40, -1.2
I -0.8
1 0.4
I 0
I 0.4
STRAIN Fig. 2.
~ = E 1 c + E2
/
STRAIN
HARDENING
HYSTERESIS
LOOPS
-{,(RI-, (R', I e
#, = '/2 ( E I + E 2) c 2 f(R)
offO
e
=
1
E1
=
27400
E2
-
17772600
Jo ~~ t ~ d t ' ;
, t = 72 E 1 c 2 + '/~ E 2
I 1.2
(%)
o
v
I 0.8
-t2,~I-,R')-,R")
= (n 1 + n 2 R ) l n
t a~ ae ()t'
__
C';')t" dt dt
__
r #t
;
(R+I).
a
=
.01
b
=
.005
n1
=
36
n2
=
4
190
K.P.
simulate steel.
W A L K E R and E. KREMPL
a strongly cyclic h a r d e n i n g b e h a v i o r The equations
were n u m e r i c a l l y
calculator
using a trapezoidal
Concluding
Remarks
The c o n t i n u o u s l y
rule.
integrated
The results
evolving m i c r o s t r u c t u r e
effects of p l a s t i c i t y
similar to type 304 stainless
in this theory.
on a p r o g r a m m a b l e are shown in Fig. 2.
parameter
It grows
is responsible
for the
slowly with d e f o r m a t i o n
upon
first loading and then more rapidly as loading proceeds.
Immediately
each successive
than before unload-
unloading
ing.
The d i s s i p a t i o n
i.e.,
during
initial
under fast loading equations
of plastic work into heat is small where R is small, loading and u n l o a d i n g
conditions
stress-strain
curves
for finite
the e q u i l i b r i u m loading
the stress
(3) - (6), by replacing
we assume that f(R) > g(R)
curves
the growth of R is much smaller
response
after
(instantaneous
rates of loading
It is also shown in
strain curve
f(R) with g(R)
7~ 0, these
stress-strain
[4].
is determined,
in equations
two functions
and equilibrium)~
form a limiting
Since
set of
with s t r e s s - s t r a i n
It is also possible
to be elastic by setting g(R)
from
(7) - (I0).
falling below the i n s t a n t a n e o u s
curves.
[4] that
and above
to assume
the fast
= 0.
Acknowledgement The research
reported
on in this paper was supported by the National
Science
Foundation.
References
[i] [2] [3] [4] [5] [6]
K.P. W a l k e r and E. Krempl, An Implicit Functional Theory of Viscoplasticity: preceding paper in this issue. E. Krempl, On the Interaction of Rate- and H i s t o r y - D e p e n d e n c e in Structural Metals, Acta M e c h a n i c a ~ 53 (1975). K.C. Valanis, On the Foundations of the E n d o c h r o n i c Theory of Plasticity, Archives of Mechanics 2 7 7 857 (1975). K.P. Walker~ Ph.D. Thesis, R e n s s e l a e r Polytechnic Institute, Troy, New York (1976) . R.M. Christensen, "Theory of Viscoelasticity. An Introduction," Academic Press (1971). K.C. Valanis, A New Intrinsic Time Measure for the E n d o c h r o n i c Description of Plastic Behavior, D i v i s i o n of Materials Engineering, The University of Iowa, Iowa City, Iowa 52242, Report G-123-DME-77-003, April 1977.