Int. J. NonLmear
Mechonrcs,
Vol. 6, pp. 435450
A FINITE
STRAIN
Pergamon Press 1971. Printed in Great Britain
THEORY FOR ELASTICPLASTIC DEFORMATION
J. B. HADDOW and T. M. HRUDEY University of Alberta, Edmonton, Canada
AbstractAn elasticplastic theory that is applicable when the elastic part of the strain is finite is proposed. A flow rule for an incompressible solid is obtained from Drucker’s postulate 111.Isothermal simple shear of a material which is neoHookean both before yielding and during elastic unloading after yielding is considered as an application of the theory. The problem is solved for two yield conditions and associated flow rules.
1. INTRODUCTION CLASSICAL elasticplastic
theory is largely based on the assumption that all deformations are small and that the strain is described by the infinitesimal approximation [2], although it has been applied to some problems that involve finite elasticplastic deformation with the elastic part of the strain infinitesimal, for example the expansion of a spherical cavity in an infinite medium [3]. Clearly, application of the classical theory is restricted to materials for which the shear yield stress k is very small compared with the modulus of rigidity p. For most metals k/p is of the order of magnitude of 10e3, consequently the elastic shear strain during plastic flow is considered infinitesimal. However, if the negative hydrostatic part of the stress is not negligible compared with the bulk modulus, the dilatational elastic strain is finite since it has been found experimentally that hydrostatic stress does not produce plastic flow. Another situation that involves finite elastic strain during plastic flow, is the plastic deformation of a material that yields after finite elastic shear strain. Whiskers of certain nonmetallic crystalline materials have been found to behave elastically at tensile strains up to 4 per cent, and whiskers of tin have been bent elastically to a strain of 2 per cent [4]. These elastic strains are too large for the infinitesimal approximation to be satisfactory, and classical elasticplastic theory is not strictly applicable to consideration of any elasticplastic flow that occurs between the elastic limit and fracture. Some plastics and elastomers undergo permanent set after finite elastic shear deformation, and a finite strain elasticplastic theory may be applicable to such materials for certain ranges of temperature and deformation rate. The purpose of this paper is to present a theory for elasticplastic flow when the elastic part of the strain is finite. The kinematic and thermoelastic theory is valid for finite elastic shear and dilatational strains, and for both perfectly plastic and workhardening solids that may be plastically anisotropic but the yield conditions discussed and the problem which is solved are for an incompressible perfectly plastic solid. An extension of the theory to consider elastically anisotropic solids is possible but is not considered here. Lee [5] has proposed a finite deformation elasticplastic theory which uses a different approach from the theory presented here although both theories are based on three 435
436
J. B. HADDOW and T. M. HRUDEY
configurations of the body. Green and Naghdi [6] have also proposed an elasticplastic theory and this theory appears to be fundamentally different from that in this paper. Certain of the assumptions of classical elasticplastic theory are retained in this paper, namely that there is no plastic volume change. and that the elastic constants are unchanged ‘myprevious plastic straining. 2. KINEMATICS
Three configurations of an initially homogeneous body. the initial unstrained and unstressed configuration at the uniform reference temperature, the unstressed plastically strained configuration at the reference temperature and finally the stressed elasticplastic configuration. are henceforth called C.l. C.2, and C.3 respectively. C.2 differs from C.l by at most a rigid body displacement if no plastic deformation has occurred. The concept of the unstressed C.2 at uniform temperature was introduced by Lee [5] and the use of three configurations to consider plastic deformation has also been suggested by Backman [7] and Sedov [8]. If nonuniform elasticplastic deformation has occurred, in general. residual stresses will remain after the body has been unloaded and returned to the reference temperature. C.2 is then obtained by imagining every infinitesimal element of the body to be isolated from the surrounding material when unloaded from the elasticplastic state so that it is stress free. A distribution of residual plastic strain is then obtained which, in general, does not satisfy compatibility relations and the space corresponding to C.2 is nonEuclidean. Let the coordinates of a particle with respect to a fixed set of rectangular Cartesian axes be X1, xcr. xi for C.l. C.2, and C.3 respectively. Upper case Latin subscripts refer to Cl, lower case Greek subscripts to C.2 and lower case Latin subscripts to C.3. The deformation gradient for the elasticplastic deformation of C.3 is
(1) If the space corresponding to C.2 is Euclidean the deformation C.2 and C.2 to C.3 are the plastic deformation gradient F(P) = % &I ?X, and the thermoelastic
deformation
gradients
relating
C.l to
(2)
gradient
Also F,, = Fiz’ F(“) ar
(4)
If the space corresponding to C.2 is nonEuclidean the displacement gradients Fi!,) and Fjz’ cannot be expressed as partial derivatives as in equations (2) and (3) and the Pfaffian forms F$)dX, and Fi”,)dXmare not exact differentials, but equations (1) and (4) are still valid. To simplify the presentation, the remainder of this paper is restricted to consideration of uniform plastic deformation and temperature distribution. consequently the space corresponding to C.2 is Euclidean and C.2 and C.3 are uniform. If this restriction is removed and the space corresponding to C.2 is nonEuclidean, the results obtained in this section are
Strain theory for elasticplastic
437
deformation
valid provided velocity and deformation gradients that involve partial derivatives. Since the material is assumed to be plastically incompressible,
C.2 are not expressed
(5)
[1*
d& e ?X,
as
=1
and
where pO is the density corresponding to C. 1 and C.2 and p is the density corresponding to C.3. The (Lagrangian) strain tensors. referred to C.l, for the deformations of C.2 and C.3 are the plastic strain
(6) and the total strain 1
(7)
E,,y
respectively. Material differentiation, of equation (7) gives
that is differentiation
with respect to time? with x1 held constant,
where
is the rate of deformation tensor and ui = .ti. A superimposed tiation. Material differentiation of equation (6) gives
dot denotes
material
differen
where
d$‘=;($+$) is the plastic rate of deformation
(9)
tensor and v, = 2,. Since there is no plastic volume change d(p) aa = 0 .
7 Since elasticplastic deformation is inviscid, monotically varyingparameterifinertiaeffectsare
material differentiation neglected.
may be with respect
to any sutable
HADDOW
438
J. B.
This relation can be obtained referred to C.2 is
from equation
and material
differentiation
and T. M.
and substitution
HRUDEY
(5). The elastic
of equation
(Lagrangian)
strain
(8) gives the elastic
tensor
strain
rate
(11) C.2 is defined
only to within a rigid body rotation,
consequently
the plastic spin
is not uniquely defined. Since I?$ given by equation (11) depends on the plastic spin W$ it is not a satisfactory elastic strain rate. For example if dij = 0 and d$ = 0 but W$ # 0 then I$$ # 0 although neither elastic nor plastic deformation is occurring. An elastic strain rate tensor which is independent of W$ and vanishes if dij = 0 and dhP,’= 0 is obtained as follows. The plastic deformation gradient (2) may be expressed in the polar form.
where Rpd is a proper such that
orthogonal
tensor
and
[email protected] IS . a symmetric
The components of elastic strain given by equation of C.2 but the elastic strain components
(10) depend
positive
definite
tensor
on the rigid body rotation
EgL = R$ Rf” ,E$
(12)
are unchanged by rigid body rotation of C.2. Material differentiation of (12) gives
where
is an antisymmetric
tensor and _Q$ zz &$ +
[email protected]! zy E!$ 91 W!$  W(P)
is the corotational
derivative
of E$’ and is independent
of the spin of C.2.
(14)
Struin theory for elusticplustic
Using equations
(8). (9) and (11) equation
439
deform&on
(14) becomes
(15) The elastic rate of deformation
tensor is defined
as (16)
andsubstituting(l5)gives d!” = d..1, _ d?. 11’ 1, where (17) The components of d,*ido not depend on the rotation of C.2, nor does dt”jdepend on the spin flow with no of either C.2 or C.3. Also d$ = 0 if ,!?$1 = 0 and d$ = 0 for elasticplastic plastic volume change. These properties indicate that d; is a suitable definition of plastic rate of deformation. However it is shown in the next section that if the solid is elastically anisotropic the definitions of elastic and plastic rates of deformation given by equations (16) and (17) are not suitable.
3. THERMOELASTIC
It is convenient given by
to introduce
CONSIDERATIONS
the PiolaKirchhoff
where aEj is the true stress referred depend on the rigid body rotation
stress tensor
S,, referred
to C.2 and
to C.3. The components of stress given by equation of C.2, but the stress components s,,
= R$jj $2
s,,
are unchanged by rigid body rotation of C.2. It is assumed that the elastic constants and heat capacity are unchanged plastic flow, consequently there is an elastic free energy function unit mass,
[email protected]) =
[email protected]) (E&
T).
(18)
(19) by previous
(20)
that is independent of previous plastic straining and the free energy is the sum of two parts, A’“’ and a part which is a function of the plastic strain history and perhaps T but not Egi. For an elastically isotropic solid
[email protected]) is a function of the invariants of EEL and T and it may then be represented by
[email protected]) = ,J’“‘(Eb’B, 7) since Rzd is a proper
orthogonal
tensor.
J. B. HADDOW and T. M. HKUDEY
440
If unloading elasticity that
from an elasticplastic
state is considered
s,,
it follows
from the theory
= P”;;
of
(21) KL
and
(22) where SC” is the elastic part of the entropy. may be replaced by
From
For an elastically
isotropic
solid equation
(21)
(20)
and substituting
equations
(21) and (22) gives
[email protected]’ = S,,_&po
 Se’ i:
(23)
When the solid is elastically isotropic the tensors Ej$ and S,, are coaxial and E$ and S,, are coaxial consequently from (13). (14) and (19). s,, and equation
l!& = s,, E$’ = s,, ;Ij$’
(24)
(23) becomes
[email protected]’ = s,, rj$j /p,  Se’ ,i.
For isothermal
deformation
it follows from equation
(23) that
$e’ = /+” where &’ is the rate of increase of elastic,strain this may be expressed as
= S
KL
$e’
energy/unit
KLIPO
mass. If the solid is elastically
[email protected]’ = 0 ,J.d!“/ I, ! P by using equation
(16). (18) and (24). The total rate of stress work/unit
isotropic (25)
mass is
l+” .+ tip’ = aijdij/p, where tip’ = Oijd&‘p is the rate of plastic energy dissipation/unit
mass.
(26)
Strain theory for elsticphstic
deform&on
If the solid is elastically anistropic the rate of increase of strain energy plastic energy dissipation are no longer given by equations (25) and (26).
4. YIELD
CONDITION
Perfectly plastic solids are considered tesimal. as in the classical elasticplastic function
AND FLOW
441
and the rate of
RULE
in this section. If strains and rotations are infinitheory. the yield condition may be described by a f(0ij.T)
= 0
(27)
and the sign of f is chosen so that f < 0 represents the elastic domain. If the solid is plastically isotropic f is a function of the stress invariants. consequently it is independent of the rotation of the body and the yield condition may be represented by an equation of the form (27) when large strain and rotation occurs. However, if the solid is plastically anisotropic the form of the function f depends on the rigid body rotation of C.3, consequently equation (27) is not a satisfactory representation of the yield condition when deformation with finite rotation occurs. This difficulty may be avoided by referring the true stress to axes fixed in the material. The deformation gradient (1) may be expressed in the polar form 2.K ’ = Ri, u,,, ?XK where U,,
is a symmetric
positive
definite
tensor
such that
and Ri, is a proper orthogonal second order tensor which represents the rigid body rotation that follows the irrotational deformation described by ULK. The yield condition for an anisotropic solid may then be described by f
bKL?
T) = 0,
(28)
where gKL = RiKRjL~ij. and this form is independent of the rigid body rotation of C.3. This procedure was suggested by Green and Naghdi [6]. The components of the rate of deformation tensors may also be referred to axes fixed in the material,
and since Ri, is an orthogonal
tensor
o,,[d,,.
d;ZiL.dly”L] = oij[dij, d$.d$)].
The plastic potential flow rule of classical plasticity may be obtained from Drucker’s postulate [l] which states that if a body is in an equilibrium state under an arbitrary set of body forces and surface tractions, and an external agency applies and removes an additional load under isothermal conditions, the work done by the external agency is positive during the loading and nonnegative during the complete cycle of application and removal of the additional load. A plastic flow rule can be obtained from this postulate, for an incompressible elastically isotropic solid that undergoes finite elastic deformation, by modifying slightly the reasoning used to obtain the classical flow rule. Consider the body to be at
J. B. HADDOW and T. M. HRUDEY
442
temperature T and stress G& which is just within the yield surface in stress space, so that when the stress point moves to a stress G;(~ which is on the yield surface, as shown diagrammatically in Fig. 1. the additional elastic strain increment is infinitesimal. Let the external agency be such as to move the stress point isothermally from C& at time 0 to aZL at time t and then along the yield surface as shown in Fig. 1 to oLL at time t + 6t. Removal
FIG. 1. Diagrammatic representation of isothermal yield surface in stress space.
of the external agency returns the stress point to a& at time t*. The cycle ‘of loading and unloading of the external agency is such as to produce infinitesimal increments of strain so that changes in the configuration of the body can be neglected during the cycle of loading and unloading by the external agency. The total work done/unit volume by both the external agency and the original loading during the cycle is W =
i a,,d,,dz
+ ‘;f
o,,d,,dz
+
[
tibt 1. i
o,,d,,dr
Then. since d KL = d$$ + d& and d&, # 0 only from t to t + 6t.
f+dt
= where the symbol $ denotes integration ments during the cycle are infinitesimal
$a,,dj$.dr
+
;c o,,d&dr
1 ,
(29)
around the complete cycle. Since the strain increthe elastic term in (29) is zero and
A part
of W is due to the original loading. From Drucker’s postulate ww;30
&ruin theory for elusticpbstic
deformution
443
consequently
and lim
w 
__~_
df0
W”
=
6t
(4~

&)d&,
3 0.
According to equation (30) the yield surface must be convex and the vector in stress space that represents the diL must be parallel to the normal to the yield surface at the corresponding stress point. Thus
where 3, is a nonnegative factor of proportionality. rule may be expressed in the form
When
d?F = & II
aqj
the solid is isotropic
the flow
(31)
The condition d& = 0 and the flow rule imply that the yield condition does not involve the hydrostatic part of the stress tensor. Two yield conditions are now considered for isothermal deformation of a material whose elastic behaviour is neoHookean. The strain energy function? for such a material is W = 2CEk’,
(32)
where C is an elastic constant and 2C is equivalent to the classical shear modulus for small deformation. If yielding occurs when the true octahedral shearing stress reaches a critical value the von Mises yield condition +aijgij
= k2,
Or
f = Oijaij  2k2 = 0,
(33)
where alj = gij  (crkk/3)hij is the deviatoric part of the stress and k is the shearing stress which acting alone would produce yielding, is valid as in classical plasticity. In classical plasticity another interpretation of the Mises yield condition is that it implies that yielding begins when the shear strain energy reaches a critical value. If yielding of a neoHookean solid begins when the shear strain energy/unit volume reaches a critical value W* then the yield condition may be expressed in the form w = w*,
(34)
where the strain energy must be expressed as function of cij. It is more convenient to express this yield condition in a different form. Since the neoHookean solid is incompressible and t For the incompressible rather
than per unit mass.
neoHookean
solid the symbol
Wis used here to denote strain energy per unit volume
J. B. HADDOW and T. M. HRUDEY
444
isotropic. the maximum shear strain stress deviator. The relation
energy yield condition
involves
the invariants
of the
where J, = aij&/(8C2),
J, = &,~?,~5~~/(24C~)
and Icp) = 2E”’ aa + 3 * is easily obtained for a neoHookean solid. Using this relation and equations (32) and (34) the maximum shear strain energy yield condition may be expressed in the form
(35) It is easily verified that f < 0 if W < W*. For a solid that yields after infinitesimal strain
elastic
W N aijoij/(8C) and W* < C. Equation
(35) then becomes 35,  3 7
and becomes
equivalent
+ 0( W*2/C2)
to the Mises yield condition
5. SIMPLE
SHEAR
= 0
when terms 0( W*2/C2) are neglected.
PROBLEM
Simple shear of a cuboid is considered. The sides in the undeformed C.l are parallel to the axes of a fixed rectangular Cartesian coordinate system. After deformation the position of a particle originally at X, is given by x1
=x, + KX,.
x2
=x2,
x3 =x3.
This is a homogeneous isochoric deformation and the parameter K = tan 0, where 8 is the angle of shear as shown in Fig. 2, is used as a time scale. The deformation gradient and its material derivative with respect to K are
Strain theory for ehsticplustic
It is assumed that the deformation Hookean elastic behavior so that
is isothermal
aij = 2C[;$’
deform&on
and
the solid
445
is isotropic
with neo
\F$)  3 (\
[email protected] FQ)&j].
(36)
The problem is solved for both the Mises yield condition and the maximum shear strain energy condition, and the associated plastic flow rules are obtained from equation (31).
FIG. 2. Simple shear ofcuboid.
A significant simplification in the analysis tensor is expressed in the polar form
results
if the elastic
deformation
F+” = l,‘/!:)R!“’ 111 LJ P’
gradient
(37)
where P$’ is a proper orthogonal tensor which represents the rigid body rotation relative to C.2 and Vi;’ is a symmetric positive definite tensor such that
of C.3
l$’ @? = F!“’ F(e), la ,a I
Substituting
(37) in (36) gives oij = 2C{ V$’ V8)  4 (YYi Vi5$Jj>
which shows that the principal
axes of Oij and Vjy’ coincide.
Substituting
(38) (37) in (17) gives
d; = f[ l$’ d& Vyj  1 + vj, d&j vi”,  ‘1,
(39)
where
[email protected] =
RIP,' R'f; d$/PB.
(40)
For the isotropic yield conditions considered, flow rule (31) indicates that the principal axes of d$ and ~ij coincide, consequently for the special case when the solid is both elastically and plastically isotropic, the principal axes of d,*jand ViJ’ coincide and it can then be deduced from (39) that diy’ also has the same principal axes and that d?. = d!P’ EJ 13. The
(41)
equation [d$‘]
= +[V(“‘ l+)
+ (V (e)  $7(@)f _ V”’  1&F  lV(‘4 _ (V”’  1i;F  lV(e))f] ,
(42)
J. B. HADDOW and T. M. HRUDEY
446
which is expressed in matrix notation, with a prime denoting from equations (39), (40). (41) and the relations
the transpose,
can be obtained
and
Matrix notation is used henceforth in this section. (31) and equations (38) and (42) give V(e) +e)
+ (V”‘ IV’@‘) = V&‘ 1kF Ijib’) + (V”‘
For the Mises yield condition, ‘i;F
flow rule
IV”‘)
 4XC[V’“‘2  f fr(VCeJ2)I], matrix and tr denotes
where I is the identity
(43)
the trace of the matrix argument.
1.0 Y
0.8
’
z ;
0.6
5
FJ0.4 z E 0.2 z 0
0.2
L
0
0.1
0.2
0.3
0.5
0.4
0.6
0.7
0.8
09
10
K FIG. 3. Stress deviatorK
Expressing gives
the yield condition
curves.
in terms
Mises yield condition
and k:C = 0.1.
of V”’ and differentiating
ry(V(@3V(C))_ ftr(V(e)‘)tr(V(r)\i(~))
= 0.
with respect
to K (44)
When K < K,, where K, = {  3 + $9
+ 3(k2/C2)]+}+
is the elastic limit value of K for the Mises yield condition, the stress deviator is obtained from equation (36). When K > K,, the system of nonlinear ordinary differential equations (43) and (44), must be solved for I/@) and x as functions of K using the known value of I’@) at the elastic limit K, as an initial condition. The stress deviator as a function of K is then obtained from equation (38).
Struin theory for elusticplustic
deformution
447
1.0
0.8 2 g
2 a
0.6
0.4
3 E 0.2 2 z z
0
 0.2
0.4
0
0.1
0.2
0.4
0.3
0.5
K FIG. 4. Mises yield condition
For the maximum
shear strain
energy
cdg’] = i, [lz and substituting v”‘
[email protected])
+
equations
(V(e)
1+9)’
0.6
0.7
0.9
1.0
and k/C = 0.5.
yield condition
(w* c
0.8
the plastic flow rule (31) gives
+
(38) and (42) gives _
V”‘
‘kF
IV(e)
_
(V”‘
1
@F
‘V’“‘)
0.8
= E a
0.6
0.4
> ;
0.2
2 ii
0
z 0.2
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
K FIG. 5. Mises yield condition
and k/C = 0.1.
1.6
1.8
2.0
J. B. HADDOW and T. M. HKUDEY
448
2 E z
0.2
0 0.2
I I
’
0
‘_f
b,z/k=.?,,,/k
01
02
03
04
05
06
07
0.8
yield
condition
K FIG.
6.
Maximum
shear strain energy k/C = 0.1.
Expressing the maximum shear strain energy yield condition tiating with respect to K gives tr(VW3+e))
_
~t,(vce)2)tr(vcp’irce,)
W* c + 3  ‘{ 3tr(V(45V(9
(
in terms of I/@) and differen
+ [+trqV’y

,~v(e’“)ltr(v(c)irc~))
1

2tr(Vce)2)tr(Vce)3~ce))j = 0.
(46)
Equations (45) and (46) must be solved simultaneously to determine I/@) as a function of K after yielding, and the stress deviator is then obtained from equation (38). Numerical solutions were obtained for both yield conditions, and the results are shown graphically in Figs. 3, 4, 5, 6, 7, and 8. The stress, k, used to nondimensionalize the stress deviator for the maximum shear strain energy yield condition material, is the shear yield stress of the Mises material with same elastic constant C and the same uniaxial yield stress. 1.0
1
0.8
.
W./C = 0.0566
DC
0
0.6
2 5
w 0
0.4
2 0.2 kY zl 0 02 t 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
K FIG. 7. Maximum
shear strain
energy
yield condition
k/C
= 0.5.
Struin theory
for elusticplastic
deformation
449
The numerical results indicate that as k/C becomes smaller the normal stresses become smaller and become negligible for k/C g 1. Also for k/C G 1 the results approach those obtained from the classical theory.
0
0.2
0.4
FIG. 8. Maximum
0.6
0.8
1.0
1.2
1.4
K shear strain energy yield condition
1.6 k/C
1.8
2.0
= 1.0.
For the Mises yield condition the deviator component 5 33 approaches zero for values of K large compared with K, and if gZ3 = 0 then @I 1 =  oz2 = O11 for large K. The maximum shear strain energy yield condition results in quite different normal stress effects during elasticplastic flow. The relation 5 22 = a,, which holds for elastic deformation before yielding also holds for the elasticplastic deformation for all K and if CJ~~= 0, oZ2 = Oanda,, =  3~7~~.
REFERENCES approach to plastic stressstrain relations. Proc. First U.S. Nutionul Ul D. C. DRUCKER, A more fundamental Congress of Applied Mechunics, 487 (1952). I21 W. T. KOITER, General theorems for elsticplastic solids, Progress in Solid Mechunics, ed. R. HILL and I. N. SNEDDON, Vol. 1. North Holland, Amsterdam (1960). Theory of Plasticity, Oxford University Press (1950). [31 R. HILL, Muthemuticul [41J. E. GORDON, The New Science of Strong Muteriuls, Penguin Books (1968). deformation at finite strains, J. uppl. Mech. 36, 1 (1969). I51E. H. LEE, Elasticplastic continuum, Archs rution. Mech. b1 A. E. GREEN and P. M. NAGHDI, A general theory of an elasticplastic Anulysis 18. 251 (1965). [71M. E. BACKMAN, Form for the relation between stress and finite elastic and plastic strains under impulsive loading, J. uppl. Phys. 35,2524(1964). (1955). PI L. I. SEDOV, Introduction to the Mechanics of u Continuous Medium, AddisonWesley (Received 24 Februury 1970 ; revised 1 June 1970)
R&urn&On propose une theorie elastoplastique qui est applicable lorsque la partie Clastique de la deformation est finie. On obtient une loi d’ecoulement pour un solide incompressible a partir du postulat de Drucker [I]. On examine, comme application de la theorie, le cisaillement simple isotherme d’un materiau qui est neoHookeen a la fois avant la limite tlastique et durant la decharge Clastique apres la deformation plastique. Le probleme est resolu pour deux conditions de limite Clastique et les lois d’koulement assocites.
450
J.B.
HADDOW
andT.M.
HRUDEY
ZnsammenfassungEine elastischplastische Theorie, die angewandt werden kann, wenn der elastische Teil der Deformation endlich ist, wird entwickelt. Ausgehend von Druckers Postulat [l] wird eine Regel fiir das Fliessen eines inkompressiblen Festkiirpers erhalten. Die isotherme einfache Scherung eines Materials. das sich sowohl vor dem Fliessen als such wahrend der Entlastung nach dem Fliessen ,,neoHookisch” verhalt, wird als eine Anwendung der Theorie erortert. Das Problem wird fur zwei Fliessbedingungen und damit verbundene Fliessregeln gel&t.
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