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On the limit case of endochronic theory
OK%ibtl3 p? U.OO+ .Mf C I W I Perpmon Pm3 plc
ON THE LIMIT CASE OF ENDOCHRONIC THEORY HAN C. WV and C. KOMARAKULNANARORN Departmentof Civil Engineering. The Univrrsity of Iowa. Iowa City, IA 52,742. U.S.A.
Abstract-An endochronic model is described and shown to lead to elastic as well as plastic behavior in the limit case of k = I. in contrast to Fazio’s claim that the limit case cannot lead to plasticity. The slope of the stress-strain curve is represented by a function F(X) whose functional form is numerically investigated. The study includes straining. unsttaining and cyclic straining. The equations of the general form of this model have also been derived.
I. 1NTRODUCTION
A modified endochronic theory was presented by Valanis (1980). The modification was based on an intrinsic time defined by a plastic-strain-like tensor, which in the one-dimensional case is given by
where r~and i: arc stress and strain. rcspcctively ; Q is the pl~tstic-strain-IiLc quantity; E is Young’s modulus; and X-is a constant p~tr~Irn~t~rhaving a vniuc bctwccn ircro and one. In the cast ofk = 0, the 1980 theory rcduccs to the carlicr theory ot‘ Valanis (1971). In the CilSC Of k = 1. Valanis (1s)XO)shows that the t9SO rhcory lccl to the cl>tssici~ltheory of plasticity with yield surface dcfinod anti suhjcctcd to ii combincd isotropic kincmatiu strainhardening rule. The normality condition of the plastic strain rate with rcspcct to the yield surfiloc was also dcrivcd from this theory. The plastic-strain-likr increment dQ rcduccs to the plastic strain inurement dt? when k = 1, and is given by
in the treatment of the case of k: = 1, Valanis (1980) cautioned that care must be exercised in the derivation, otherwise erroneous results might be obtained. In a recent paper, Fazio (1989). in a one-dimensional study of the limit case of the endochronic theory, claimed that, in the case of k = I. the endochronic theory could only lead to elastic behavior and that plastic behavior could not develop according to the 1980 theory. The purpose of this paper is to show that Fuzio’s claim is not valid. Furthermore, additional details for the cast of iE_= 1 are presented. The one-dimensional case is first discussed, and a discussion of the three-dimensional case follows. in this paper, explicit equations using r number of internal variables have been derived. 2. ONE-DIMENSIONAL
ENDOCHRONIC
CONSTITUTIVE
EQUATION
Following the 1985 theory, the constitutive equation in the Gibbs formulation (see Wu and Aboutorabi, 1988) is given by dr: = $ + f: Ci dc/ I- I
(3)
where d are r number of internal variables. and C, are material constants (see the Appendix SAS19:1-CI 135
H. C. WV and C. KOMARAKCLXANAKORN
1x7
for additional discussion concerning the Helmholtz and Gibbs formulation). The internal variables q’ may be interpreted as the descriptors for the internal structure of a material during plastic deformation, and these variables evolve with respect to an intrinsic time z, which is used to register the deformation history during deformation. The rate of change of the internal variables is determined by the current state of stress and internal variables. In this investigation. a linear evolution equation for the internal variables q’ is assumed such that for each internal variable q’. dy’ I - = T (C,o--F,q’) dz ,
(i not summed).
where N, and F, are material constants. In order to account for strain-hardening.
the intrinsic
(4)
time : is scaled by
where the function/represents isotropic hardening. The increment defined in terms of the plastic-strain-like quantity. so that
4 = WI
of intrinsic
time dc is
(6)
or dn d< = & dC - k E [ where d< > 0. By combining
(4) and (5) it is obtained
dr/’ =;)i
(C,CT-Ey’) ,
1
(7)
that
= s d; I
(inot
summed),
(8)
where
,I’, = 2 (C,aJ’N Therefore,
by the substitution
F,y’)
(i not summed).
(9)
of (7) and (g), eqn (3) becomes
where
(11) Equation
(IO) is further
written
as da $1
This equation
provides
+_kX] = dc[l &xl.
the basis for the discussion
to follow.
(12)
Endochroaic theory
Fazio (1989), in his eqn (24) (see the Appendix),
da
137
rewrote the above equation as
I&X EI+_kX [ 1
iii=
(13)
with no mention of the condition
IkkX#O.
(14)
The condition given by (14) should be satisfied in order for (I 3) to be valid. Fazio went on to say that when k = I. (13) reduces to
da iG= E
(15)
and has thus arrived at an erroneous conclusion that, in the case of k = I, the endochronic theory could only lead to elastic behavior and that plastic behavior could not develop. A careful observation
would lead to the result I +X
= 0, when
k = I. Therefore
Fazio, in
fact. dealt with the case of O/O. The fact that it is possible to deduce the elastic as well as the plastic behavior from the 1980 theory will now be shown. Equation (I 2) may be rewritten as (I 3) in the case of k # I, because in this case (14) is satisfied. But for the case of k = I, (I 2) should be rewritten as
Thus,
dc-%=O
When dr:-da/E stress-strain
or
1*X=0.
(17)
= 0, the behavior is elastic and generally X # + I. Hence, the slope of the curve is E. But. when I *X
= 0, i.e. X = + I. generally d&---da/E
# 0, and
the slope is not E. In fact, the condition X = + I with the help of (9) and (I I) leads to
(18) which can then be rewritten as
u = +cYr/+av
’ GF,q’
2 1-1 N,
(19)
whcrc
,$
cz
aY =
-I
( >. ,;,
Note that the first term on the right hand side of (19) represents isotropic hardening the second term the kinematic hardening. By writing (I 3) as
(20) and
138
where
F(X) =
1+x -;m. -
(22)
’
the variation of the value of function F(.Y) may now be investigated. Note that the form of the isotropic hardening functionf(z) affects the value of X. The following expression used by Wu and Yip (198 If has been used in the present computation : f(z) = A-(tl-
l)exp(-/?c).
(231
A hypothetical material with the following material constants: 1V= 3.64 MPa, C = 0.54, F = 140 MPa, A = 3.3, p = 150 and E = 28000 MPa is now considered to demonstrate the applicability of the present model (with only one internal variable for simplicity) and to show the form of the function F(X). This function has been calculated for k = 0.5, 0.95, and I, and plotted versus .Y in Fig. I. It is seen from the figure that, for the cases of k f I, the value of F(X) changes gradually with X. But. for the case of k = I. F(X) remains one in the elastic range as X increases; as soon as X reaches one, plastic ~ieform~~tion occurs and F(X) decreases as the plastic def~~rmiItion ~lccurnul~~tcswith X staying at one The step-by-step numerical proccdurcs arc : knowing dr: and the current vafucs of :, rr, and q. eqn (12) may be used to iind drr ; cyns (I), (33) itnd (6) can then be used to dctcrminc dQ and dc, and oqn (8) Icads to dq. Thus z, 6. and y can bc updated. In the case of strain-controlled cyclic loading with strain limits of + I%, the cyclic stress -strain curve is shown in Fig. 2 for this hypothetical material. The corresponding I*‘(X) versus X diagram is shown in I:ig. 3. For the citsc of k = 0.95, and similarly for k = 0.5. the followingscqucnoe is followed : ALCDAMFGALCD.. . . Note that thechange between CD and f:G is abrupt; while that between AC and AF is gradual. In the cxx of k = I, the following sequence is foflowud : ABCBAEFEABC. _. . The change from B to C and from E to F is gradual; while from C to B and F to E is abrupt. The corresponding plot for F(X) versus intrinsic time z is shown in Fig. 4. The letters denote c~rrespondjn~ points in these figures.
2” I
X Fig. 1. Function F(.t’) frtr varinus ~cmst~~ntI;.
Endochronic
thuory
139
-25
-60 -1.2
-0.8
-0.4
straPn.
0.4
0.1
1.2
K.
Fig. 2. Onedimensional
cyclic stress-strain
curve for a hypothetical
material.
k = 0.95.
k=l --k - 0.95 k-_- - 0.5 1.6
1
R
z
0.5
L 0
C I-
-1
1.6
-0.6
Fig. 3. Function
3. THE
ii
F(X)
0.1
,
1.b
for cyclic straining.
THREE-DIMENSIONAL
EQUATIONS
In this paper, the material is assumed to be plastically incompressible. For a discussion of material subjected to volumetric plastic deformation, the reader is referred to Wu and Aboutorabi (1988). For plastically incompressible materials, it suffices to consider only the deviatoric response. In the deviatoric response, the internal variables are p”, with n = I to r. The deviatoric strain increment de is expressed in terms of the increments of deviatoric stress ds and internal variables dp”, i.e. d.q de,, = c+
0
’ c c”dp:,, n- I
(24)
where C” are material constants and 11”is the shear modulus. The evolution of p” is defined
H. C. Wu
and C. KOMARAKL-LNXGAKORS
1.2 , / (A 0.0 -
z ii:
0.3
0.3 -
L
Fig. 4. F(x) versus z for cyclic
with respect to intrinsic
straining.
k = 0.95.
time 2. By assuming a linear evolution equation, the rate ofchangc
of’ p” is given by
whore IV”, F” are material constants. A deviatoric strain-like
dQ,, = d<,,,-k
tensor Q is now dcfincd by
d.s,, kI
where k is again ;L constant parameter such that 0 < k < I. Note that when k = I, Q is equal to the plastic strain c*. An intrinsic
time increment d< is now dcfincd as
di’ = dQ,,
de,,,
(27)
which is related to z through the relation given by (5). It will be shown subsequently that functionj’represents
isotropic
hardening.
In a strain-controlled
test, de,, is a known and
input quantity. Rewriting
eqn (24) as
,
this equation together with eqn (25) may be substituted
dQ,, = ( -k)
(28)
into eqn (26) to yield
de,, + k i C” dp;, n- I
(2%
or
dQ,, = zt,+ fli, dwhere
(30)
En&chronic
theory
141
a,, = (I -k)de,,
(31)
(32)
Finally,
by substituting
eqns (5) and (30) into (27), the following
quadratic
equation
is
obtained : Pd?+Qdc+R
=0
(33)
where
’ =
BtjBi,-f’
(34)
Q = h,Btj
(35)
R = a,,a,j.
(36)
The functions P, Q and R are defined in terms of the constants of the model and increments of input variables. When dc is solved in eqn (33). eqn (25) can be used to obtain dp:,. By substitution. the deviatoric stress increment ds,, can then be found from eqn (28). In the case of Xr = I. eqns (31). (35) and (36) give a,! = R = Q = 0. Thus, eqn (33) is reduced to Pd? = 0. The condition d: = 0 is associated with the elastic state. while the condition
d: # 0 is associated with the plastic state. Hence, in the plastic state, P must be
equal to zero and eqn (34) lcads to
h --r,,)(%,-r,j) = (SJ’)
(37)
where
(38)
and
(39)
Equation (37) is the von Mises yield criterion with combined isotropic-kinematic in which
f describes
hardening,
the isotropic hardening and r,, the kinematic hardening in thedcviatoric
behavior. S,, is the initial yield stress. From eqns (30). (32), (38) and (39). the plastic strain increment is now given by dc = dQv = fi,j
$ (So -r,/) J
(40)
which is normal to the deviatoric yield surface as is seen by comparing
4. CONCLUDING
REMARKS
It has been shown in this paper that. in contrary
endochronic
with (37).
to Fazio’s claim, the limit case of the
theory is capable of describing plastic deformation.
The reason that has led
to Fazio’s erroneous conclusion has been pointed out. The slope of the stress-strain curve is represented by the function F(x). The form of this function has been numerically investigated for the case of straining, unstraining and
I42
H.
C. WL
and C
KOMAKAKULNANAMHW
cyclic straining.
For mathematical simplicity. this calculation has been performed based on the assumption that only one internal statr variable is important. Endochronic equations for the general case using r number of internal variables have also been derived. The derivation is based on the Gibbs formulation. It has been shown that. for the limit case of k = I. the equations reduce to the van Mires yield criterion with combined isotropic-kinematic hardening and that the plastic strain rate obeys the normality rule. in agreement with results obtained by Valanis (1980) using the Helmholtz formulation.
REFERESCES Fazio. C. J. ( 1989).A one-dlmcnsional study of the limit C;LSCSof the endochronic theory. fnr. J. Solttlr S!rttcrwe.c 25,86X477. Valanis. K. C. (1971). A theory of visco-plasticity without a yield surface. .4rc+1. ,tlt~ch. 23. 517-551. Valanis. K. C. (1975). On the loundations of the endochronic theory of viscoplasticity. .Arch. Mech. 27. X57-868. Valanis. K. C. (IY,YO). Fundamental consequence of a new intrinsic time measure-plasticity as ;L limtt of the endochronic theory. :(rc.h. .I/~*&. 32. I71 IY I. Wu. H. C. and Ahoutorahi. Al. R. (IWS). Endochronic modelins of coupled volumetric~dcviatorlc lxhavior of porous and granular materials. III!. 1. Pbr.vr. 4, 163 -181. Wu. H. C . Wang. P. T.. Pan. W’. F and XII. Z. !‘. (IYYO). Cyclic stress-strain response of porous aluminum fr1r. J. Pltrsl. 6. 107 ??O Wu, H. C. and YIP. M. C. ( IYXI ). Cyclic hardening hehavwr of metallic material. J. Engng .C/trwr. Twhno/. 103. 21’m217.
Note that eqn (A?) IS (14) 111I:a/io (IYYY).
In ths caw that
E<,,./‘kkf’i(E,c-6) (A2) leads to an elastic slope as in Fazio (IYXY). &j+kPi(E,E-o)
# 0,
However,
wlthk =
I.
(A3)
I
(AJJ
in the USC of = 0.
(AZ) leads to plastic behavior which was nrglcctcd by F;i/io.
with k =
In fact. (AJ) can be rewritten
as
which dcscribcs plastic &formation as prsdwtcd by 1:wio‘s model. t‘quatwn (AS) does not provide .I good description of the enpcrlmsntal qtrcss strain rcsponsc due to the re;tson given in the beginning paragraph of this Appcndtr. More internal wkthlcs arc nccdcd in I’afio’s approxh. l’in;tlly, cqn (60) of F.wio ( 1%‘)) reads
where 0 .~nd W, are constants dclincd in that paper. The functions R, .we reLIted to the evolution of Internal wri;lhles q’. Farw was correct in pomting out that thas equation signilics ;I constraint condition placed on the cndochronw m<,del. In fact. this cquatinn is prcciscly the constitutivc equation which governs the plastic behavior. In the case that the R,F art linear in 0 and q’ as tn rqn (4). (A6) is reduced to (27). which is the condition of
Endochronic theory
143
X = + I discussedin Section 2 of this paper. While Fazio recognized the importance of the condition of X = + I in his eqn (60). he did not apply this condition in !he discussion of his eqn (24). The f sign appearing in (A6) merely distinyishes straining from unstraining. The fact that (27) correctly describes the straining. unstraining. and cyclic behavior is demonstrated in Fig. 2.
ON THE LIMIT CASE OF ENDOCHRONIC THEORY HAN C. WV and C. KOMARAKULNANARORN Departmentof Civil Engineering. The Univrrsity of Iowa. Iowa City, IA 52,742. U.S.A.
Abstract-An endochronic model is described and shown to lead to elastic as well as plastic behavior in the limit case of k = I. in contrast to Fazio’s claim that the limit case cannot lead to plasticity. The slope of the stress-strain curve is represented by a function F(X) whose functional form is numerically investigated. The study includes straining. unsttaining and cyclic straining. The equations of the general form of this model have also been derived.
I. 1NTRODUCTION
A modified endochronic theory was presented by Valanis (1980). The modification was based on an intrinsic time defined by a plastic-strain-like tensor, which in the one-dimensional case is given by
where r~and i: arc stress and strain. rcspcctively ; Q is the pl~tstic-strain-IiLc quantity; E is Young’s modulus; and X-is a constant p~tr~Irn~t~rhaving a vniuc bctwccn ircro and one. In the cast ofk = 0, the 1980 theory rcduccs to the carlicr theory ot‘ Valanis (1971). In the CilSC Of k = 1. Valanis (1s)XO)shows that the t9SO rhcory lccl to the cl>tssici~ltheory of plasticity with yield surface dcfinod anti suhjcctcd to ii combincd isotropic kincmatiu strainhardening rule. The normality condition of the plastic strain rate with rcspcct to the yield surfiloc was also dcrivcd from this theory. The plastic-strain-likr increment dQ rcduccs to the plastic strain inurement dt? when k = 1, and is given by
in the treatment of the case of k: = 1, Valanis (1980) cautioned that care must be exercised in the derivation, otherwise erroneous results might be obtained. In a recent paper, Fazio (1989). in a one-dimensional study of the limit case of the endochronic theory, claimed that, in the case of k = I. the endochronic theory could only lead to elastic behavior and that plastic behavior could not develop according to the 1980 theory. The purpose of this paper is to show that Fuzio’s claim is not valid. Furthermore, additional details for the cast of iE_= 1 are presented. The one-dimensional case is first discussed, and a discussion of the three-dimensional case follows. in this paper, explicit equations using r number of internal variables have been derived. 2. ONE-DIMENSIONAL
ENDOCHRONIC
CONSTITUTIVE
EQUATION
Following the 1985 theory, the constitutive equation in the Gibbs formulation (see Wu and Aboutorabi, 1988) is given by dr: = $ + f: Ci dc/ I- I
(3)
where d are r number of internal variables. and C, are material constants (see the Appendix SAS19:1-CI 135
H. C. WV and C. KOMARAKCLXANAKORN
1x7
for additional discussion concerning the Helmholtz and Gibbs formulation). The internal variables q’ may be interpreted as the descriptors for the internal structure of a material during plastic deformation, and these variables evolve with respect to an intrinsic time z, which is used to register the deformation history during deformation. The rate of change of the internal variables is determined by the current state of stress and internal variables. In this investigation. a linear evolution equation for the internal variables q’ is assumed such that for each internal variable q’. dy’ I - = T (C,o--F,q’) dz ,
(i not summed).
where N, and F, are material constants. In order to account for strain-hardening.
the intrinsic
(4)
time : is scaled by
where the function/represents isotropic hardening. The increment defined in terms of the plastic-strain-like quantity. so that
4 = WI
of intrinsic
time dc is
(6)
or dn d< = & dC - k E [ where d< > 0. By combining
(4) and (5) it is obtained
dr/’ =;)i
(C,CT-Ey’) ,
1
(7)
that
= s d; I
(inot
summed),
(8)
where
,I’, = 2 (C,aJ’N Therefore,
by the substitution
F,y’)
(i not summed).
(9)
of (7) and (g), eqn (3) becomes
where
(11) Equation
(IO) is further
written
as da $1
This equation
provides
+_kX] = dc[l &xl.
the basis for the discussion
to follow.
(12)
Endochroaic theory
Fazio (1989), in his eqn (24) (see the Appendix),
da
137
rewrote the above equation as
I&X EI+_kX [ 1
iii=
(13)
with no mention of the condition
IkkX#O.
(14)
The condition given by (14) should be satisfied in order for (I 3) to be valid. Fazio went on to say that when k = I. (13) reduces to
da iG= E
(15)
and has thus arrived at an erroneous conclusion that, in the case of k = I, the endochronic theory could only lead to elastic behavior and that plastic behavior could not develop. A careful observation
would lead to the result I +X
= 0, when
k = I. Therefore
Fazio, in
fact. dealt with the case of O/O. The fact that it is possible to deduce the elastic as well as the plastic behavior from the 1980 theory will now be shown. Equation (I 2) may be rewritten as (I 3) in the case of k # I, because in this case (14) is satisfied. But for the case of k = I, (I 2) should be rewritten as
Thus,
dc-%=O
When dr:-da/E stress-strain
or
1*X=0.
(17)
= 0, the behavior is elastic and generally X # + I. Hence, the slope of the curve is E. But. when I *X
= 0, i.e. X = + I. generally d&---da/E
# 0, and
the slope is not E. In fact, the condition X = + I with the help of (9) and (I I) leads to
(18) which can then be rewritten as
u = +cYr/+av
’ GF,q’
2 1-1 N,
(19)
whcrc
,$
cz
aY =
-I
( >. ,;,
Note that the first term on the right hand side of (19) represents isotropic hardening the second term the kinematic hardening. By writing (I 3) as
(20) and
138
where
F(X) =
1+x -;m. -
(22)
’
the variation of the value of function F(.Y) may now be investigated. Note that the form of the isotropic hardening functionf(z) affects the value of X. The following expression used by Wu and Yip (198 If has been used in the present computation : f(z) = A-(tl-
l)exp(-/?c).
(231
A hypothetical material with the following material constants: 1V= 3.64 MPa, C = 0.54, F = 140 MPa, A = 3.3, p = 150 and E = 28000 MPa is now considered to demonstrate the applicability of the present model (with only one internal variable for simplicity) and to show the form of the function F(X). This function has been calculated for k = 0.5, 0.95, and I, and plotted versus .Y in Fig. I. It is seen from the figure that, for the cases of k f I, the value of F(X) changes gradually with X. But. for the case of k = I. F(X) remains one in the elastic range as X increases; as soon as X reaches one, plastic ~ieform~~tion occurs and F(X) decreases as the plastic def~~rmiItion ~lccurnul~~tcswith X staying at one The step-by-step numerical proccdurcs arc : knowing dr: and the current vafucs of :, rr, and q. eqn (12) may be used to iind drr ; cyns (I), (33) itnd (6) can then be used to dctcrminc dQ and dc, and oqn (8) Icads to dq. Thus z, 6. and y can bc updated. In the case of strain-controlled cyclic loading with strain limits of + I%, the cyclic stress -strain curve is shown in Fig. 2 for this hypothetical material. The corresponding I*‘(X) versus X diagram is shown in I:ig. 3. For the citsc of k = 0.95, and similarly for k = 0.5. the followingscqucnoe is followed : ALCDAMFGALCD.. . . Note that thechange between CD and f:G is abrupt; while that between AC and AF is gradual. In the cxx of k = I, the following sequence is foflowud : ABCBAEFEABC. _. . The change from B to C and from E to F is gradual; while from C to B and F to E is abrupt. The corresponding plot for F(X) versus intrinsic time z is shown in Fig. 4. The letters denote c~rrespondjn~ points in these figures.
2” I
X Fig. 1. Function F(.t’) frtr varinus ~cmst~~ntI;.
Endochronic
thuory
139
-25
-60 -1.2
-0.8
-0.4
straPn.
0.4
0.1
1.2
K.
Fig. 2. Onedimensional
cyclic stress-strain
curve for a hypothetical
material.
k = 0.95.
k=l --k - 0.95 k-_- - 0.5 1.6
1
R
z
0.5
L 0
C I-
-1
1.6
-0.6
Fig. 3. Function
3. THE
ii
F(X)
0.1
,
1.b
for cyclic straining.
THREE-DIMENSIONAL
EQUATIONS
In this paper, the material is assumed to be plastically incompressible. For a discussion of material subjected to volumetric plastic deformation, the reader is referred to Wu and Aboutorabi (1988). For plastically incompressible materials, it suffices to consider only the deviatoric response. In the deviatoric response, the internal variables are p”, with n = I to r. The deviatoric strain increment de is expressed in terms of the increments of deviatoric stress ds and internal variables dp”, i.e. d.q de,, = c+
0
’ c c”dp:,, n- I
(24)
where C” are material constants and 11”is the shear modulus. The evolution of p” is defined
H. C. Wu
and C. KOMARAKL-LNXGAKORS
1.2 , / (A 0.0 -
z ii:
0.3
0.3 -
L
Fig. 4. F(x) versus z for cyclic
with respect to intrinsic
straining.
k = 0.95.
time 2. By assuming a linear evolution equation, the rate ofchangc
of’ p” is given by
whore IV”, F” are material constants. A deviatoric strain-like
dQ,, = d<,,,-k
tensor Q is now dcfincd by
d.s,, kI
where k is again ;L constant parameter such that 0 < k < I. Note that when k = I, Q is equal to the plastic strain c*. An intrinsic
time increment d< is now dcfincd as
di’ = dQ,,
de,,,
(27)
which is related to z through the relation given by (5). It will be shown subsequently that functionj’represents
isotropic
hardening.
In a strain-controlled
test, de,, is a known and
input quantity. Rewriting
eqn (24) as
,
this equation together with eqn (25) may be substituted
dQ,, = ( -k)
(28)
into eqn (26) to yield
de,, + k i C” dp;, n- I
(2%
or
dQ,, = zt,+ fli, dwhere
(30)
En&chronic
theory
141
a,, = (I -k)de,,
(31)
(32)
Finally,
by substituting
eqns (5) and (30) into (27), the following
quadratic
equation
is
obtained : Pd?+Qdc+R
=0
(33)
where
’ =
BtjBi,-f’
(34)
Q = h,Btj
(35)
R = a,,a,j.
(36)
The functions P, Q and R are defined in terms of the constants of the model and increments of input variables. When dc is solved in eqn (33). eqn (25) can be used to obtain dp:,. By substitution. the deviatoric stress increment ds,, can then be found from eqn (28). In the case of Xr = I. eqns (31). (35) and (36) give a,! = R = Q = 0. Thus, eqn (33) is reduced to Pd? = 0. The condition d: = 0 is associated with the elastic state. while the condition
d: # 0 is associated with the plastic state. Hence, in the plastic state, P must be
equal to zero and eqn (34) lcads to
h --r,,)(%,-r,j) = (SJ’)
(37)
where
(38)
and
(39)
Equation (37) is the von Mises yield criterion with combined isotropic-kinematic in which
f describes
hardening,
the isotropic hardening and r,, the kinematic hardening in thedcviatoric
behavior. S,, is the initial yield stress. From eqns (30). (32), (38) and (39). the plastic strain increment is now given by dc = dQv = fi,j
$ (So -r,/) J
(40)
which is normal to the deviatoric yield surface as is seen by comparing
4. CONCLUDING
REMARKS
It has been shown in this paper that. in contrary
endochronic
with (37).
to Fazio’s claim, the limit case of the
theory is capable of describing plastic deformation.
The reason that has led
to Fazio’s erroneous conclusion has been pointed out. The slope of the stress-strain curve is represented by the function F(x). The form of this function has been numerically investigated for the case of straining, unstraining and
I42
H.
C. WL
and C
KOMAKAKULNANAMHW
cyclic straining.
For mathematical simplicity. this calculation has been performed based on the assumption that only one internal statr variable is important. Endochronic equations for the general case using r number of internal variables have also been derived. The derivation is based on the Gibbs formulation. It has been shown that. for the limit case of k = I. the equations reduce to the van Mires yield criterion with combined isotropic-kinematic hardening and that the plastic strain rate obeys the normality rule. in agreement with results obtained by Valanis (1980) using the Helmholtz formulation.
REFERESCES Fazio. C. J. ( 1989).A one-dlmcnsional study of the limit C;LSCSof the endochronic theory. fnr. J. Solttlr S!rttcrwe.c 25,86X477. Valanis. K. C. (1971). A theory of visco-plasticity without a yield surface. .4rc+1. ,tlt~ch. 23. 517-551. Valanis. K. C. (1975). On the loundations of the endochronic theory of viscoplasticity. .Arch. Mech. 27. X57-868. Valanis. K. C. (IY,YO). Fundamental consequence of a new intrinsic time measure-plasticity as ;L limtt of the endochronic theory. :(rc.h. .I/~*&. 32. I71 IY I. Wu. H. C. and Ahoutorahi. Al. R. (IWS). Endochronic modelins of coupled volumetric~dcviatorlc lxhavior of porous and granular materials. III!. 1. Pbr.vr. 4, 163 -181. Wu. H. C . Wang. P. T.. Pan. W’. F and XII. Z. !‘. (IYYO). Cyclic stress-strain response of porous aluminum fr1r. J. Pltrsl. 6. 107 ??O Wu, H. C. and YIP. M. C. ( IYXI ). Cyclic hardening hehavwr of metallic material. J. Engng .C/trwr. Twhno/. 103. 21’m217.
Note that eqn (A?) IS (14) 111I:a/io (IYYY).
In ths caw that
E<,,./‘kkf’i(E,c-6) (A2) leads to an elastic slope as in Fazio (IYXY). &j+kPi(E,E-o)
# 0,
However,
wlthk =
I.
(A3)
I
(AJJ
in the USC of = 0.
(AZ) leads to plastic behavior which was nrglcctcd by F;i/io.
with k =
In fact. (AJ) can be rewritten
as
which dcscribcs plastic &formation as prsdwtcd by 1:wio‘s model. t‘quatwn (AS) does not provide .I good description of the enpcrlmsntal qtrcss strain rcsponsc due to the re;tson given in the beginning paragraph of this Appcndtr. More internal wkthlcs arc nccdcd in I’afio’s approxh. l’in;tlly, cqn (60) of F.wio ( 1%‘)) reads
where 0 .~nd W, are constants dclincd in that paper. The functions R, .we reLIted to the evolution of Internal wri;lhles q’. Farw was correct in pomting out that thas equation signilics ;I constraint condition placed on the cndochronw m<,del. In fact. this cquatinn is prcciscly the constitutivc equation which governs the plastic behavior. In the case that the R,F art linear in 0 and q’ as tn rqn (4). (A6) is reduced to (27). which is the condition of
Endochronic theory
143
X = + I discussedin Section 2 of this paper. While Fazio recognized the importance of the condition of X = + I in his eqn (60). he did not apply this condition in !he discussion of his eqn (24). The f sign appearing in (A6) merely distinyishes straining from unstraining. The fact that (27) correctly describes the straining. unstraining. and cyclic behavior is demonstrated in Fig. 2.