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A viscoplastic model with a smooth transition to describe rate-independent plasticity
Pergamon
International Journal of Plasticity, Vol. 10, No. 4, pp. 347-362, 1994 Copyright © 1994 Elsevier Science Ltd Printed in the USA. All rights reserved 0749-6419/94 $6.00 + .00
0749-6419(94)E0003-W
A VISCOPLASTIC MODEL WITH A SMOOTH TRANSITION TO DESCRIBE R A T E - I N D E P E N D E N T PLASTICITY O.T. BRUHNS and U. ROTT* Ruhr-University Bochum Abstract-A concept for the analysis of viscoplastic structures is introduced that allows to develop viscoplasticmaterial models for steel based on "established" elastic-plasticconstitutive equations. The attainment of an exact limiting value for quasistatic processespermits utilizing an efficient strategy to determine the material functions. Structural analysesare conducted with an iterative and incremental finite element algorithm. The efficiency of the proposed concept is assessed by comparison with experimental and numerical results taken from the literature.
I. INTRODUCTION M a n y technically important alloys exhibit quite complex behaviour if they are loaded beyond their yield limit. Movement of dislocations is impeded by long-ranging obstacles that can be overcome only by increasing the stress level. These athermic mechanisms result in strain-hardening, the Bauschinger effect, and the attainment of saturation at cyclic loading. At higher homologous temperatures, dislocation movement is supported by thermally activated mechanisms, for example, diffusion of vacancies, leading to a rate-dependency of the material, as, for example, the increase of stress at higher rates of loading, creep, and stress relaxation. To describe the mechanical behaviour of structures, a large number of constitutive models have been developed. Whereas in the past more or less purely rate-independent (plastic) models were used in computer codes, nowadays more rate-dependent (viscous) constitutive equations are implemented. The latter can be divided roughly into models that assume the existence of a yield surface and those that suppose appearance of inelastic deformations at every stress level. Models o f the first kind usually employ the overstress concept introduced by PERZYNA [1963]. Typical representatives are the material models of CHASOCrm [1983] and BRUnNS et aL [1984]. Models of the second group essentially can be classified as highly nonlinear viscoelastic constitutive relations that represent inelastic hysteresis by the use of absolute value functions. Classical examples are the models of HART [1976], MnLER [1976a, 1976b], and YAO and KSmlVtPL[1985]. Extensive reviews of viscous constitutive models were performed by KRm~ [1977] and KREX~PL [1987]. In m a n y cases, however, structures are loaded in a way that the material response in one region has to be treated as rate-dependent, whereas in another part o f the structure the deformation is influenced only by athermal mechanisms. A typical example is the problem of a vertical impact on a plate. Very high deformation rates can be observed *Currently affiliated with Bremer Woll-K~immereiAG, Bremen, Germany. 347
348
O . T . BRUHNS and U. ROTT
in a narrow zone around the point of impact, whereas in the neighborhood both the deformations and their rates can become relatively small. In the case of even more complicated structures under complex loading, the regimes of pronounced rate-dependent and of more and less rate-independent material responses, respectively, are generally not known in advance. Therefore, extensive (and expensive) modifications and new computations have to be performed if distinct constitutive equations are used to model the different material responses of regions with initially unknown boundaries. The purpose of this article is to present a concept for the development of material models suited for the description of rate-dependent as well as rate-independent behaw iour. A rather wide variety of structural analyses can be performed with the help of just one constitutive law. To describe both aforementioned micromechanical mechanisms, the viscoplastic model transforms into a purely plastic one in the case of quasistatic processes. The attainment of an exact limiting value at vanishing rates of loading is assumed in many viscoplastic models with yield-surfaces but cannot be achieved by ordinary constitutive laws of the overstress type. The proposed separation of rate-dependent from rate-independent effects leads to yet two further advantages: First, this procedure allows to extend established elastic-plastic models into elastic-viscoplastic constitutive relations. Second, the material functions of the plastic model can be used without any modifications to describe quasistatic behaviour. Due to the concept of plastic deformations as a limit of viscoplastic ones, only the additional viscous material functions have to be determined. il. C O N S T I T U T I V E E Q U A T I O N S
II. 1. Basic assumptions For the proposed concept, the following assumptions are made: 1. The response is purely elastic until a certain stress level is reached. 2. Beyond this limit the material behaviour depends on both the history and the rate of loading. 3. A state of balance exists that is attained at quasistatic processes. 4. Only isothermal loading, moderate rates of deformation (_< 10 -2 I/s), and small deformations are taken into account. 5. The material is isotropic apart form the anisotropy induced by the inelastic deformations. 6. It can be treated as inelastically incompressible. 7. The influence of phase transformations and damage is not considered. The first three assumptions are essential for this concept, whereas the following ones will partially be omitted in the future. 11.2. Evolution equations The basic idea is to split the rate of deformation tensor into a reversible (elastic) and two inelastic parts so as to separate purely rate-dependent effects from rate-independent ones = ~+
e~ = ~ +
~p÷ ~.
(1)
Rate-independentplasticity
349
During quasistatic loading ~ vanishes, and thus the inelastic part of eqn (1) transforms into a purely plastic deformation rate. Accordingly, the applied stress a is separated into an athermal part # corresponding with the quasistatic state of balance and a thermally activated overstress
a = 6 + ~.
(2)
If now the normality rule of classical plasticity is adopted for/:u, and the overstress model of PERZYNA [1963] is employed for (iv, the following relationship is obtained from eqn (1)
1 (
"-
v
1+--S
tr(#)l) +Afi+((#(A)))n,
(3)
where referring to the rate-independent as well as the rate-dependent descriptions the different inelastic terms are subject to additional yield and/or loading conditions. Here G denotes the elastic shear modulus related to Young's modulus by E = 2G(1 + v), and v is Poisson's ratio. A and ~ - a s function of the overstress A - d e t e r m i n e the different amounts of (:p and (iv, contributing to the deformation. Furthermore ~ is the unit normal of the yield surface at point ~, whereas unit tensor n is calculated from the projection of u on the yield surface. The description of the material behaviour is here performed within the concept of internal variables. For the evolution of the internal variables, the differential equations of the chosen basic elastic-plastic model are used. These variables describe hardening due to the history of loading. As an example for application of the proposed concept the elastic-plastic formulation of the INTERATOM model (BRtrHNS et al. [1982,1989]) is employed. This model is especially suited to characterize the hardening of austenitic steels as well as its saturation due to cyclic loading with a small number of material functions. Isotropic as well as kinematic hardening are represented by internal variables K and ~, respectively, whereas a process variable za is used in connection with an updating procedure to model the material response after drastic (nonradial) changes of the loading direction
= (#, _/~) .e~
(4)
/i =p(fi) g ~ ° k ~] g(r)
Herein, #' is the deviatoric part of the athermal stresses #, and g(x), c(x), and p(fi) are material functions that can be determined from quasistatic uniaxial standard tests. Index 0 furthermore denotes initial values. A more detailed description is given with the Appendix. It should be noted, further, that here increments of the internal variables depend only on that part of the deformation rate which transforms into a purely plastic one during quasistatic processes. To allow a smooth transition from elastic-viscoplastic to elastic-plastic behaviour as a function of the applied rate of deformation, a third internal variable, the balance stress
350
o.T. BRUI-INSand U. ROTT
#, is introduced in addition to the two internal variables ~ and ~j. During inelastic deformations, these stresses have to fulfill the generalized v. Mises yield condition F =f(#'
- ~) - g(K) = ( # ' - ~ ) . ( 0 ' - ~) - g(K) = 0.
(5)
Based on this equation, the unit normal fi can be determined according to
~=~
-
,;~
'
(6)
where 1[all = axfau.ais the Euclidian norm of tensor a. The excess of the yield surface by the applied stresses a can be characterized by a scalar valued overstress A =
I1•' - ~11 - I1~' - ~11,
(7)
determining/~v via overstress function ~. The occurrence of this part of the inelastic rate of deformation, however, is related to nonnegative values of A. Thus, this loading condition simply becomes x, i f A _ > 0 (8)
<> =
O, otherwise.
The remainder/:p depends on scalar factor A, which, as usual, is evaluated by applying the condition of consistency F ( # . . . . ) = 0 for the balance stresses #. We thus find A=
#.fi 1 Og c-F-
-
d.n
K
(9) '
-2 OK
where K is a hardening modulus related with functions c(K) and g(K). It should be emphasized here that a constraint including yield condition eqn (5) and a loading condition for the actual stresses a, for example, x, if F ( # ) = 0 and # . n > 0 =
(lO)
0, otherwise,
is not attached to the total expression of Aft in eqn (3). This deviates somewhat from classical approaches and is here a consequence of the request for uniqueness of the constitutive relations that will be discussed later. It m a y furthermore be explained through the fact that according to the concept Afi contains rate,independent as well as rate-dependent contributions. The latter ones are assumed to vanish during quasistatiC processes together with ~v, and only the first part can be subject to eqn (10). The evolution equation for the athermal part of stresses now will be introduced in a most general f o r m as
Rate-independent plasticity
351
~, = 116' - ~II re, _ (e.n)n] + (A(~.n)>n + ((B(a'
-
6'))),
(11)
and with the initial condition 6(A = 0) = ~. From the first part of this relation it can be deduced that the unit normals corresponding to a and 6 have to coincide n
= ft.
(12)
The second part of eqn (11) allows 6 to grow during inelastic loading where the material function A (A,v) is a function of overstress A and the rate of loading which here m a y be defined as (13)
v = (~'.n) = Ok.n).
Creep finally can be described with the third term of eqn (11) with an appropriate choice o f the recovery function B( f(x/f~, IIo' II)This system o f constitutive equations is illustrated by use o f the Haigh-Westergaard stress space (see Fig. 1). The stresses 6 that correspond with the quasistatic state of balance are obtained from the actual stresses a by projection on the yield surface. The origin of this surface with coordinates ~ moves in the direction o f unit normal n during inelastic deformations. In this case, the yield surface expands so that the distance between ~r and 6 varies according to 3"." = o . n ( l - (A>) - ((BA)).
(14)
03
^/~
(~'.n dtn
Fig. 1. Representation of the proposed concept in Haigh-Westergaard stress space.
352
O.T. BRUHNSand U. ROTT
Prescribing here, for example, creep through # = 0 appears in a reduction of overstress A until B vanishes. Consequently, the rates of the internal variables become zero as well, and stationary creep takes place according to i:* = ~(A*)n,
(15)
where the * marks the stationary values of the variables. Introducing now eqn (11) into eqn (9) results in
and we finally, from eqn (3), arrive at
1[
e = 2 G- -
O - -l-+t rv( O ) l
+
K (o.n)
/// +
~(A)
+
A
n.
(17)
By comparing this relation with classical plasticity, it becomes evident that eqn (3) transforms into a purely elastic-plastic model during quasistatic processes e=i:e+
(~)n,
(18)
provided the following conditions for the material functions hold for vanishing overstresses A =0:
4,(A) = 0 ,
(19)
0 < A(A,v) <_ l ] (20) lim A = 1 U ~ U0
and where v0 is considered as the rate of quasistatic loading. For practical purposes, this value will later be related to a strain rate of (say) k = 10 -8 1/s of the uniaxial tension test. The limiting value Vo = 0, however, is not excluded from the approach. II.3.
Uniqueness of constitutive equations
Any system of constitutive equations has to fulfill the requirements of uniqueness, that is, a unique relationship between stress and strain increments must exist for all possible loading paths starting from an arbitrary initial state. Conventional material models are linear in these increments and therefore satisfy this condition a priori. In the present case, however, the material function A depends on the rate of loading and thus on the stress rate. This implies that eqn (17) cannot be solved explicitly for the stress increments
I
B n'
(21)
Rate-independent plasticity
353
whereas the constraint (10) is unique due to the special form o f eqn (11)
(x) =( x' ifotherwi F(#) se. =Oand ~'n- ((B A +
>0
(22)
The difficulty to prove uniqueness of constitutive equations generally prevents ratedependent formulations of material functions. Here this problem can be solved if function A is introduced as a product of the form A = al (A)a2 (v),
(23)
where a~ and a2 are exclusive functions of A and v, respectively. Introducing now this relation into (17) and solving this equation for a2 (v) we find
aE(v) =
K [(e.n)_ B A _ # I al(A----S K
1
(6.n)
K al(A)2G'
(24)
provided both yield and loading conditions are satisfied. Processes with identical loading histories display the same values of state variables, but can differ with respect to the process variable/:. If such processes of identical initial state but different continuation are considered, the right-hand side of eqn (24) can be considered as a set of hyperbolas in v with i:. n as parameter al az(v) - b v
1
b"
(25)
Different deformation rates in general lead to distinct instantaneous values of (6. n), which appear as different solutions of eqn (25). A sufficient condition for uniqueness o f the incremental stress-strain relation at a prescribed loading history can be obtained if just one such solution exists for every deformation rate. If the independent variable v in eqn (25) is transformed according to 1
v = -, x
(26)
the hyperbolas constitute straight lines emanating from the ordinate - ( l / b ) . The condition o f uniqueness o f the constitutive relations is now fulfilled, if a function a2(v) can be found which only once intersects with each of the straight lines (see Fig. 2). Since a2(v) furthermore has to satisfy the requirements a2(vo) = a2o = 1 da 2 --<--0 dv
lim a2(v) = a2~ _> 0 O~o~
(27)
354
O . T . BRUHNS and U. ROTT
o2(x)
Q2~
x
!
b Fig. 2.
Material function a 2 and right-hand side of eqn (24) after transformation (Vo = 0).
only nonconvex functions according to d2a2 - dv 2
> 0
(28)
can be considered. II.4. A strategy to determine the material functions The procedure of finding appropriate material functions and their parameters for a set of constitutive equations is in general a significant problem. Several approaches for solving this problem can be found in the literature. Trial and error procedures (cf. EFTIS et al. [1989]) have been found troublesome and unsatisfactory. Due to the stiffness of the system, they sometimes furthermore are not very successful. Optimization techniques represent more elaborate procedures. Mainly deterministic algorithms are employed. More recently, however, useful results were also obtained by stochastic optimization ( M O L L E R and H A R T M A N N [1989]; FORNEFELD [1990]). All these techniques require the specification of the structure of the material functions. Usually they are very time consuming because a whole system of differential equations has to be solved to get one single value of the weight function. The large expenditure of optimization techniques becomes evident when it is considered that the integration has possibly to be performed several times, for example, with different rates of loading. Another principal problem that arises only with deterministic algorithms is the choice of an initial estimate of the parameters. The advantage of optimization techniques is their extensive independence of the kind of the problem and their suitability for programming. Here, another method is used to some extent that can be characterized as incremental reduction of material functions to experimentally observable quantities. The decisive advantage of this procedure is that the structure of the material functions does not need to be assumed in advance, but is obtained by a set of discrete values of arbi-
Rate-independent plasticity
355
trary density. Problems may arise, however, if the differential equations are extremely sensitive to small variations of parameters. It is not an easy task to make a simple decision for the "best" strategy of identification for the material functions. The following general recommendations, however, can be made: If the structure of the material functions is not known at all, incremental reduction of these functions to experimentally observable quantities should be employed. If only material parameters have to be fitted, optimization techniques may have advantages. In the case of a large number of parameters or if their initial values are not known accurately, stochastical optimization algorithms like the evolution strategy developed by RECn~E~G [1973] are recommended. If, on the other hand, only a few parameters are to be determined with fairly good initial values, as is the case with the functions g(r) and c(r), a deterministic algorithm can be used. A decisive feature of the proposed concept is the attainment of the underlying rateindependent material model in the case of quasistatic processes. Therefore, the functions describing hardening can be decoupled from those that model rate-dependent phenomena and thus can be taken from the elastic-plastic model. They were evaluated by using a deterministic optimization algorithm to fit approved hardening functions to experimental data. The viscous function A (A,v), however, is totally unknown except for the restrictions of eqns (27) and (28). For that reason, it is determined by its reduction to experimentally observable quantities. Solving the differential equations for the one-dimensional tension problem at the yield limit for A and considering al (A = O) = a~o = 1 results in the following relationship Eto E - Eto ( V ) az ( v ) = - , E,o(V) E - Eto
(29)
where E and Eto ( v ) are the elastic and the instantaneous inelastic moduli at a special rate of loading v, respectively, and Et0 denotes the instantaneous inelastic modulus for quasistatic processes. Because all quantities on the right-hand side of eqn (29) are known from experimental data, a2 (v) can be calculated. With the knowledge of this function al (A) can be determined. Supposing that the values of the state variables and material functions at arbitrary times t are given together with the time histories of stress and strain, the corresponding quantities at t + A t can be evaluated by explicit integration of the constitutive relations 6(,+at) = 6(t) + ( A o + B ( a - O) ) ( t ) A t
~(,+~,~ =
~(,7 + c(,7 ~ ~ (,7 (30)
K('+4°=~c'7+(~O- 0
A(t+,~t) -= A(t) + p
(t)
(K)At (t)
g/~ [~(t)At (tl
In these equations, abbreviations of the form c(t) = c ( r ( t ) ) and so forth have been used. Moreover, overstress A and al are computed through
356
O.T. BRUHNSand U. ROT'[
A(t+At) = ~
~ ° - - ~ (t+At) -- x/g(t+,at) (31)
al(t+at) =
k
E
3 K
(t+at)
(t+At)
by means of which a~ (A) can be expressed by an analytical function (see Appendix). As a first approximation, functions B and • are neglected during this procedure, because they have been introduced to model the rate-dependent relaxation and creep behaviour of the material. Accordingly, they are ascertained by comparison with creep tests. In this case, however, the proposed procedure fails due to stability problems. Therefore, both functions have been calculated by employing optimization techniques (see Roxx [1991]). II.5. Results As an example of application, austenitic stainless steel AISI 316L mod was examined. All simulations were performed at a temperature of 550°C. Figure 3 shows a comparison of computation and experimental results taken from the French standard RCC-MR [1985]. Calculated cyclic saturation curves for three strain-controlled tests are presented together with the experimentally obtained saturation function. These quasistatic computations emphasize the quality of the fitting of both hardening functions c (K) and g (K). The smooth transition at the onset of inelastic deformations and the symmetry of the saturation curves should be noticed. In Fig. 4, calculated monotonic tension tests at different strain rates are compared with those curves that were used for determining function A. Both the curves for high and quasistatic (here, ~ = 10 -8 l/s) strain rates are almost identical. Only in the range
o
IMeo]
300
-0,5
0,4
/~0~2
I ?
0,5
e 1"/.1
-300 Fig. 3. Calculatedcyclicsaturationcurvesand experimentallyobtainedsaturationfunction(quasistaticcase).
Rate-independent plasticity
357
0 [MPo]
.I. 10.2 s"I
200-
Jl.10"~S "1
~ _
100.
i
,
,
0
~
~
1
-
1
0
-
%
'
1 ~1
J
I
~
I
i
~5
i
I
I
lOSs 1
i
I
1,0
I
I
1,5 E[o/,]
Fig. 4. Uniaxial tension tests at different strain-rates, experiments(dashed lines) and calculations (full lines).
of ~ -- 10 -6 1/s can more pronounced deviations be observed, which certainly can be reduced by further improvement of the material parameters. Creep tests are considered as a more severe examination of both the constitutive equations and the fitting strategy. Therefore, calculated and experimentally obtained creep curves are compared in Figs. 5 and 6.
E
['/=]
0,20 160 MPo
o,1s 140 MPo
0,10 .............
----"
"'"
120 MPo / / 0
100 MPo
l
I
=
I
0
0,3
0,6
0,9
I
1,2
I
1,5
I
1,8 t[s] .107
Fig. 5. Creep curves from experiments(dashed lines) and calculations (full lines).
O.T. BRUHNS and U. ROTT
358
0,20-
0,15 -
150 MPa 0,10 -
130 MPo 0,05-
110 MPa
f
o,3
r
0,6
T
7
m
o,9
1,2
1,5
~-I1~
1,8
t [s]
.107 Fig. 6. Creep curves from experiments (dashed lines) and calculations (full lines).
The stress levels of Fig. 5 were used to determine functions B and ~, whereas Fig, 6 shows real predictions. Both strain level and strain rate for secondary creep agree rather well. More distinct deviations, however, can be found in the region of primary creep. This is here a consequence of the employed weight function; it can be improved by further modification, though with a loss of the agreement at secondary creep. Another approach, which may be more successful, consists of a modification of the functions themselves. As a whole, comparisons between one-dimensional computations and experiments show encouraging results. FE calculations were performed to check the suitability of the proposed concept for the analysis of structures. From a number of results, only one example is presented: The expansion of the inelastic region of a rectangular plate with a circular hole is demonstrated with Fig. 7. On the left-hand side, the analysis based on the proposed model is displayed and compared with an analysis for ideal-plastic material first examined by ZI~NKmWICZ[1975]. The loading parameter p is here defined as ratio of tension stress applied at the upper boundary of the plate and (instantaneous) yield limit. In the analysis, the expansion of the inelastic region starting from the hole is similarly reproduced. The same loading parameter causes a smaller inelastic domain in the case of AISI 316 than for the ideal-plastic material, due to the hardening :of the material. The bulge and the development of a second inelastic region are features that can be observed in both analyses. 1II. CONCLUSIONS
A new concept is presented that allows the extension of classical plastic material models into rate-dependent descriptions by definition of an evolution equation for the atherreal stresses. The appropriate choice of a rate-dependent material function enables the
Rate-independent plasticity
I~13
359
,r('-v-f
0,92
[ ii iJ llj IJ¢l I I'%11 I I /[/I////V/Z//
~
1,13
Fig. 7. Expansion of inelastic region due to increase of loading parameter p.
attainment of an exact limiting value in the case of quasistatic processes. This is essential not only from a micromechanical point of view but also for the analysis of structures displaying regions of both very low and high deformation rates. Furthermore, this feature provides the basis for an efficient strategy to determine material functions where rate-independent hardening functions can be separated from those reproducing viscous behaviour. The practicability of this concept is assessed by comparing the calculations with experimental and numerical results from the literature. In the case o f uniaxial processes, a good agreement for cyclic loading and creep tests is emphasized. This is of particular importance for application to many technical problems. The qualification for implementation into FE-codes was proved by analysis o f a standard benchmark test. It is intended to employ this concept on further plastic material models, for example, on multiyield surface models, to accommodate even better cyclic behaviour. In addition, nonisothermal processes will be taken into account.
REFERENCES
1963
PE~X'NA,P., "The Constitutive Equations for Rate Sensitive Plastic Materials," Quart. Appl. Mech., 20, 321. 1973 RECrmr~BERG,I., Evolutionsstrategie, Friedrich Frommann Verlag, Stuttgart. 1975 ZmNKmW]cz,O.C., Methode der Finiten Elemente, Carl Hanser Verlag, Miinchen. 1976 HART,E.W., "Constitutive Relations for the Non-Elastic Deformation of Metals," J. Engng. Mat. Techn., 98, 193. 1976a MrZLBR,A.K., "An Inelastic Constitutive Model for Monotonic, Cyclic and Creep Deformation: Part I--Equations Developmentand Analytical Procedures," J. Engng. Mat. Techn., 98, 97. 1976b MrtLER, A.K., "An Inelastic Constitutive Model for Monotonic, Cyclic and Creep Deformation: Part II-Application to Type 304 Stainless Steel," J. Engng. Mat. Techn., 98, 106. 1977 K~G, R.D., "Numerical Integration of Some New Unified Plasticity-Creep Formulations," Proc. 4 th SMIRT Conference, M 6/4.
360
1982 1983 1984 1985 1985 1987 1987 1989 1989 1989 1990 1991
O.T. BRUHNSand U. ROTT
BRUHNS,O.T., et al., "New Constitutive Equations to Describe Infinitesimal Elastic-Plastic Deformations," ASME, 82-PVP-71, 1. Ct~mocrm, J.L., "On the Plastic and Viscoplastic Constitutive Equations-Part l: Rules Developed With Internal Variable Concept," J. Press. Vessel Techn., 105, 153. BRtmNS, O.T., BOEC•E, B., LINK, F., "The Constitutive Relations of Elastic-Inelastic Materials at Small Strains," Nucl. Engng. Design, 83, 325. RCC-MR, "Design and Construction Rules for Mechanical Components of FBR Nuclear Islands," AFCEN. YAO, D., and KREMPL, E., "Viscoplasticity Theory Based on Overstress. The Prediction of Monotonic and Cyclic Proportional and Nonproportional Loading Paths of an Aluminium Alloy," Int. J. Plast., 1,259. BRUHNS,O.T., PITZER, M., "Constitutive Modelling in the Range of Inelastic Deformations- Evaluation of Parameters," in P.S. WroTE (ed.), Constitutive Modelling in the Range of Inelastic Deformations-Uniaxial Evaluation, GEC-Report (W) C. 12.224, Whetstone, Annex C. KREMPL,E., "Models of Viscoplasticity, Some Comments on Equilibrium (Back) Stress and Drag Stress," Acta Mech., 69, 25. BRUHNS,O.T. et al., "The lnteratom Model," in J. FAN, S. MURAKAMI(eds.), Advances in Constitutive Laws for Engineering Materials, Vol. 1, Pergamon Press, New York, pp. 16-21. EFTIS,J., ABDEL-KADER,m.s., JONES, D.L., "Comparisons Between the Modified Chaboche and Bodner-Partom Viscoplastic Constitutive Theories at High Temperature," Int. J. Plast. 5, 1. MOLLER, D., HARTMANN, G., "Identifikation yon Werkstoffparametern mit einem numeriscben Optimierungsverfahren nach der Evolutionsstrategie," in O.T. BRUHNS(ed.), GroBe Plastische Form/inderungen, Bad Honnef 1988, Mitt. Inst. Mech. 63, Bochum, 67. FORNEFELD,W., ,'Zur Parameteridentifikation und Berechnung yon Hochgeschwindigkeitsdeformationen metallischer Werkstoffe anhand eines Kontinuums-Damage-Modells," Mitt. Inst. Mech. 73, Bochum. ROTT,U., "Ein neues Konzept zur Berechnung viskoplastischer Strukturen," Mitt. Inst. Mech. 76, Bochum.
Institute of Mechanics Ruhr-University D-44780 Bochum, Germany (Received in final revised form 15 June 1993)
APPENDIX: MATERIAL FUNCTIONS A N D THEIR PARAMETERS
For the purely plastic material functions c (r) and g (K), the formulations proposed by BRUHNS and PITZER [1987] were employed g(K) = go + (g~ -- g0)(1 -- e-C'~),
(A1)
c(K) -
(A2)
2
EEt
3 E - Et
1 dg
2 dK'
with the initial value go = -~°2
(A3)
goo = czgo.
(A4)
and the limiting case
I n the r a n g e o f inelastic d e f o r m a t i o n s , the t a n g e n t m o d u l u s Et = do/de is a s s u m e d to
be a function of internal variable K
Rate-independent plasticity Et = E t ~ +
c4(c3 - E~ c4)
/
c4 + h +
~
c4A(K) /2'
361 (AS)
h2 + ao(1 - ~b~)
with abbreviations
c4 =
1
~ 1c3 -- ao(1 - ~b~.)],
¢~ =
Et~/E
(A6)
and
h[,~(K)] =
~/ao A(r) 1 E + ~o(1 - ~ ) )
(A7)
The viscous function A (A,v) was found to be A = al (A)a 2 (v),
(A8)
with
a~(A)
=
1-L [tanh(DA + C) - 1] + L, tanh C - 1
(A9)
and
=~rl e~lvz' + r2e~2vz~, if a2(v)
Vo < v < v~
L 1, if v --< Vo.
(A10)
Parameter r~ herein has to be chosen according to rl = 11 -
r2e ~:g' }e -~:g~,
(A1 l)
such that Vo can be looked upon as the rate o f quasistatic loading. Functions ~ and B finally describe the creep behaviour by means o f ~ ( A ) .~ e dl(A/'~)d~ - 1
(A12)
and d3 tanh(d4xff + ds) + d6, if , i f > x/f* B ( i f , ][tr'[[ ) =
0, otherwise,
where x/f* denotes the value of ~ from
(A13)
at stationary creep and which can be calculated
362
O . T . BRUHNS and U. ROTq"
l/ar tanh( •
t
Herein coefficients d 3 t o d 6 are assumed to be quadratic polynomials of
di
=Pi]
+Pi211,~'ll +Pi31t,T'll 2, i = 3,4,5,6.
II-' II (A15)
The related material parameters of AISI 316L mod at a temperature of 550°C are compiled in Table 1.
Table 1. Elastic and inelastic material parameters of AISI 316L mod at 550°C Elastic material parameters E [MPa] 1.492-105 Plastic material parameters a o [MPa] 74.5 c2 [--] 9.12 Eto [MPa] 1.417.105 Function A L [-] C [-] al [ - ] /~1 [ - ] r2 [ - ] Vo [MPa/s]
75.0 -3.901 -5.182.101 1.734-10 2 9.691 • 10 --I 1.2.10 -3
[-l ct [MPa] c 3 [MPa] E~o~ [MPa] D [MPa-I] a2 [ - ] 132 [ _ ]
0.3 7.300.10 -2 151.0 1.57.103 0.269 -4.268 4.982.10-3
v~ [MPa/s]
1.3.103
6.117.10 -15 1
d2 [ - ]
3.326
Function B P31 [l/s] P33 [MPa-2/s]
3.521.10 -8 3.287-10 -14
P32 [M P a - l / s ]
3.526.10 - t °
P41 IMP a - l ] P43 [M Pa-3]
9.547-10 -1 3.316.10 -7
P42 [MPa-2]
2.307-10 -5
Psm [ - ] P53 [M Pa-2]
-4.985 "101 -3.780.10 -5
P52 [MPa-1]
Function dl [--] [MPa]
P61 [l/s] P63 [MPa-2/s]
2.324.10 -8 4.018.10 -14
P62 [M P a - l / s ]
-1.705.10
1
2.747.10 - l °
International Journal of Plasticity, Vol. 10, No. 4, pp. 347-362, 1994 Copyright © 1994 Elsevier Science Ltd Printed in the USA. All rights reserved 0749-6419/94 $6.00 + .00
0749-6419(94)E0003-W
A VISCOPLASTIC MODEL WITH A SMOOTH TRANSITION TO DESCRIBE R A T E - I N D E P E N D E N T PLASTICITY O.T. BRUHNS and U. ROTT* Ruhr-University Bochum Abstract-A concept for the analysis of viscoplastic structures is introduced that allows to develop viscoplasticmaterial models for steel based on "established" elastic-plasticconstitutive equations. The attainment of an exact limiting value for quasistatic processespermits utilizing an efficient strategy to determine the material functions. Structural analysesare conducted with an iterative and incremental finite element algorithm. The efficiency of the proposed concept is assessed by comparison with experimental and numerical results taken from the literature.
I. INTRODUCTION M a n y technically important alloys exhibit quite complex behaviour if they are loaded beyond their yield limit. Movement of dislocations is impeded by long-ranging obstacles that can be overcome only by increasing the stress level. These athermic mechanisms result in strain-hardening, the Bauschinger effect, and the attainment of saturation at cyclic loading. At higher homologous temperatures, dislocation movement is supported by thermally activated mechanisms, for example, diffusion of vacancies, leading to a rate-dependency of the material, as, for example, the increase of stress at higher rates of loading, creep, and stress relaxation. To describe the mechanical behaviour of structures, a large number of constitutive models have been developed. Whereas in the past more or less purely rate-independent (plastic) models were used in computer codes, nowadays more rate-dependent (viscous) constitutive equations are implemented. The latter can be divided roughly into models that assume the existence of a yield surface and those that suppose appearance of inelastic deformations at every stress level. Models o f the first kind usually employ the overstress concept introduced by PERZYNA [1963]. Typical representatives are the material models of CHASOCrm [1983] and BRUnNS et aL [1984]. Models of the second group essentially can be classified as highly nonlinear viscoelastic constitutive relations that represent inelastic hysteresis by the use of absolute value functions. Classical examples are the models of HART [1976], MnLER [1976a, 1976b], and YAO and KSmlVtPL[1985]. Extensive reviews of viscous constitutive models were performed by KRm~ [1977] and KREX~PL [1987]. In m a n y cases, however, structures are loaded in a way that the material response in one region has to be treated as rate-dependent, whereas in another part o f the structure the deformation is influenced only by athermal mechanisms. A typical example is the problem of a vertical impact on a plate. Very high deformation rates can be observed *Currently affiliated with Bremer Woll-K~immereiAG, Bremen, Germany. 347
348
O . T . BRUHNS and U. ROTT
in a narrow zone around the point of impact, whereas in the neighborhood both the deformations and their rates can become relatively small. In the case of even more complicated structures under complex loading, the regimes of pronounced rate-dependent and of more and less rate-independent material responses, respectively, are generally not known in advance. Therefore, extensive (and expensive) modifications and new computations have to be performed if distinct constitutive equations are used to model the different material responses of regions with initially unknown boundaries. The purpose of this article is to present a concept for the development of material models suited for the description of rate-dependent as well as rate-independent behaw iour. A rather wide variety of structural analyses can be performed with the help of just one constitutive law. To describe both aforementioned micromechanical mechanisms, the viscoplastic model transforms into a purely plastic one in the case of quasistatic processes. The attainment of an exact limiting value at vanishing rates of loading is assumed in many viscoplastic models with yield-surfaces but cannot be achieved by ordinary constitutive laws of the overstress type. The proposed separation of rate-dependent from rate-independent effects leads to yet two further advantages: First, this procedure allows to extend established elastic-plastic models into elastic-viscoplastic constitutive relations. Second, the material functions of the plastic model can be used without any modifications to describe quasistatic behaviour. Due to the concept of plastic deformations as a limit of viscoplastic ones, only the additional viscous material functions have to be determined. il. C O N S T I T U T I V E E Q U A T I O N S
II. 1. Basic assumptions For the proposed concept, the following assumptions are made: 1. The response is purely elastic until a certain stress level is reached. 2. Beyond this limit the material behaviour depends on both the history and the rate of loading. 3. A state of balance exists that is attained at quasistatic processes. 4. Only isothermal loading, moderate rates of deformation (_< 10 -2 I/s), and small deformations are taken into account. 5. The material is isotropic apart form the anisotropy induced by the inelastic deformations. 6. It can be treated as inelastically incompressible. 7. The influence of phase transformations and damage is not considered. The first three assumptions are essential for this concept, whereas the following ones will partially be omitted in the future. 11.2. Evolution equations The basic idea is to split the rate of deformation tensor into a reversible (elastic) and two inelastic parts so as to separate purely rate-dependent effects from rate-independent ones = ~+
e~ = ~ +
~p÷ ~.
(1)
Rate-independentplasticity
349
During quasistatic loading ~ vanishes, and thus the inelastic part of eqn (1) transforms into a purely plastic deformation rate. Accordingly, the applied stress a is separated into an athermal part # corresponding with the quasistatic state of balance and a thermally activated overstress
a = 6 + ~.
(2)
If now the normality rule of classical plasticity is adopted for/:u, and the overstress model of PERZYNA [1963] is employed for (iv, the following relationship is obtained from eqn (1)
1 (
"-
v
1+--S
tr(#)l) +Afi+((#(A)))n,
(3)
where referring to the rate-independent as well as the rate-dependent descriptions the different inelastic terms are subject to additional yield and/or loading conditions. Here G denotes the elastic shear modulus related to Young's modulus by E = 2G(1 + v), and v is Poisson's ratio. A and ~ - a s function of the overstress A - d e t e r m i n e the different amounts of (:p and (iv, contributing to the deformation. Furthermore ~ is the unit normal of the yield surface at point ~, whereas unit tensor n is calculated from the projection of u on the yield surface. The description of the material behaviour is here performed within the concept of internal variables. For the evolution of the internal variables, the differential equations of the chosen basic elastic-plastic model are used. These variables describe hardening due to the history of loading. As an example for application of the proposed concept the elastic-plastic formulation of the INTERATOM model (BRtrHNS et al. [1982,1989]) is employed. This model is especially suited to characterize the hardening of austenitic steels as well as its saturation due to cyclic loading with a small number of material functions. Isotropic as well as kinematic hardening are represented by internal variables K and ~, respectively, whereas a process variable za is used in connection with an updating procedure to model the material response after drastic (nonradial) changes of the loading direction
= (#, _/~) .e~
(4)
/i =p(fi) g ~ ° k ~] g(r)
Herein, #' is the deviatoric part of the athermal stresses #, and g(x), c(x), and p(fi) are material functions that can be determined from quasistatic uniaxial standard tests. Index 0 furthermore denotes initial values. A more detailed description is given with the Appendix. It should be noted, further, that here increments of the internal variables depend only on that part of the deformation rate which transforms into a purely plastic one during quasistatic processes. To allow a smooth transition from elastic-viscoplastic to elastic-plastic behaviour as a function of the applied rate of deformation, a third internal variable, the balance stress
350
o.T. BRUI-INSand U. ROTT
#, is introduced in addition to the two internal variables ~ and ~j. During inelastic deformations, these stresses have to fulfill the generalized v. Mises yield condition F =f(#'
- ~) - g(K) = ( # ' - ~ ) . ( 0 ' - ~) - g(K) = 0.
(5)
Based on this equation, the unit normal fi can be determined according to
~=~
-
,;~
'
(6)
where 1[all = axfau.ais the Euclidian norm of tensor a. The excess of the yield surface by the applied stresses a can be characterized by a scalar valued overstress A =
I1•' - ~11 - I1~' - ~11,
(7)
determining/~v via overstress function ~. The occurrence of this part of the inelastic rate of deformation, however, is related to nonnegative values of A. Thus, this loading condition simply becomes x, i f A _ > 0 (8)
<
O, otherwise.
The remainder/:p depends on scalar factor A, which, as usual, is evaluated by applying the condition of consistency F ( # . . . . ) = 0 for the balance stresses #. We thus find A=
#.fi 1 Og c-F-
-
d.n
K
(9) '
-2 OK
where K is a hardening modulus related with functions c(K) and g(K). It should be emphasized here that a constraint including yield condition eqn (5) and a loading condition for the actual stresses a, for example, x, if F ( # ) = 0 and # . n > 0
(lO)
0, otherwise,
is not attached to the total expression of Aft in eqn (3). This deviates somewhat from classical approaches and is here a consequence of the request for uniqueness of the constitutive relations that will be discussed later. It m a y furthermore be explained through the fact that according to the concept Afi contains rate,independent as well as rate-dependent contributions. The latter ones are assumed to vanish during quasistatiC processes together with ~v, and only the first part can be subject to eqn (10). The evolution equation for the athermal part of stresses now will be introduced in a most general f o r m as
Rate-independent plasticity
351
~, = 116' - ~II re, _ (e.n)n] + (A(~.n)>n + ((B(a'
-
6'))),
(11)
and with the initial condition 6(A = 0) = ~. From the first part of this relation it can be deduced that the unit normals corresponding to a and 6 have to coincide n
= ft.
(12)
The second part of eqn (11) allows 6 to grow during inelastic loading where the material function A (A,v) is a function of overstress A and the rate of loading which here m a y be defined as (13)
v = (~'.n) = Ok.n).
Creep finally can be described with the third term of eqn (11) with an appropriate choice o f the recovery function B( f(x/f~, IIo' II)This system o f constitutive equations is illustrated by use o f the Haigh-Westergaard stress space (see Fig. 1). The stresses 6 that correspond with the quasistatic state of balance are obtained from the actual stresses a by projection on the yield surface. The origin of this surface with coordinates ~ moves in the direction o f unit normal n during inelastic deformations. In this case, the yield surface expands so that the distance between ~r and 6 varies according to 3"." = o . n ( l - (A>) - ((BA)).
(14)
03
^/~
(~'.n dtn
Fig. 1. Representation of the proposed concept in Haigh-Westergaard stress space.
352
O.T. BRUHNSand U. ROTT
Prescribing here, for example, creep through # = 0 appears in a reduction of overstress A until B vanishes. Consequently, the rates of the internal variables become zero as well, and stationary creep takes place according to i:* = ~(A*)n,
(15)
where the * marks the stationary values of the variables. Introducing now eqn (11) into eqn (9) results in
and we finally, from eqn (3), arrive at
1[
e = 2 G- -
O - -l-+t rv( O ) l
+
K (o.n)
/// +
~(A)
+
A
n.
(17)
By comparing this relation with classical plasticity, it becomes evident that eqn (3) transforms into a purely elastic-plastic model during quasistatic processes e=i:e+
(~)n,
(18)
provided the following conditions for the material functions hold for vanishing overstresses A =0:
4,(A) = 0 ,
(19)
0 < A(A,v) <_ l ] (20) lim A = 1 U ~ U0
and where v0 is considered as the rate of quasistatic loading. For practical purposes, this value will later be related to a strain rate of (say) k = 10 -8 1/s of the uniaxial tension test. The limiting value Vo = 0, however, is not excluded from the approach. II.3.
Uniqueness of constitutive equations
Any system of constitutive equations has to fulfill the requirements of uniqueness, that is, a unique relationship between stress and strain increments must exist for all possible loading paths starting from an arbitrary initial state. Conventional material models are linear in these increments and therefore satisfy this condition a priori. In the present case, however, the material function A depends on the rate of loading and thus on the stress rate. This implies that eqn (17) cannot be solved explicitly for the stress increments
I
B n'
(21)
Rate-independent plasticity
353
whereas the constraint (10) is unique due to the special form o f eqn (11)
(x) =( x' ifotherwi F(#) se. =Oand ~'n- ((B A +
>0
(22)
The difficulty to prove uniqueness of constitutive equations generally prevents ratedependent formulations of material functions. Here this problem can be solved if function A is introduced as a product of the form A = al (A)a2 (v),
(23)
where a~ and a2 are exclusive functions of A and v, respectively. Introducing now this relation into (17) and solving this equation for a2 (v) we find
aE(v) =
K [(e.n)_ B A _ # I al(A----S K
1
(6.n)
K al(A)2G'
(24)
provided both yield and loading conditions are satisfied. Processes with identical loading histories display the same values of state variables, but can differ with respect to the process variable/:. If such processes of identical initial state but different continuation are considered, the right-hand side of eqn (24) can be considered as a set of hyperbolas in v with i:. n as parameter al az(v) - b v
1
b"
(25)
Different deformation rates in general lead to distinct instantaneous values of (6. n), which appear as different solutions of eqn (25). A sufficient condition for uniqueness o f the incremental stress-strain relation at a prescribed loading history can be obtained if just one such solution exists for every deformation rate. If the independent variable v in eqn (25) is transformed according to 1
v = -, x
(26)
the hyperbolas constitute straight lines emanating from the ordinate - ( l / b ) . The condition o f uniqueness o f the constitutive relations is now fulfilled, if a function a2(v) can be found which only once intersects with each of the straight lines (see Fig. 2). Since a2(v) furthermore has to satisfy the requirements a2(vo) = a2o = 1 da 2 --<--0 dv
lim a2(v) = a2~ _> 0 O~o~
(27)
354
O . T . BRUHNS and U. ROTT
o2(x)
Q2~
x
!
b Fig. 2.
Material function a 2 and right-hand side of eqn (24) after transformation (Vo = 0).
only nonconvex functions according to d2a2 - dv 2
> 0
(28)
can be considered. II.4. A strategy to determine the material functions The procedure of finding appropriate material functions and their parameters for a set of constitutive equations is in general a significant problem. Several approaches for solving this problem can be found in the literature. Trial and error procedures (cf. EFTIS et al. [1989]) have been found troublesome and unsatisfactory. Due to the stiffness of the system, they sometimes furthermore are not very successful. Optimization techniques represent more elaborate procedures. Mainly deterministic algorithms are employed. More recently, however, useful results were also obtained by stochastic optimization ( M O L L E R and H A R T M A N N [1989]; FORNEFELD [1990]). All these techniques require the specification of the structure of the material functions. Usually they are very time consuming because a whole system of differential equations has to be solved to get one single value of the weight function. The large expenditure of optimization techniques becomes evident when it is considered that the integration has possibly to be performed several times, for example, with different rates of loading. Another principal problem that arises only with deterministic algorithms is the choice of an initial estimate of the parameters. The advantage of optimization techniques is their extensive independence of the kind of the problem and their suitability for programming. Here, another method is used to some extent that can be characterized as incremental reduction of material functions to experimentally observable quantities. The decisive advantage of this procedure is that the structure of the material functions does not need to be assumed in advance, but is obtained by a set of discrete values of arbi-
Rate-independent plasticity
355
trary density. Problems may arise, however, if the differential equations are extremely sensitive to small variations of parameters. It is not an easy task to make a simple decision for the "best" strategy of identification for the material functions. The following general recommendations, however, can be made: If the structure of the material functions is not known at all, incremental reduction of these functions to experimentally observable quantities should be employed. If only material parameters have to be fitted, optimization techniques may have advantages. In the case of a large number of parameters or if their initial values are not known accurately, stochastical optimization algorithms like the evolution strategy developed by RECn~E~G [1973] are recommended. If, on the other hand, only a few parameters are to be determined with fairly good initial values, as is the case with the functions g(r) and c(r), a deterministic algorithm can be used. A decisive feature of the proposed concept is the attainment of the underlying rateindependent material model in the case of quasistatic processes. Therefore, the functions describing hardening can be decoupled from those that model rate-dependent phenomena and thus can be taken from the elastic-plastic model. They were evaluated by using a deterministic optimization algorithm to fit approved hardening functions to experimental data. The viscous function A (A,v), however, is totally unknown except for the restrictions of eqns (27) and (28). For that reason, it is determined by its reduction to experimentally observable quantities. Solving the differential equations for the one-dimensional tension problem at the yield limit for A and considering al (A = O) = a~o = 1 results in the following relationship Eto E - Eto ( V ) az ( v ) = - , E,o(V) E - Eto
(29)
where E and Eto ( v ) are the elastic and the instantaneous inelastic moduli at a special rate of loading v, respectively, and Et0 denotes the instantaneous inelastic modulus for quasistatic processes. Because all quantities on the right-hand side of eqn (29) are known from experimental data, a2 (v) can be calculated. With the knowledge of this function al (A) can be determined. Supposing that the values of the state variables and material functions at arbitrary times t are given together with the time histories of stress and strain, the corresponding quantities at t + A t can be evaluated by explicit integration of the constitutive relations 6(,+at) = 6(t) + ( A o + B ( a - O) ) ( t ) A t
~(,+~,~ =
~(,7 + c(,7 ~ ~ (,7 (30)
K('+4°=~c'7+(~O- 0
A(t+,~t) -= A(t) + p
(t)
(K)At (t)
g/~ [~(t)At (tl
In these equations, abbreviations of the form c(t) = c ( r ( t ) ) and so forth have been used. Moreover, overstress A and al are computed through
356
O.T. BRUHNSand U. ROT'[
A(t+At) = ~
~ ° - - ~ (t+At) -- x/g(t+,at) (31)
al(t+at) =
k
E
3 K
(t+at)
(t+At)
by means of which a~ (A) can be expressed by an analytical function (see Appendix). As a first approximation, functions B and • are neglected during this procedure, because they have been introduced to model the rate-dependent relaxation and creep behaviour of the material. Accordingly, they are ascertained by comparison with creep tests. In this case, however, the proposed procedure fails due to stability problems. Therefore, both functions have been calculated by employing optimization techniques (see Roxx [1991]). II.5. Results As an example of application, austenitic stainless steel AISI 316L mod was examined. All simulations were performed at a temperature of 550°C. Figure 3 shows a comparison of computation and experimental results taken from the French standard RCC-MR [1985]. Calculated cyclic saturation curves for three strain-controlled tests are presented together with the experimentally obtained saturation function. These quasistatic computations emphasize the quality of the fitting of both hardening functions c (K) and g (K). The smooth transition at the onset of inelastic deformations and the symmetry of the saturation curves should be noticed. In Fig. 4, calculated monotonic tension tests at different strain rates are compared with those curves that were used for determining function A. Both the curves for high and quasistatic (here, ~ = 10 -8 l/s) strain rates are almost identical. Only in the range
o
IMeo]
300
-0,5
0,4
/~0~2
I ?
0,5
e 1"/.1
-300 Fig. 3. Calculatedcyclicsaturationcurvesand experimentallyobtainedsaturationfunction(quasistaticcase).
Rate-independent plasticity
357
0 [MPo]
.I. 10.2 s"I
200-
Jl.10"~S "1
~ _
100.
i
,
,
0
~
~
1
-
1
0
-
%
'
1 ~1
J
I
~
I
i
~5
i
I
I
lOSs 1
i
I
1,0
I
I
1,5 E[o/,]
Fig. 4. Uniaxial tension tests at different strain-rates, experiments(dashed lines) and calculations (full lines).
of ~ -- 10 -6 1/s can more pronounced deviations be observed, which certainly can be reduced by further improvement of the material parameters. Creep tests are considered as a more severe examination of both the constitutive equations and the fitting strategy. Therefore, calculated and experimentally obtained creep curves are compared in Figs. 5 and 6.
E
['/=]
0,20 160 MPo
o,1s 140 MPo
0,10 .............
----"
"'"
120 MPo / / 0
100 MPo
l
I
=
I
0
0,3
0,6
0,9
I
1,2
I
1,5
I
1,8 t[s] .107
Fig. 5. Creep curves from experiments(dashed lines) and calculations (full lines).
O.T. BRUHNS and U. ROTT
358
0,20-
0,15 -
150 MPa 0,10 -
130 MPo 0,05-
110 MPa
f
o,3
r
0,6
T
7
m
o,9
1,2
1,5
~-I1~
1,8
t [s]
.107 Fig. 6. Creep curves from experiments (dashed lines) and calculations (full lines).
The stress levels of Fig. 5 were used to determine functions B and ~, whereas Fig, 6 shows real predictions. Both strain level and strain rate for secondary creep agree rather well. More distinct deviations, however, can be found in the region of primary creep. This is here a consequence of the employed weight function; it can be improved by further modification, though with a loss of the agreement at secondary creep. Another approach, which may be more successful, consists of a modification of the functions themselves. As a whole, comparisons between one-dimensional computations and experiments show encouraging results. FE calculations were performed to check the suitability of the proposed concept for the analysis of structures. From a number of results, only one example is presented: The expansion of the inelastic region of a rectangular plate with a circular hole is demonstrated with Fig. 7. On the left-hand side, the analysis based on the proposed model is displayed and compared with an analysis for ideal-plastic material first examined by ZI~NKmWICZ[1975]. The loading parameter p is here defined as ratio of tension stress applied at the upper boundary of the plate and (instantaneous) yield limit. In the analysis, the expansion of the inelastic region starting from the hole is similarly reproduced. The same loading parameter causes a smaller inelastic domain in the case of AISI 316 than for the ideal-plastic material, due to the hardening :of the material. The bulge and the development of a second inelastic region are features that can be observed in both analyses. 1II. CONCLUSIONS
A new concept is presented that allows the extension of classical plastic material models into rate-dependent descriptions by definition of an evolution equation for the atherreal stresses. The appropriate choice of a rate-dependent material function enables the
Rate-independent plasticity
I~13
359
,r('-v-f
0,92
[ ii iJ llj IJ¢l I I'%11 I I /[/I////V/Z//
~
1,13
Fig. 7. Expansion of inelastic region due to increase of loading parameter p.
attainment of an exact limiting value in the case of quasistatic processes. This is essential not only from a micromechanical point of view but also for the analysis of structures displaying regions of both very low and high deformation rates. Furthermore, this feature provides the basis for an efficient strategy to determine material functions where rate-independent hardening functions can be separated from those reproducing viscous behaviour. The practicability of this concept is assessed by comparing the calculations with experimental and numerical results from the literature. In the case o f uniaxial processes, a good agreement for cyclic loading and creep tests is emphasized. This is of particular importance for application to many technical problems. The qualification for implementation into FE-codes was proved by analysis o f a standard benchmark test. It is intended to employ this concept on further plastic material models, for example, on multiyield surface models, to accommodate even better cyclic behaviour. In addition, nonisothermal processes will be taken into account.
REFERENCES
1963
PE~X'NA,P., "The Constitutive Equations for Rate Sensitive Plastic Materials," Quart. Appl. Mech., 20, 321. 1973 RECrmr~BERG,I., Evolutionsstrategie, Friedrich Frommann Verlag, Stuttgart. 1975 ZmNKmW]cz,O.C., Methode der Finiten Elemente, Carl Hanser Verlag, Miinchen. 1976 HART,E.W., "Constitutive Relations for the Non-Elastic Deformation of Metals," J. Engng. Mat. Techn., 98, 193. 1976a MrZLBR,A.K., "An Inelastic Constitutive Model for Monotonic, Cyclic and Creep Deformation: Part I--Equations Developmentand Analytical Procedures," J. Engng. Mat. Techn., 98, 97. 1976b MrtLER, A.K., "An Inelastic Constitutive Model for Monotonic, Cyclic and Creep Deformation: Part II-Application to Type 304 Stainless Steel," J. Engng. Mat. Techn., 98, 106. 1977 K~G, R.D., "Numerical Integration of Some New Unified Plasticity-Creep Formulations," Proc. 4 th SMIRT Conference, M 6/4.
360
1982 1983 1984 1985 1985 1987 1987 1989 1989 1989 1990 1991
O.T. BRUHNSand U. ROTT
BRUHNS,O.T., et al., "New Constitutive Equations to Describe Infinitesimal Elastic-Plastic Deformations," ASME, 82-PVP-71, 1. Ct~mocrm, J.L., "On the Plastic and Viscoplastic Constitutive Equations-Part l: Rules Developed With Internal Variable Concept," J. Press. Vessel Techn., 105, 153. BRtmNS, O.T., BOEC•E, B., LINK, F., "The Constitutive Relations of Elastic-Inelastic Materials at Small Strains," Nucl. Engng. Design, 83, 325. RCC-MR, "Design and Construction Rules for Mechanical Components of FBR Nuclear Islands," AFCEN. YAO, D., and KREMPL, E., "Viscoplasticity Theory Based on Overstress. The Prediction of Monotonic and Cyclic Proportional and Nonproportional Loading Paths of an Aluminium Alloy," Int. J. Plast., 1,259. BRUHNS,O.T., PITZER, M., "Constitutive Modelling in the Range of Inelastic Deformations- Evaluation of Parameters," in P.S. WroTE (ed.), Constitutive Modelling in the Range of Inelastic Deformations-Uniaxial Evaluation, GEC-Report (W) C. 12.224, Whetstone, Annex C. KREMPL,E., "Models of Viscoplasticity, Some Comments on Equilibrium (Back) Stress and Drag Stress," Acta Mech., 69, 25. BRUHNS,O.T. et al., "The lnteratom Model," in J. FAN, S. MURAKAMI(eds.), Advances in Constitutive Laws for Engineering Materials, Vol. 1, Pergamon Press, New York, pp. 16-21. EFTIS,J., ABDEL-KADER,m.s., JONES, D.L., "Comparisons Between the Modified Chaboche and Bodner-Partom Viscoplastic Constitutive Theories at High Temperature," Int. J. Plast. 5, 1. MOLLER, D., HARTMANN, G., "Identifikation yon Werkstoffparametern mit einem numeriscben Optimierungsverfahren nach der Evolutionsstrategie," in O.T. BRUHNS(ed.), GroBe Plastische Form/inderungen, Bad Honnef 1988, Mitt. Inst. Mech. 63, Bochum, 67. FORNEFELD,W., ,'Zur Parameteridentifikation und Berechnung yon Hochgeschwindigkeitsdeformationen metallischer Werkstoffe anhand eines Kontinuums-Damage-Modells," Mitt. Inst. Mech. 73, Bochum. ROTT,U., "Ein neues Konzept zur Berechnung viskoplastischer Strukturen," Mitt. Inst. Mech. 76, Bochum.
Institute of Mechanics Ruhr-University D-44780 Bochum, Germany (Received in final revised form 15 June 1993)
APPENDIX: MATERIAL FUNCTIONS A N D THEIR PARAMETERS
For the purely plastic material functions c (r) and g (K), the formulations proposed by BRUHNS and PITZER [1987] were employed g(K) = go + (g~ -- g0)(1 -- e-C'~),
(A1)
c(K) -
(A2)
2
EEt
3 E - Et
1 dg
2 dK'
with the initial value go = -~°2
(A3)
goo = czgo.
(A4)
and the limiting case
I n the r a n g e o f inelastic d e f o r m a t i o n s , the t a n g e n t m o d u l u s Et = do/de is a s s u m e d to
be a function of internal variable K
Rate-independent plasticity Et = E t ~ +
c4(c3 - E~ c4)
/
c4 + h +
~
c4A(K) /2'
361 (AS)
h2 + ao(1 - ~b~)
with abbreviations
c4 =
1
~ 1c3 -- ao(1 - ~b~.)],
¢~ =
Et~/E
(A6)
and
h[,~(K)] =
~/ao A(r) 1 E + ~o(1 - ~ ) )
(A7)
The viscous function A (A,v) was found to be A = al (A)a 2 (v),
(A8)
with
a~(A)
=
1-L [tanh(DA + C) - 1] + L, tanh C - 1
(A9)
and
=~rl e~lvz' + r2e~2vz~, if a2(v)
Vo < v < v~
L 1, if v --< Vo.
(A10)
Parameter r~ herein has to be chosen according to rl = 11 -
r2e ~:g' }e -~:g~,
(A1 l)
such that Vo can be looked upon as the rate o f quasistatic loading. Functions ~ and B finally describe the creep behaviour by means o f ~ ( A ) .~ e dl(A/'~)d~ - 1
(A12)
and d3 tanh(d4xff + ds) + d6, if , i f > x/f* B ( i f , ][tr'[[ ) =
0, otherwise,
where x/f* denotes the value of ~ from
(A13)
at stationary creep and which can be calculated
362
O . T . BRUHNS and U. ROTq"
l/ar tanh( •
t
Herein coefficients d 3 t o d 6 are assumed to be quadratic polynomials of
di
=Pi]
+Pi211,~'ll +Pi31t,T'll 2, i = 3,4,5,6.
II-' II (A15)
The related material parameters of AISI 316L mod at a temperature of 550°C are compiled in Table 1.
Table 1. Elastic and inelastic material parameters of AISI 316L mod at 550°C Elastic material parameters E [MPa] 1.492-105 Plastic material parameters a o [MPa] 74.5 c2 [--] 9.12 Eto [MPa] 1.417.105 Function A L [-] C [-] al [ - ] /~1 [ - ] r2 [ - ] Vo [MPa/s]
75.0 -3.901 -5.182.101 1.734-10 2 9.691 • 10 --I 1.2.10 -3
[-l ct [MPa] c 3 [MPa] E~o~ [MPa] D [MPa-I] a2 [ - ] 132 [ _ ]
0.3 7.300.10 -2 151.0 1.57.103 0.269 -4.268 4.982.10-3
v~ [MPa/s]
1.3.103
6.117.10 -15 1
d2 [ - ]
3.326
Function B P31 [l/s] P33 [MPa-2/s]
3.521.10 -8 3.287-10 -14
P32 [M P a - l / s ]
3.526.10 - t °
P41 IMP a - l ] P43 [M Pa-3]
9.547-10 -1 3.316.10 -7
P42 [MPa-2]
2.307-10 -5
Psm [ - ] P53 [M Pa-2]
-4.985 "101 -3.780.10 -5
P52 [MPa-1]
Function dl [--] [MPa]
P61 [l/s] P63 [MPa-2/s]
2.324.10 -8 4.018.10 -14
P62 [M P a - l / s ]
-1.705.10
1
2.747.10 - l °