A thermodynamic foundation using the concept of internal state variables is presented for the kinematic description of a viscoplastic material. Three different evolution equations for the back stress are considered. The first is that of classical, nonlinear, kinematic hardening. The other two include a contribution that is linear in stress rate. Choosing an appropriate change in variables can remove this stress rate dependence. As a result, one of these two models is shown to be equivalent to the classical, kinematic hardening model; while the other is a new model, one which seems to have favorable characteristics for representing ratchetting behavior. All three models are thermodynamically admissible

Chaboche, J.-L.

Freed, Alan D.

Walker, K. P.

English

NASA Technical Reports Server

//v-39
NASA Technical Memoran_d_um 103794
=__
On the Thermodynamics of Stress
Rate in the Evolution of Back
Stress in Viscoplasticity
A.D. Freed
Lewis Research center
Cleveland, Ohio
J.-L. Chaboche .....
Office National d'Etudes et_de RecherchesA_rospatiales .....
Ch_ttillon, Cedex, France
and
K.P. Walker
Engineering Science Software, bw.
Smithfield, Rhode Island .....
Prepared for the
Third International Symposium on P!asticjty
sponsored by the Institute de M_canique de GrEnoble
Grenoble, France, August 12-16, I991
m
(NASA-TU-IO_794) ON THE THERMODYNAMICS OF
STRFSS RATE IN THE FVOLUTION OF BACK STRESS
IN VISCOPLASTICITY (NASA) 8 p CSCL 20K
_ j _7 :
Ngl-1947b .....
Uncl_s
G3/39 OOOIoB3
https://ntrs.nasa.gov/search.jsp?R=19910010163 2020-03-19T18:30:04+00:00Z

On the Thermodynamics of Stress Rate
in the Evolution of Back Stress in Viscoplasticity
A. D. Freed
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, OH 44135, USA
J.-L. Chaboche
Office National d'Etudes et de Recherches A_rospatiales
92322 Ch£tillon, Cedex, France
K. P. Walker
Engineering Science Software, Inc.
Smithfield, RI 02917, USA
Abstract
A thermodynamic foundation using the concept of internal state variables is presented for the
kinematic description of a viscoplastic material. Three different evolution equations for the
back stress are considered. The first is that of classical, nonlinear, kinematic hardening. The
other two include a contribution that is linear in stress rate. Choosing an appropriate change in
variables can remove this stress rate dependence. As a result, one of these two models is shown
to be equivalent to the classical, nonlinear, kinematic hardening model; while the other is a new
model--one which seems to have favorable characteristics for representing ratchetting behavior.
All three models are thermodynamically admissible.
Let us consider a complementary (Gibbs) free energy for the thermodynamic potential function,
i.e.
¢ = gJ[T,a_,,fluv] , flii= 0, (1)
where square brackets [.] are used to denote 'function of'. The temperature T and stress _rij
are external variables whose variations can, in principle, be controlled by an observer. However,
the evolution of the internal state variable flij--associated with kinematic hardening--cannot be
controlled by an observer; it is a material response. This variable must therefore evolve as a function
of state. Furthermore, if this kinematic variable is to remain invariant under transformation to
another kinematic variable, then its evolution will, in general, also depend upon the rates of change
in the external variables [1]. For simplicity, let us consider the following equation for the evolution
of internal state, i.e.
/)ij = xo[T,S,,,,,/_,,,] + (g/2H),_ij, (2)
where Sij = ffij -- 1/3 ffkkt_ij is the deviatoric stress with _ij representing the Kronecker delta, and
where N and H are material constants. In general, a term that is linear in T may also be introduced
into (2) [2], but it is ignored in this paper for the purpose of simplification. Notice that flij, and
therefore X.ij, are deviatoric by definition. The function :_ij describes the irreversible contribution
to the evolution of glj.
Given (1) and (2), one can derive [1, 2] relationships that govern the entropy,
-O9
s- OT ' (3)
the strain,
-09 N O_
- _ =0 (4)
eij Oaij 2H O_ij _- e_j , eii ,
and the intrinsic dissipation,
_/f_ _:,i > 0, (5)O_j
where Q_ is the plastic strain, which is deviatoric. It evolves according to a separate evolution
equation (13).
If we assume that the complementary free energy (1) is given by
thermoelastic contribution
- 4--_ - 18---_°iiajj -- OtAT 6rii -- SoAT - C T - To 1
+ tt_i_j + N _ SijS_j - _#&j + A[S,.,,_] ,
viscoplastic contribution
+
(6)
then we obtain--from (3) and (4)--the constitutive equations for an isotropic Hookean material ;
they are:
AS = a aii+ (C/T)AT , (7)
ai_ = 3_ (e. - aAT&i) , (8)
and
provided that
Si, = 2# (Eij - Q_.) , (9)
OA -N OA
- (10)
OSij 2H O_i.i "
Here/_ and a are the elastic shear and bulk moduli, a is the coefficient of thermal expansion, C is
the specific heat, So and To are the initial values of entropy and temperature with AS = S - So
and AT = T - To, and Eij = eij- 1/3emdflj is the deviatoric strain. The first three terms in
the viscoplastic contribution to • are introduced to remove unwanted cross products that would
otherwise appear in (9) because of (4) [2]. One also obtains
O/3ij - 2H lJij _-_Sij = 2H Xij O-_ij ' (11)
which defines the thermodynamic force conjugate to _ij. As a consequence, (5) becomes
( 0n)aijgi_ - 2H Xij _ fCij >_ O, (12)
which describesthe intrinsic dissipationpropertiesof our material. The function A, which is
constrained by (10), is introduced into _ to affect the intrinsic dissipation (12); its form is model
dependent.
For the evolution of plastic strain, we shall consider that
where the back stress,
Sij - Bij (13)
Bij = 2Hflij = 2Hxij + NSij , (14)
accounts for kinematic behavior, with H > 0 being its modulus and 0 _< N < 1. The norms used are
defined by III II= ¢1/2 IijIij where Iij is any deviatoric stress-like quantity, and by HJ II=
where Jij is any deviatoric strain-like quantity. These von Mises norms are scaled for shear.
1 Model I
The classical, nonlinear, kinematic hardening model [3, 4] has an evolution of internal state de-
scribed by
.p HXij ][_p 1[ (15)fCij = eij L
where L is the limiting state for the back stress Bij. There is no stress rate term in the evolution
of flij for this model, i.e. N = 0 in (2), and consequently
(., B,, ]]_p[,) (16)Bij = 2H cij 2L
describes its evolution for the back stress. Because N = 0, the last three terms in the complementary
free energy (6) do not contribute to the dissipation in this model; its intrinsic dissipation (12) is
therefore given by
-- II 'll >--0, (17)
and it is always satisfied. Hence, Model I is thermodynamically admissible.
2 Model II
This model uses the same description for the irreversible evolution of internal state that Model I
uses, viz. (15). In addition, it assumes that 0 < N < 1, which introduces a reversible attribute to
the evolution of internal state. As a result,
describes this model's evolution for the back stress. Model II reduces to Model I when N = 0.
As Lubliner [1] discussed in his paper, one can always (in principle) transform from an internal
state variable whose evolution contains terms that are linear in the external variable rates, to
another internal state variable where there are no external variable rates present in the evolution
equation. This is accomplished in our case by considering the linear transformation
2H
Xij - 1 - N Xij = 2Hrxij , (19)
wherethevariableXij is a back stress, but different from Bij, with H _ = H/(1 -N) as its associated
hardening modulus. This transformation enables the flow law (13) to be rewritten in an equivalent
form as
g_ = 1/2II_'li • (20)
'_ xll
Likewise, it allows the evolution equation for back stress (18) to be rewritten in the equivalent form
£,j = 2H' (_., X,J2L,I1_11) ' (21)
where L _ = L/(1 - N) is the limiting state for the back stress Xij. Upon comparing (20) and
(21) with (13) and (16), one observes that they are identical in mathematical structure, but with
different values for their constants. These differences lead to differences in the intrinsic dissipation
properties of the material [2].
From the perspective of material science [5], the physically correct, internal, state variable has an
evolution equation with no Sij or T dependence. This coincides with the experimental observation
that an instantaneous change in either stress or temperature does not produce an instantaneous
change in a material's internal structure, i.e. its dislocation structure. Consequently, the back
stress Xij is the physically correct back stress for Model II, and it is referred to as the physical
hack stress. (The back stress Bij is the physical back stress of Model I.)
A description of Model II's dissipation response requires knowledge of its material function A,
which is found in the expression for the complementary free energy (6). This function is evaluated
by combining (12), (14), (15), (19), and (20), and then using A to cancel out those terms in
the dissipation inequality that can become negative valued. This process leads to the differential
equation )o&--_- 1 - N _-_ S_j - _,j , (22)
which is the simplest of several possible solutions. This equation can be integrated, in conjunction
with the constraint given in (10), to produce
which when substituted into (6) defines • for Model II; it is the basic constitutive equation of this
particular model. It follows then that the intrinsic dissipation (12) for Model II is given by the
inequality
(,,S-X,,+ ''X''_)L-_ _ II_nll > 0, (24)
and it is always satisfied. Hence, Model II is thermodynamically admissible. Notice the similarity
between (17)and (24).
3 Model III
This model considers the irreversible evolution of internal state to be described by
)_i/= _,Pj L ' (25)
4
whichdiffersfrom (15)by theexchangeof Xij with fllj in the dynamic recovery term. Like Model II,
this model takes 0 < N < 1, and therefore
/_ij = 2H Qj 2L II_PII + NSij (26)
describes the evolution for the back stress Bij. Equation 26 was first proposed by Ramaswamy
et al. [6]. Notice that the only difference between (16) and (26) for Models I and III is the presence
of the term NSij found in (26). Model III reduces to Model I when N = 0.
Using the same linear transformation that was used in Model II, i.e. (19), one determines that
Model III has the same transformed flow law (20) as Model II, but it has a different evolution
equation for the physical back stress Xij, viz.
J(iJ -12H-N- ( _i_- (1- N)X'j + NS'j2L ']kn][) . (27)
To the best of our knowledge, this ia a new expression for the evolution of the physical back stress.
Here N proportions the dynamic recovery between the physical back stress Xij and the applied
stress Sij.
We set out to derive the dissipation response of Model III as we did for Model II. We begin by
considering a decomposition
A = A_ + A2, (28)
where A1 is taken to be given by (23), i.e. A1 is the A of Model II. The remaining function is
evaluated by combining (12), (14), (19), (20), and (25), and then using A2 to cancel out those
remaining terms in the dissipation inequality that can become negative valued. Because of the
constraint equation (10), one obtains two partial differential equations, viz.
OA_ N - 2Hflij ij (29)
0S_---_= 2L(1 - N) Ske_ke 2H/3II ,
0A2 _ -H - 2Hi3ij ij (30)
Ol3ij L(1 - N) Skel3ke -- 2H/3[[
which when integrated will lead to the complementary free energy • for Model III, i.e. its fun-
damentai constitutive equation. We have not integrated these equations. For our purpose, it is
sufficient to know only that A exists. The intrinsic dissipation (12) for Model III is therefore given
by the inequality
I[n II_IlS - Xll + LCi--=N)JI1+_11---O, (31)
and it is always satisfied. Hence, Model III is also thermodynamically admissible.
A preliminary study of these three models [7] indicates that Model III may be able to predict
realistic ratchetting behavior; whereas, Model I (and therefore Model II) is known to overpre-
dict ratchetting behavior [8]. Continued research is required to better understand the predictive
capabilities of Model III.
Acknowledgment
This work is an outgrowth of a personnel exchange program between the agencies of NASA and
ONERA.
References
[1] Lubliner, J., "On the Structure of the Rate Equations of Materials with Internal Variables,"
Aeta Mech., 17, 1973, 109-119.
[2] Freed, A. D., Chaboche, J.-L., and Walker, K. P., "A Viscoplastic Theory with Thermodynamic
Considerations," in press: Acta Mech.
[3] Armstrong, P. J. and Frederick, C. 0., "A Mathematical Representation of the Multiaxial
Bauschinger Effect," CEGB, RD/B/N731, Berkeley Nuclear Laboratories, 1966.
[4] Chaboche, J.-L., "Viscoplastic Constitutive Equations for the Description of Cyclic and
Anisotropic Behavior of Metals," Bull. Acad. Pol. Sci., Ser. Sci. Tech., 25, 1977, 33[39]-42[48].
[5] Nix, W. D., Gibellng, J. C., and Hughes, D. A., "Time-Dependent Deformation of Metals,"
Metall. Trans. A, 16A, 1985, 2215-2226.
[6] Ramaswamy, V. G., Stouffer, D. C., and Laflen, J. H., "A Unified Constitutive Model for tile
Inelastic Response of Rend 80 at Temperatures Between 538°C and 982°C," J. Engr. Mater.
Tech., 112, 1990, 280-286.
[7] Freed, A. D. and Walker, K. P., "Model Development in Viscopla.stic Ratchetting," NASA TM
102509, April, 1990.
[8] Inoue, T., Yoshida, F., Ohno, N., Kawai, M., Niitsu, Y., and Imatani, S., "Plasticity-Creep
Behavior of 21/4Cr-lMo Steel at 600°C in Multiaxial Stress State," in: Structural Design for El-
evated Temperature Environments--Creep, Ratchet, Fatigue, and Fracture, PVP 163, C. Becht
IV et al. eds., ASME, New York, 1989, 101-107.

Report Documentation Page
Naliona] Aeronaulics and
Space Admlnistralion
1. ReportNo.
NASA TM- 103794
2. GovernmentAccessionNo.
4. Title and Subtitle
On the Thermodynamics of Stress Rate in the Evolution
of Back Stress in Viscoplasticity
7. Author(s)
A.D. Freed, J.-L. Chaboche, and K.P. Walker
g. PerformingOrganization Nameand Address
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135-3191
12. SponsoringAgencyNameand Address
National Aeronautics and Space Administration
Washington, D.C. 20546-0001
3. Recipient'sCatalogNo.
5. ReportDate
6. PerformingOrganizationCode
8. PerformingOrganizationReportNo.
E-607 !
10. WorkUnit No.
505-63-52
1t. Contract or Grant No.
13. Type of I_eportandPeriodCovered
Technical Memorandum
14. SponsoringAgency Code
15. SupplementaryNotes
Prepared for the Third International Symposium on Plasticity sponsored by the Institute de _¢_anique de Grenoble,
Grenoble, France, August i2-16, 1991. A.D. Freed, NASA Lewis Research Center. J.-L. Chaboche, Office
National d'Etudes et de Recherches A_rospatiales, 92322 ChNillon, Cedex, France. K.P. Walker, Engineering
Science Software, Inc., Smithfield, Rhode Island 02917. Responsible person, A.D. Freed (216) 433-3262.
t6. Abstract
A thermodynamic foundation using the concept of internal state variables is presented for the kinematic description
of a viscoplastic material. Three different evolution equations for the back stress are considered. The first is that
of classical, nonlinear, kinematic hardening. The other two include a contribution that is linear in stress rate.
Choosing an appropriate change in variables can remove this stress rate dependence. As a result, one of these
two models is shown to be equivalent to the classical, nonlinear, kinematic hardening model; while the other is a
new model--one which seems to have favorable characteristics for representing ratchetting behavior. All three
models are thermodynamically admissible.
17. Key Words iSuggested by Author(s)i
Thermodynamics; Internal state variables;
Plasticity; Viscoplasticity; Back stress
!18. Distribution Statement
Unclassified - Unlimited
Subject Category 39
19. SecurityClassif. (of this report)
Unclassified
20. Security Classif. (of this page)
Unclassified
21, No. of pages
7
22. Price*
A02
NASAFORM'1626OCT86 *For sale by the National Technical Information Service, Springfield, Virginia 22161


On the thermodynamics of stress rate in the evolution of back stress in viscoplasticity

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