A constitutive model for rate dependent and rate independent inelasticity. Application to IN718

International Journal of Plasticity 27 (2011) 596–619

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International Journal of Plasticity

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A constitutive model for rate dependent and rate independentinelasticity. Application to IN718

Martin Becker ⇑, Hans-Peter HackenbergMTU Aero Engines GmbH, Dachauer Straße 665, 80995 München, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 July 2010Received in final revised form 11 August 2010Available online 21 August 2010

Keywords:B. Constitutive behaviorCyclic loadingElastic-viscoplastic materialC. Finite elementsNumerical algorithms

0749-6419/$ - see front matter � 2010 Elsevier Ltddoi:10.1016/j.ijplas.2010.08.005

⇑ Corresponding author. Tel.: +49 89 1489 3567;E-mail addresses: [email protected] (M. Bec

In this paper, a constitutive model for the unified description of rate dependent and rateindependent material behavior is proposed. It is applicable to isotropic metals subjectedto arbitrary thermomechanical loading conditions at small strains. The focus of the modelformulation is its validity for the full range of thermal and mechanical loading conditionsto be covered in an industrial context by e.g. modern aero-engine designs. Consequently,the proposed model describes the material behavior on a macroscopic level covering thefull temperature range from room temperature to the upper application limit under mono-tonic as well as cyclic loading. Special emphasis is also put on the correct representation ofthe observed ratcheting behavior. The most prominent features of the model are the com-bined treatment of both, rate dependent, as well as rate independent inelasticity through alimit surface concept, the description of primary, secondary and tertiary creep behaviorand the application of an appropriate backstress evolution equation. A fully implicit inte-gration algorithm for the proposed model is developed and implemented in connectionwith uniaxial integration point drivers as well as three alternative Finite Element packages.The parameters of the proposed model can be identified based on a limited number of com-plex cyclic tests and monotonic creep tests. After reporting on the fitting results for suchtests for the Nickel-base superalloy IN718 the predictive capabilities of the proposed modelare assessed for a number of isothermal and nonisothermal tests. Finally, the performanceof the algorithmic implementation into the Finite Element packages is briefly addressed.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In view of more efficient designs of structural components, the material capability in withstanding various loading re-gimes is exploited further and further. Thus, in order to allow for lighter designs at ever increasing temperature and/or loadlevels, material scientists eagerly develop high-end superalloys with improved capabilities. However, in order to safely profitfrom these capabilities during the design phase as well as to allow for an exploitation of the entire potential of already exist-ing alloys it is at least of the same importance to enable a precise and efficient description of the material response withinthe relevant loading regime.

Therefore, the scope of this paper is the proposal of a constitutive material model allowing for an efficient applicationthroughout the design of structural components of e.g. modern aero-engines. Many constitutive material models for thedescription of cyclic inelasticity have been proposed in the literature over the past few decades (Armstrong and Frederick,1966; Mróz, 1967; Bodner, 1987; Contesti and Cailletaud, 1989; Nouailhas, 1989; Chaboche et al., 1991; Freed and Walker,1993; Ohno and Wang, 1993; Chaboche, 1991; Chaboche, 1994; Jiang and Sehitoglu, 1996; Auricchio, 1997; Ohno, 1998;

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fax: +49 89 1489 97757.ker), [email protected] (H.-P. Hackenberg).

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M. Becker, H.-P. Hackenberg / International Journal of Plasticity 27 (2011) 596–619 597

Abdel-Karim and Ohno, 2000; Bari and Hassan, 2000; Chen et al., 2005; Kang and Kan, 2007; Zhang and Jiang, 2008; Taleband Cailletaud, 2010; Krishna et al., 2009; Bai and Chen, 2009; Abdel-Karim, 2010) see also the review by Chaboche (2008)for a detailed discussion of some of these and other models. Recent models for certain Ni-base superalloys can be found inManonukul et al. (2005), Mücke and Bernhardi (2006), Shenoy et al. (2006). However, none of the beforementioned modelsappears to be general enough in the previously addressed context. Except for the latter three models, most of them addressonly a limited temperature range, i.e. they are not intended to cover the full range from nearly rate independent behaviorwithin the low temperature regime to mainly rate dependent behavior at the upper temperature applicability limit. Someare not capable of describing monotonic as well as cyclic behavior including softening. And other models do not allow foran efficient integration algorithm which can be implemented into modern commercially available Finite Element (FE) pack-ages such as ABAQUS, MARC or ANSYS or the freeware package CalculiX.

In order to cover all of these features we propose a new constitutive model which also incorporates successful aspects ofsome of the abovementioned formulations. Therefore, after almost 20 years, we pick up again the idea of Freed and Walker(1993) to bound viscoplasticity by creep and plasticity and integrate it into our limit surface concept. This concept is the keyto a unified description of both, rate dependent and fully rate independent inelasticity. In this context we derive, for the firsttime and in contrast to Freed and Walker (1993), a consistent evolution equation for the inelastic strain in the rate indepen-dent case through the principle of maximum dissipation. Further, we improve the model by inclusion of a backstress formu-lation with scalable ratcheting rate, owing to Ohno and Wang (1993). Additionally we extend the concept to hardening/softening behavior and propose a new creep formulation which covers primary, secondary and tertiary creep behavior. Fi-nally we develop a new, consistent and very efficient integration algorithm for the abovementioned model. Here we extendthe classical return mapping scheme by replacing the elastic predictor step by a rate dependent inelastic predictor step.

The model can be classified as a unified multimechanism model. Examples of other recent multi-mechanism models areManonukul et al. (2005), the SPC-model in Kang and Kan (2007) and Taleb and Cailletaud (2010). However, these are non-unified theories where the first two address, like our model, rate independent as well as rate dependent inelasticity while thelatter one can only be formulated for one or the other case. Comparing with Kang and Kan (2007), the model proposed in thefollowing might be considered as a unified incorporation of their SVC-model into the SPC-model. Hence, we expect it to over-come the deficiencies of the SPC-model and to have the same potential as the SVC-model. This seems to be supported bysome results presented later, namely the strain rate dependency under monotonic loading.

Let us address some further points which distinguish our model from recently investigated formulations. Concerning theratcheting behavior many promising approaches are available (see Kang, 2008, for an overview). However, the underlyingmodels are usually restricted to either rate dependent or rate independent inelasticity. This may owe to the fact, that in mostof the metals investigated in this context (steels, aluminum, copper or solders) ratcheting is observed over the full temper-ature applicability range (Kang et al., 2006, e.g.; Hassan et al., 2008; Bai and Chen, 2009). In contrast, in the current inves-tigation we especially address the fact, that in the Ni-base superalloy IN718 at room temperature no significant ratchetingseems to be observable while with increasing temperature cyclic creep or relaxation becomes quite relevant. Concerning thedescription of the softening behavior we rely, in line with Krishna et al. (2009), on a two term hardening/softening relationallowing to model both, monotonic and cyclic inelasticity. But, in contrast to Krishna et al. (2009) who model the softeningbehavior through an evolution of the yield surface size, we incorporate it into the backstress evolution equation. Here weaccount for softening through evolution of the parameters Hi which is in contrast to Zhang and Jiang (2008) and Abdel-Karim(2010) where an evolution of the critical backstress values ri is assumed in connection with an additional term in the back-stress evolution equation, compare also Döring et al. (2003). The softening evolution itself is, like in Abdel-Karim (2010), as-sumed to be of isotropic character and hence driven by the amount of accumulated inelastic strain whereas in otherapproaches, like e.g. Zhang and Jiang (2008), it depends on the strain memory surface size.

Clearly our model still exhibits some limitations, the most obvious ones being the restriction to (1) small strains and (2)isotropy and also the limited predictive capabilities for the (3) ratcheting or (4) creep behavior under certain multiaxial load-ing conditions. Further, (5) strain range effects or (6) tension compression asymmetries are not addressed here. However, iffound necessary, the modular structure of the presented model should allow for an extension beyond any of the aforemen-tioned limitations. E.g. through the concepts of (1) multiplicative plasticity (see e.g. Miehe, 1993 for a detailed discussion)and (2) multisurface crystal plasticity. In view of an appropriate description of the (3) ratcheting behavior under nonpropor-tional multiaxial loading, the model might be improved through the concepts investigated e.g. in Chen et al. (2005), Rahmanet al. (2008) or Abdel-Karim (2009). Or, if in the same context a large strain extension of the backstress evolution equation isattempted, the framework provided by Johansson et al. (2005) might be employed and a more precise distinction of backstress related kinetics might be covered through the concept of Bai and Chen (2009). A promising extension for (4) multiaxialcreep in a Ni-base superalloy, including stress state sensitive cavitation damage, can be found in Manonukul et al. (2005). If itis found necessary to include (5) strain range effects, the introduction of a memory surface (Zhang and Jiang, 2008, e.g. re-cently in; Krishna et al., 2009; Taleb and Cailletaud, 2010) can be employed, especially in order to account for nonpropor-tional hardening. Regarding (6) tension compression asymmetries Iyer and Lissenden (2003) developed a promisingmodel for IN718 based additionally on the third effective stress invariant. The proposed model should be general enoughto capture the constitutive response of many metals. But even for the same alloy, but different microstructure the materialparameters vary significantly. Such a sensitivity of the macro-model parameters with respect to grain size and precipitateconfiguration has been investigated for a Ni-base superalloy in Shenoy et al. (2008) based on an experimentally calibratedrate dependent micro-level crystal plasticity model.

598 M. Becker, H.-P. Hackenberg / International Journal of Plasticity 27 (2011) 596–619

The paper is organized as follows. After laying out the principal model ingredients in their continuous form in Section 2, afully implicit integration algorithm, ready for implementation into any FE package, is developed in Section 3. Next, the iden-tification of the model parameters is discussed in Section 4 and subsequently presented for IN718. In Section 5, some fittingresults and predictions for isothermal as well as non-isothermal monotonic and cyclic tests are presented for IN718. At theend of this section the algorithmic implementation into ABAQUS, MARC and CalculiX is briefly validated. The paper closeswith some concluding remarks in Section 6.

2. Continuous formulation of the constitutive model

In the following, a constitutive material model is presented, valid throughout the entire temperature applicability range(from room temperature up to temperatures providing a sufficient safety margin to the temperature limit for the stability ofthe microstructure-for IN718>ca. 680�C). In view of numerical efficiency and the length scale of the components for whichthe model shall be employed, the model is formulated on the macroscale based on phenomenological expressions reassem-bling the mechanisms on the microscale. Furthermore, the formulation is limited to small strains and isotropic materials.

Bold letters T indicate second order tensors, a colon between two tensors denotes their inner product T : T , the symbol �denotes the dyadic product and jT j ¼

ffiffiffiffiffiffiffiffiffiffiffiT : Tp

denotes the norm of the tensor T . Through a volumetric-isochoric split, we cansplit the total strains � into a deviatoric part E and a volumetric part 1

3 tr½��1

Fig. 1.inelasti

� ¼ dev½�� þ 13

tr½��1 ¼ E þ 13

tr½��1 ð1Þ

With the classical assumption of purely isochoric inelasticity for metals, the additive split of the total deviatoric strain E intoa deviatoric elastic part Ee and an inelastic part �in and an assumption of isotropic Hookean elastic material response thestresses read

r ¼ 2l½E � �in� þ jtr½��1 ¼ S þ rvol ð2Þ

in terms of the shear and bulk moduli l and j and the deviatoric and volumetric stresses S ¼ dev ½r� and rvol. In order todetermine the stress response of the material to an arbitrary, prescribed strain � we need to determine the evolution ofthe a priori deviatoric inelastic strain �in. In the following, the ingredients of the model leading to a specification of theinelastic strain evolution will be first laid out separately. Then, at the end of this section, we will assemble all governingequations in a summary of the complete model.

2.1. The limit surface concept – key to unified rate dependent and rate independent inelasticity

Neglecting highly dynamic loading conditions like impact events, we can understand the inelastic behavior of manysuperalloys at low temperatures to be rate independent over a wide range of practically relevant stress or strain rates. Undersuch conditions the most appropriate description of inelasticity is achieved through the classical definition of a convex elas-tic domain

E :¼ fCjU ¼ffiffiffiffiffiffiffiffiffiffiffiC : Cp

� K < 0g ð3Þ

Through (3) U defines a limit surface between purely elastic behavior inside the elastic domain (i.e. for U < 0) and rate inde-pendent inelastic (i.e. purely plastic) behavior for stress states on the limit surface (i.e. for U ¼ 0), see Fig. 1a. In (3) C denotesthe effective stress

C ¼ S � B ð4Þ

a b

Limit surface concept in deviatoric stress plane. (a) Purely rate independent inelasticity with elastic domain E and yield stress K. (b) Generalcity with elastic limit stress Y, viscoplastic domain V and inelastic limit stress K.


in terms of the (deviatoric) backstress tensor B which describes the translation of the elastic domain, i.e. the yield surfaceU ¼ 0, within the deviatoric stress space and K denotes the yield stress which itself might be a function of the inelastic defor-mation state, i.e. the internal variables, describing isotropic hardening or softening.

Now, as higher temperatures are considered, time dependent deformation mechanisms start to play a more and moredominant role resulting in a rate dependent inelastic material response. However, beyond certain high stress rates or strainrates the inelastic response can again be assumed to be rate independent, i.e. a further increase of the stress rate or the strainrate, does not change the material response significantly and inelasticity is bound again by the limit surface U. But for effec-tive stress states residing inside the limit surface (i.e. for U < 0) now the material shows rate dependent inelastic (i.e. visco-plastic) behavior. Thus a reasonable assumption is to now interprete U as a limit surface separating rate dependent (i.e.viscoplastic) from rate independent (i.e. plastic) inelastic behavior and thus to introduce the viscoplastic domain

V :¼ fCjU ¼ffiffiffiffiffiffiffiffiffiffiffiC : Cp

� K < 0g ð5Þ

in terms of the inelastic limit stress K. The inelastic limit surface U exists throughout the whole temperature range. Only thegoverning equations describing the rate dependency will predict a higher or a lower amount of rate dependency of the mate-rial behavior just depending on the current temperature level. For very low effective stress or temperature levels rate inde-pendent elastic behavior might again be observed. Thus, we introduce the nested purely elastic domain

E :¼ fCj/ ¼ffiffiffiffiffiffiffiffiffiffiffiC : Cp

� Y < 0g ð6Þ

in terms of the elastic limit surface / and the elastic limit stress Y 6 K (at low temperatures a reasonable assumption wouldbe Y � K). Consequently we observe rate independent elastic behavior for

ffiffiffiffiffiffiffiffiffiffiffiC : Cp

< Y , rate dependent inelastic behavior forY 6


< K and rate independent inelastic behavior forffiffiffiffiffiffiffiffiffiffiffiC : Cp

¼ K , see Fig. 1b for a visualization.On the one hand, the purely elastic regime E provides us with some additional flexibility in the description of the material

response and may be beneficial regarding computational costs. On the other hand, it requires the effort of a further limitsurface check. In the present context we will not consider the purely elastic domain and just represent it by sufficientlylow creep/relaxation rates at low stress/strain or temperature levels. This corresponds to setting Y ¼ 0.

A combination of rate dependent and rate independent inelasticity similar to the previously discussed limit surface con-cept has already been introduced by Contesti and Cailletaud (1989) and later Freed and Walker (1993). However, in contrastto Contesti and Cailletaud (1989), the model proposed here is based on one unique inelastic strain and on one unique back-stress only. This is in complete analogy to the formulation of the latter two authors. Thus, the current formulation can beclassified as a so-called unified theory. Therein, a unique inelastic strain evolves differently depending on the regime the cur-rent stress state resides in-with the regimes being characterized by the limit surface concept. The main differences betweenour model and the one by Freed and Walker (1993) are (1) the derivation of a consistent evolution equation for the inelasticstrain in the rate independent case through the classical principle of maximum dissipation, (2) the incorporation of soften-ing/hardening effects and (3) the introduction of an alternative backstress evolution equation resulting in a more adequatedescription of the ratcheting behavior.

In contrast to classical overstress models (e.g. Chaboche, 1977) the limit surface concept introduced above allows to de-scribe in a unified manner both, rate dependent as well as fully rate independent inelasticity exactly. This is of special impor-tance if a wide range of temperatures is to be covered where at high temperatures both rate dependent as well as rateindependent behavior can be observed while at low temperatures rate independent behavior is dominant. The limit surfaceconcept allows for an algorithmic formulation which is stable for any applied stress or strain rate in contrast to the overstressmodels where convergence difficulties can be experienced in connection with the necessarily high viscosity exponents atlow temperatures. Furthermore, the limit surface concept covers, as special cases, both, classical purely rate independent‘plasticity’ for Y ¼ K > 0 and rate dependent (overstress-) ‘viscoplasticity’ for Y > 0 and K !1.

2.2. Evolution equations for the inelastic strains

In line with the limit surface concept, the strains evolve according to the respective case.

2.2.1. The case of rate independent inelasticityFor the rate independent case, i.e. effective stress states C satisfying U ¼ 0, we derive the evolution equation for the

inelastic strain �in from the classical principle of maximum dissipation in connection with (5). The resulting constrained opti-mization problem can be solved by a Lagrange multiplier method giving the evolution equation

_�in ¼ _cCffiffi

Jp ¼ _cn ð7Þ

and the Karush–Kuhn–Tucker (KKT) loading–unloading conditions

_c P 0; U ¼ffiffiJ

p� K 6 0; _cU ¼ 0 ð8Þ

where the plastic slip rate _c ¼ j _�inj, often also denoted as plastic multiplier, specifies the amount of the inelastic strain rate _�in

and n ¼ C=ffiffiJ

pdetermines its direction. Therein

ffiffiJ

p¼


is the norm of the effective stress. The plastic slip rate _c itself is


implicitly specified through the loading–unloading conditions (8) by enforcing the stress state to lie on the yield surfaceU ¼ 0 if plastic slip occurs, i.e. if _c > 0, see also Section 3.2.1. In this framework of associative rate independent inelasticitythe flow direction n obeys the normality rule, i.e. the direction of inelastic straining is always perpendicular to the yield sur-face U ¼ 0.

2.2.2. The case of rate dependent inelasticityIn the rate dependent case we assume the inelastic deformation to comprise of three parts. These are related to primary,

secondary and tertiary creep, indicated by superscripts pc, sc and tc

_�in ¼ _�pc þ _�scþtc ð9Þ

For the part of the inelastic strain evolution associated with primary creep, we use exactly the same structure (7) as for therate independent case, i.e.

_�pc ¼ _cCffiffi

Jp ¼ _cn ð10Þ

Only the slip rate _c is now determined on basis of a phenomenological function f describing transient creep behavior insteadof using the KKT. Following the approach of Freed and Walker (1993) we transform the steady-state Zener parameter ofGarofalo (1963) into a transient one by relating it to the norm of the effective stress giving the function f

_c ¼ f ¼ Apcsinhnpc

ffiffiJ

pCpc

" #ð11Þ

in terms of the material constants Apc; Cpc and npc which might be temperature dependent. Alternatively, (11) might be ex-

tended by e.g. an Arrhenius function to capture the temperature dependency. The incorporation of the sinh-function in (11)reflects the power-law breakdown generally observed in the creep response: At low stress levels the rate controlling defor-mation mechanism is diffusion controlled dislocation climb and changes to obstacle controlled dislocation glide at highstress levels, thus the creep response is altered from power-law to exponential-type behavior and finally the material re-sponse is practically rate independent at very high stress levels.

We describe the part of the rate dependent inelastic strain evolution associated with secondary and tertiary creep in afashion similar to (10), however now we relate it to the norm

ffiffiIp¼

ffiffiffiffiffiffiffiffiffiffiS : Sp

of the deviatoric stress S and its direction m ¼ S=ffiffiIp

_�scþtc ¼ em ð12Þ

instead of the corresponding quantities related to the effective stress C in (10). This implicitly satisfies the physical fact thatat steady state the plastic strain rate _�in becomes coaxial with the deviatoric applied stress S. A formulation in terms of Cwould also be possible, but would not allow for a decoupled representation in the sense of (9) since it would introducean additional transient contribution into (12). Furthermore, such a formulation would require the inclusion of an additionalrecovery term into the backstress evolution equation discussed in Section 2.3 and constraints for the additionally introducedmaterial parameters would have to be found, so that at steady state _�in and S are again coaxial. In (12) the material function econsists of two parts, a first part esc describing secondary creep and a second part etc modeling tertiary creep

e ¼ esc � etc

with

esc ¼ffiffiffi32

rAsc

1

ffiffiffiffiffiffi32

I

r" #nsc1

þ Asc2

ffiffiffiffiffiffi32

I

r" #nsc2

8<:

9=;

etc ¼ ð1� DtcÞ�ntc

ð13Þ

in terms of the temperature dependent parameters Asc1 ; nsc

1 ; Asc2 ; nsc

2 for secondary and ntc for tertiary creep. Similar to (11)the two terms in (13)2 allow for a distinction of the low and high stress creep response. Instead of (13)2 a sinh-function like in(11) might as well be used or the classical description of Larson and Miller (1952) for the secondary creep rate could be em-ployed, thus allowing for an effective inclusion of the temperature dependency. In the tertiary creep expression (13)3 Dtc de-notes the accumulated tertiary creep damage which is assumed to evolve through

_Dtc ¼ 1tr

ð14Þ

in terms of the time to creep rupture tr . Tertiary creep in Ni-base superalloys is attributed to various damage mechanisms inthe literature. E.g. Mukherji and Wahi (1996) relate it to rafting (transformation of the c0 precipitates into plate-like struc-tures) at very high temperatures (>980 �C), Maclachlan et al. (2001) attribute it to shearing of the c0 precipitates, or Ibanez(2003) observed the formation of voids and cracks perpendicular to the loading direction. The present formulation is re-stricted to a purely phenomenological description of such mechanisms. Given several uniaxial creep tests up to specimen


failure, the rupture time tr (in [s]) can, at a certain temperature T, be described through the beforementioned relation of Lar-son and Miller (1952)

Fig. 2.

tr ¼ 3600 � 10ð1000P

T �CÞ ð15Þ

in terms of the material parameter C and the so-called Larson–Miller parameter P, which itself depends on the applied stress,e.g. through a fourth order polynomial

P ¼ A0 þ A1 þ A2 þ A3log10

ffiffiffiffiffiffiffiffiffiffi3=2I

ph i� �log10


ph in olog10


ph ið16Þ

with the material parameters A0; . . . ;A3. Thus in (13)1 the creep rate increases through (13)3 along with an increasing tertiarycreep damage Dtc . Therein, Dtc evolves according to the time spent at a certain stress level in relation to the rupture time tr

which the material can withstand at the corresponding stress and temperature level. Note, that a prerequisite of any creepformulation is its objectivity with respect to the initial time which is entirely met by the model presented above.

2.3. The backstress formulation

After discussing the limit surface concept and the inelastic strain evolution equations the third, important ingredient forthe modeling of cyclic inelasticity is the backstress formulation. For the-at most moderate-strain ranges investigated in thepresent paper, kinematic hardening (i.e. translation of the limit surfaces in the deviatoric stress space) is generally observedin metallic materials. Mostly it occurs in combination with cyclic hardening/softening (see Section 2.4) and sometimes withan additional isotropic hardening/softening (i.e. change of the limit surface diameter). The kinematic hardening is governedby the evolution equation for the backstress B.

Many kinematic hardening rules have been proposed in the literature, see Chaboche (2008) for a detailed overview.Among those are the multisurface concepts of Mróz (1967) and Garud (1982) or the models which base on extensions ofthe linear kinematic hardening law of Prager (1949) such as the improvement by Armstrong and Frederick (1966) and itsfurther enhancement by Chaboche (1977). It is well known, that in many cases the latter two models largely overpredictthe ratcheting response observed experimentally.

An ongoing inelastic strain evolution during a cyclic, stress controlled test with constant mean stress and stress amplitudeis usually denoted as a ratcheting response, see Fig. 2a for a visualization. This is characterized by open stress–strain hyster-esis loops. The underlying material behavior also results in a mean stress relaxation in a strain controlled test with constantmean strain and strain amplitude, see Fig. 2b. In order to characterize the specific material response in view of its ratchetingbehavior and to allow for an appropriate choice of the backstress formulation it is important to differentiate between lowtemperature ratcheting and the combination of ratcheting and cyclic creep or relaxation at high temperatures. In this regard,the stabilized material response, usually after completing the first few hundred cycles, should be investigated.

In view of a more accurate description of the ratcheting behavior, Chaboche (1991) and Ohno and Wang (1993) incorpo-rated two different threshold concepts for the dynamic recovery of the backstress into the Armstrong-Frederick formulation.Bari and Hassan (2000) have shown for various uniaxial and biaxial ratcheting experiments on steel that the Ohno-Wangmodel performed slightly better than the Chaboche model, see also Ohno and Wang (1995) for a comparison where theChaboche model was slightly extended. At low temperatures for IN718 basically no ratcheting or mean stress relaxationcan be observed in the uniaxial cyclic tests available to the author. Thus we use the Ohno and Wang (1993) formulationto describe the backstress evolution

B ¼XnB

i¼1

Bi

_Bi ¼ Hi ri _cn� jBijri

� �mi

_cn :Bi

jBij

� �Bi

¼ Hiri _cn� gi _cLi

ð17Þ

a b

Ratcheting and mean stress relaxation. (a) Ratcheting in a cyclic stress controlled test. (b) Mean stress relaxation in a cyclic strain controlled test.


with large values for mi. It should be noted, that we assume one unique total backstress B which evolves according to (17)during both rate dependent and rate independent inelasticity. Following Chaboche et al. (1979) the total backstress isdecomposed into nB backstresses in (17)1. In (17)2 < � > indicates the MacCauley bracket

x ¼x if x > 00 if x 6 0

¼ d½x� � x with d½x� ¼

1 if x P 00 if x < 0

ð18Þ

Hi and ri are material constants. For mi !1 (17) predicts no ratcheting or mean stress relaxation and each decomposedhardening rule (17)2 or (17)3 simulates, like the basic law of Prager (1949), linear kinematic hardening with a slope ofHi � ri until jBij reaches its limit value ri. For decreasing mi nonlinear kinematic hardening is simulated on each branch Bi

and the model predicts open hysteresis loops, i.e. increasing ratcheting rates. In (17)3 Li ¼ Bi=jBij indicates the direction ofthe backstress and the material function gi is defined as

gi ¼ HirijBijri

� �miþ1

d n : Li½ �n : Li ð19Þ

Carefully note, that (17) implies the assumption that secondary and tertiary creep do not contribute to the backstress evo-lution, vice versa reflected through (12). Although (17) allows for good approximations of uniaxial ratcheting rates twodrawbacks remain: 1. for large values of mi and small numbers of nB the predicted material behavior is significantly multi-linear. 2. under multiaxial loading conditions the formulation of Ohno and Wang (1993) was shown to overpredict the exper-imentally observed ratcheting rate, see e.g. Bari and Hassan (2000). Many modifications have been proposed in order toremedy the latter deficiency, see Chen et al. (2005), Rahman et al. (2008), Abdel-Karim (2009) for an evaluation of someof them. These might be employed instead of (17) without any problem. Due to the lack of sufficient experimental dataand the expected minor impact on the practical problems considered this is not pursued here.

Other improvements of (17), which might be relevant in simulating cyclic tests with significant dwell times or noniso-thermal behavior, should concern the inclusion of a static recovery term or a temperature rate term of the form@½Hiri�=@TBi

_T=Hiri such as discussed in Chaboche (2008), pg. 1653 or 1657 f. However, this seems unnecessary in the presentcontext since the simulation results for IN718 appear to be in sufficient agreement with the available test data, see Section 5.Furthermore, as discussed in Section 4.2.3, for IN718 we can assume no temperature dependency for Hi and ri. Hence, thetemperature rate term a priori equals zero. For other materials, which, with increasing temperature, reveal e.g. significanthysteresis loop shape variations, or changes from softening to hardening behavior, Hi and ri can be strongly temperaturedependent. Then the inclusion of a temperature rate term is crucial in order to avoid unreasonable hysteresis loop shifts dur-ing nonisothermal loading, see also Mücke and Bernhardi (2006).

2.4. Cyclic hardening/softening

A progressive change of the shape of the stress–strain curve along with the accumulation of cycles results from cyclichardening or softening. Following Jiang and Sehitoglu (1996) we incorporate cyclic hardening or softening into (17) and(19) by introducing a functional dependency for Hi on a portion �c of the accumulated inelastic strain through

Hi ¼ H0i ai þ 1� aið Þexp �v1�c½ �f g bi þ 1� bið Þexp �v2�c½ �f g ð20Þ

where �c denotes the portion of the accumulated inelastic strain collected during rate independent inelasticity or primarycreep/relaxation

�c ¼Z t

0cdt ð21Þ

The two terms in (20) allow to differentiate between a first quick and significant change of the cyclic stress–strain curveshape (basically along with the first loading up to the monotonic limit surface) and a subsequent slow softening responseduring cycle accumulation. We model cyclic softening through (20), since, as also pointed out by Jiang and Sehitoglu(1996), changing the coefficients ri might result in a previously saturated backstress Bi suddenly lying above its admissiblelimit value ri if no additional constraints are enforced. Alternatively, Abdel-Karim (2010) proposed an approach relying on anevolution of ri in order to account for isotropic hardening. This required the inclusion of the term ðjBij=riÞmi Bi _ri=ri (see alsoDöring et al., 2003) into (17) but proved to result in higher accuracy in the simulation of ratcheting experiments, especiallyunder multiaxial loading. The proposal of Abdel-Karim (2010) to include the hardening relations only for a subset of the nB

backstresses is incorporated here by simply setting the corresponding constants ai and bi to 1. If found necessary, additionalisotropic softening might be introduced easily by assuming a functional dependency for the inelastic limit stress K similar to(20).


2.5. Summary of the model

Box 1 summarizes the constitutive equations of the model for the combined description of rate dependent and rate inde-pendent inelasticity under monotonic and cyclic loading laid out in Sections 2.1, 2.2, 2.3, 2.4. The specifications of the mate-rial functions used for the modeling of IN718, see also Sections 4 and 5, are given in Box 2.

Box 1: Summary of the constitutive equations.

1. Total strain � ¼ E þ 13 tr½��1

2. Stress r ¼ S þ jtr½��1
S ¼ 2l½E � �in�; I ¼ S : S; m ¼ S=
ffiffiIp

3. Effective stress C ¼ S � B; J ¼ C : C; n ¼ C=ffiffiJ

p4. Backstress B ¼

PnB

Bi
i¼1
_Bi ¼ Hiri _cn� gi _cLi; Li ¼ Bi=jBij

Inelastic limit surface U ¼ffiffiJ

p� K
5.
6. Inelastic strain evolution

6.1. Rate dependent _�in ¼ _cnþ em; _c ¼ f ; U ¼ffiffiJ

p� K < 0ffiffip

. Rate independent _�in ¼ _cn; _c P 0; U ¼ J � K ¼ 0
6.2
Box 2: Specification of the material functions for the Ni-base superalloy IN718.

1. Hardening/Softening

Hi ¼ H0 i ai þ 1� aið Þexp �v1�c½ �f g bi þ 1� bið Þexp �v2�c½ �f g

2. Dynamic recovery

gi ¼ Hiri jBij=rið Þmiþ1d n : Li½ �n : Li

3. Primary creep

f ¼ Apcsinhnpc ffiffiJ

p=Cpc

h i
4. Secondary and tertiary creep
e ¼ffiffiffiffiffiffiffiffi3=2

pAsc

1


ph insc1 þ Asc

2


ph insc2

ð1� DtcÞ�ntc

5. Tertiary creep damage

_Dtc ¼ 3600 � 10ð1000P

T �CÞn o�1

6. Larson–Miller parameter

P ¼ A0 þ A1 þ A2 þ A3log10


ph i� �log10


ph in olog10


ph i

3. Algorithmic formulation of the constitutive model

In the following, a fully implicit strain driven integration algorithm is developed for the constitutive model discussed inthe previous section. It is ready for an implementation in a general Finite Element (FE) context where the proposed algorithmis applied locally at every integration point during each global iteration step.


3.1. Time discretization

Consider a time interval ½tn; tnþ1� 2 Rþ. Then the time increment is defined as

Dt ¼ tnþ1 � tn ð22Þ

In the following, any quantity evaluated at tn will be indicated by a subscript n. For quantities evaluated at tnþ1 the subscriptnþ 1 will be omitted. In order to determine the stress response at tnþ1, we need to integrate the evolution equations of allinternal state variables throughout the time increment Dt. Combining the rate form of the deviatoric part of (2) with the evo-lution Eq. (7) we obtain the following relation for the change in inelastic strain throughout the time step ½tn; tnþ1� in the rateindependent case

�in � �inn ¼

12l

Sn � Sf g þ DE ¼ Dcn ð23Þ

in terms of the slip increment Dc ¼ c� cn and the total deviatoric strain increment DE ¼ E � En. Similarly, based on (2) and(9)–(12), we derive for the rate dependent case

�in � �inn ¼

12l Sn � Sf g þ DE ¼ Dcnþ eDtm ð24Þ

Time discretization of (17) gives the backstress at tnþ1 and the backstress increments

B ¼XnB

i¼1

Bi

Bi � Bi;n ¼ HiriDcn� giDcLi

ð25Þ

which, as pointed out before, are common to both, the rate dependent and the rate independent case. Also the portion �c ofthe accumulated slip evolves during both, rate independent inelasticity as well as primary creep/relaxation according to thetime discrete form

�c� �cn ¼ Dc ð26Þ

Finally, for the rate dependent case, we derive the time discrete form of the evolution Eq. (14) for tertiary creep damage

Dtc � Dtcn ¼

Dttr

ð27Þ

3.2. Residual expressions

Within the strain driven algorithm under consideration the total strains � are prescribed at tnþ1 and we choose the stres-ses S, the backstress components Bi and the slip increment Dc as the basic unknowns to be determined at tnþ1. In view of acompact notation we collect the stress components together with the backstress components in the vector ½X� and their algo-rithmic increments in the vector ½DX�, each with the dimensions 1� ð6þ 6 � nBÞ

X½ � ¼ ½S B1 B2 . . . BnB �T ; DX½ � ¼ ½DS DB1 DB2 . . . DBnB �

T ð28Þ

3.2.1. The case of rate independent inelasticityBased on (23), (25) and the discrete form of (8)2 we specify for each unknown S; Bi and Dc one residual expression

RS ¼1

2lSn � Sf g þ DE � Dcn¼! 0

RBi¼ Bi � Bi;n � HiriDcnþ giDcLi¼

! 0

RDc ¼ U ¼ffiffiJ

p� K ¼! 0

ð29Þ

Summarizing (29) in matrix format reads

½R� ¼RS

RBi

� �¼

12l Sn � Sf g þ DE � Dcn

Bi � Bi;n � HiriDcnþ giDcLi

" #¼! 0½ �

RDc �

¼ffiffiJ

p� K

h i¼! ½0�

ð30Þ

where the residual vectors ½R� and ½RDc� have the dimensions 1� ð6þ 6 � nBÞ and 1� 1.


3.2.2. The case of rate dependent inelasticityIn analogy to (29) we derive the following residual expressions for the rate dependent case

RS ¼1

2lSn � Sf g þ DE � Dcn� eDtm¼! 0

RBi¼ Bi � Bi;n � HiriDcnþ giDcLi¼

! 0

RDc ¼ Dc� fDt¼! 0

ð31Þ

based on (24), (25) and the discrete form of (11). (31) reads in matrix format

½R� ¼RS

RBi

� �¼

12l Sn � Sf g þ DE � Dcn� eDtm

Bi � Bi;n � HiriDcnþ giDcLi

" #¼! 0½ �

RDc �

¼ Dc� fDt½ � ¼! ½0�

ð32Þ

3.3. Linearization and return mapping algorithm

In order to solve the system of nonlinear Eqs. (30) or (32) through a Quasi–Newton strategy they first need to belinearized. Therein, a consistent linearization with respect to all basic unknowns describing the problem is essential(Marsden and Hughes, 1983, see e.g.; Simo and Taylor, 1985). The consistent linearization of (30) or (32) with respectto ½X� and Dc reads

½R� þ ½A�½DX� þ ½F�½DDc� ¼! 0½ �

½RDc� þ ½H�T ½DX� þ ½cc�½DDc� ¼! ½0�ð33Þ

in terms of the algorithmic tangent operator

A½ � ¼

dRSdS

dRSdB1

. . . dRSdBnB

dRB1dS

dRB1dB1

. . .dRB1dBnB

..

. ... . .

. ...

dRBnBdS

dRBnBdB1

. . .dRBnBdBnB

266666664

377777775

ð34Þ

with the dimensions ð6þ 6 � nBÞ � ð6þ 6 � nBÞ and the iteration matrices

½F� ¼

dRSdDc

dRB1dDc

..

.

dRBnBdDc

266666664

377777775; H½ � ¼

dRDcdS

dRDcdB1

..

.

dRBDcdBnB

266666664

377777775; cDc �

¼ dRDc

dDc

� �ð35Þ

with dimensions 1� ð6þ 6 � nBÞ of ½F�; ½H�, and 1� 1 of ½cDc�. Furthermore, d < x > =d < y > indicates the total derivative of< x > with respect to < y > and the specific forms of RS and RDc are given by (29)1 and (29)3 in the rate independent and by(31)1 and (31)3 in the rate dependent case.

Hence the system of Eq. (33) is specified for either the rate independent or the rate dependent case. In order to solve(33) in connection with the limit surface concept of Section 2.1, we apply a return mapping scheme. The classical returnmapping algorithm (Wilkins, 1964; Krieg and Krieg, 1977; Simo and Taylor, 1985; Ortiz and Simo, 1986; Miehe, 1993;Simo and Govindjee, 1991) is generally a two step procedure with an elastic predictor step followed by an inelastic cor-rector step if necessary. Concentrating on inelasticity without a purely elastic domain (i.e. Y ¼ 0) we adjust the classicalreturn mapping scheme by replacing the elastic predictor step by a viscoplastic, i.e. rate dependent inelastic, predictorstep possibly followed by a plastic, i.e. rate independent inelastic, corrector step. The resulting algorithm is summarizedin Box 3.


Box 3: Local return mapping algorithm for combined inelasticity.

1. Given at tnþ1 : �; T; at tn : �n;�inn ; �cn;Bi;n;D

tcn

2. Initialization : evaluate Sn at tn; set S ¼ Sn;Bi ¼ Bi;n;�in ¼ �inn ; �c ¼ �cn;D

tc ¼ Dtcn ;Dc ¼ 0

Viscoplastic predictor

� assume rate dependent inelastic behavior

3.

� solve system of linear Eq. (33) through update algorithm (Box 4)

4. Check inelastic limit surface

IF ðU ¼ffiffiJ

p� K < 0Þ GO TO 6:

5. Plastic corrector

� assume rate independent inelastic behavior

� solve system of linear Eqs. (33) through update algorithm (Box 4)

6. Consistent tangent C ¼ Celvol þ Ciso

If a purely elastic domain is used, i.e. Y > 0, the return algorithm shown in Box 3 needs to be extended by an elastic pre-dictor step and a corresponding elastic limit surface check analogous to steps 3 and 4 before the viscoplastic predictor step ispossibly executed.

The plastic corrector step (step 5 in Box 3) is based on the trial solution of the viscoplastic predictor step. Alternatively,the plastic corrector step can be replaced by a plastic solution step where rate independent inelasticity is assumed and theconverged solution at the end of the last time step tn is employed as the starting value for the iterative solution procedureoutlined in Section 3.4. Thus, before executing step 5 in Box 3 step 2 would be repeated. Both algorithms generally are robustand show quadratic convergence upon approaching the solution point.

3.4. Update algorithm

In order to determine the solution of the system of Eq. (33) during the predictor or the corrector step through a Newtonmethod, we first solve (33)1 with respect to ½DX� and insert the result into (33)2 giving the incremental change of Dc asfollows

DDc ¼ 1G� ½RDc� � ½H�T ½A��1½R�n o

with G ¼ ½H�T ½A��1½F� � ½cDc�n o

ð36Þ

Backsubstitution then gives the incremental change of the stresses and backstresses ½DX�

½DX� ¼ �½A��1½R� � ½A��1½F�½DDc� ð37Þ

With (36) and (37) at hand we can now update the slip increment Dc and the stresses and backstresses ½X� through

Dc( Dcþ DDc; ½X� ( ½X� þ ½DX� ð38Þ

where < X >(< X > þ < Y > indicates that the quantity < X>ðkþ1Þ at the end of the current iteration step kþ 1 is deter-mined through < X>ðkþ1Þ ¼< X>ðkÞþ < Y>ðkþ1Þ based on the iterative solution < X>ðkÞ for < X > of the last iteration step kand the incremental change < Y>ðkþ1Þ of < X > between steps k and kþ 1, determined e.g. through (36) or (37). Thereinthe initialization of < X>ðkÞ at the beginning of the iterative solution procedure is crucial and is given through step 2 inBox 3 or the solution of the viscoplastic predictor step, as discussed at the end of the previous section. Finally, theremaining state variables are updated, the material functions are evaluated for the current solution and if the conver-gence criteria are not fulfilled the next iteration step is executed. The entire update algorithm is summarized inBox 4 where the tolerance tol appearing within the convergence criteria has a sufficiently small value, e.g. 10�12.


Box 4: Update algorithm for Predictor/Corrector step.

1. Initialization DDc ¼ 02. Current state, evaluate for current solution ½X� ¼ ½S Bi¼1;...;nB �

T and Dc� relevant material functions (Box 2) and derivatives

� residuals ½R RDc�T , tangent operator ½A� and iteration matrices ½F�; ½H�; ½cDc�3. Check convergence

IF ðj½RRDc�T j < tol AND jDDcj < tol � Dc AND Dc > 0Þ EXIT

4. Inelastic slip increment

DDc ¼ 1

½H�T ½A��1½S� � ½cDc�n o � ½RDc� � ½H�T ½A��1½R�

n o

5. Update

� inelastic slip Dc( Dcþ DDc� unknowns ½X� ( ½X� � ½A��1½R� � ½A��1½S�½DDc�

� state variables �in ( �in þ Dcn
�c( �cþ Dc
in in tc tc Dt
IF (Predictor step) � ( � þ eDtm;D ( D þ tr
6. GO TO 2.

3.5. Consistent tangent moduli

To complete the algorithmic scheme discussed above we derive in the following the tangent moduli consistent with theproposed algorithm. The tangent moduli determine the incremental change of the stresses Dr along with the incrementalchange of the total strains D� determined through the global iteration algorithm according to

Dr ¼ C : D� ð39Þ

where C represents a fourth order tensor. Since we consider purely isochoric inelasticity we can split the tangent moduli intoa purely elastic volumetric part Cel

vol and an isochoric part Ciso resulting from both elastic and inelastic material response asfollows

C ¼ Celvol þ Ciso with Cel

vol ¼drvol

d�¼ j1� 1 ð40Þ

Thus the isochoric part Ciso remains to be determined. Based on the fact, that any solution of the system of Eqs. (30) or (32) isrequired to satisfy the conditions ½R� ¼! ½0� and ½RDc� ¼! ½0� at any time tnþ1 and hence any prescribed total strain � we can write

dRd�¼! 0 ¼ @R

@�þ ½A� d

d�½X� þ ½F� d

d�½Dc� ¼! 0

dRDc

d�¼! 0 ¼ @RDc

@�þ ½H�T d

d�½X� þ ½cc�

dd�½Dc� ¼! 0

ð41Þ

where @ < x > =@ < y > indicates the partial derivative of the quantity < x > with respect to < y >. Solving (41)1 for dd� ½X� and

inserting the result into (41)2 gives

dd�½Dc� ¼ 1

G½T� � ½H�T ½A��1½P�n o

ð42Þ

with the definitions of the matrices ½P� ¼ ½@R=@�� (dimensions ð6þ 6 � nBÞ � 6) and ½T � ¼ ½@RDc=@�� (dimensions 1� 6). Finally,insertion of (42) into the expression for d

d� ½X� obtained from (41)1 gives

dd�½X� ¼ �½A��1 ½P� þ 1

G½F� � ½T � � ½H�T ½A��1½P�

� � ð43Þ

where d½X�=d� has the dimensions ð6þ 6 � nBÞ � 6 and describes the sensitivity of the solution vector ½X� ¼ ½S Bi¼1;...;nB �T with

respect to the total strains �. From (43) the isochoric part Ciso of the tangent moduli can be directly extracted simply as thepart associated with the first 6� 6 entries, i.e. in index notation


½Ciso�ij ¼d½X�d�

� �ij¼ d

d�j½X�i with i; j ¼ 1; . . . ;6 ð44Þ

Note that in the context of the algorithmic implementation the entire expression (43) needs to be evaluated at the endof the return mapping algorithm outlined in Box 3 before the required isochoric tangent moduli Ciso can be extractedaccording to (44). This is computationally far less expensive than it appears, since the necessary matrix evaluations (ex-cept for [P] and [T]) and the inversion of A have already been performed while determining the expressions for the up-date algorithm outlined in Box 4. As an alternative to (43) and (44), a closed form expression only for the isochoricmoduli could be directly determined by enforcing dRS=d�¼! 0 and dRBi

=d�¼! 0 separately. This, however, results in verylengthy expressions which require all dRBi

=dBj for i; j ¼ 1; . . . ;nB to be inverted separately thus compensating the com-putational benefits.

3.6. Specification of the algorithmic expressions

The integration algorithm for combined inelasticity has been laid out in a general form so far. Now we specify all deriv-atives occurring in the expressions ½A�; ½F�; ½H�; ½cDc� and Ciso.

3.6.1. Derivatives of the material functionsAs a prerequisite for the determination of the derivatives occurring in the tangent operator ½A� and the iteration matrices

½F�; ½H� and ½cDc� we need to determine the derivatives of all material functions (see Box 2) with respect to the selected basicunknowns S; Bi¼1;...;nB and Dc. For both, the rate independent as well as the rate dependent case, these are determined asfollows

dgi

dJ¼ g0iJ ¼ �

12J

gi

dgi

dBijJ;C ¼ g0iBi

¼ Hiðmi þ 1Þ jBijri

� �mi

d½n : Li�n : Lif gLi þ HirijBijri

� �miþ1

d n : Li½ � 1jBij

n : I� Li � Lif g

dgi

dBj–ijJ;C ¼ g0iBj–i

¼ 0

dgi

dCjJ;Bi¼ g0iC ¼

HiriffiffiJ

p jBijri

� �miþ1

d n : Li½ �Li

dgi

dDc¼ g0iDc ¼ H0iDcri

jBijri

� �miþ1

d n : Li½ �n : Li

dHi

dDc¼ H0iDc ¼ �H0iv1ð1� aiÞexp �v1�c½ � bi þ ð1� biÞexp½�v2�c�f g � H0iv2ð1� biÞexp �v2�c½ � ai þ ð1� aiÞexp½�v1�c�f g

ð45Þ

Therein, I denotes the fourth order symmetric identity tensor. d < x > =d < y > j<z> indicates that the total derivative of< x > with respect to < y > is determined for fixed < z > since the expression including the derivative with respect to< z > is specified separately. Furthermore, the abbreviations < x > 0<y> are introduced to indicate the derivativesd < x > =d < y > in view of a compact notation. Additionally, we obtain only for the rate dependent case

dfdJ¼ f 0J ¼

npcApc

2Dpc

ffiffiJ

p sinhnpc�1

ffiffiJ

pDpc

" #cosh

ffiffiJ

pDpc

" #

dedIjDtc ¼ e0I ¼

ffiffiffi32

r34

Asc1 nsc

1

ffiffiffiffiffi32

I

r" #nsc1 �2

þAsc2 nsc

2

ffiffiffiffiffi32

I

r" #nsc2 �2

8<:

9=;ð1�DtcÞ�ntc

�ffiffiffi32

rAsc

1

ffiffiffiffiffi32

I

r" #nsc1

þAsc2

ffiffiffiffiffi32

I

r" #nsc2

8<:

9=;ntcð1�DtcÞ�ntcþ1Dtc

I0

dDtc

dIjP ¼Dtc

I0 ¼ �Dtcln½10�1000

TP0I

dPdI¼ P0I ¼

12ln½10�I A1þ 2A2þ3A3log10

ffiffiffiffiffi32

I

r" # !log10

ffiffiffiffiffi32

I

r" #( )

ð46Þ

3.6.2. Derivatives for the algorithmic tangent operatorBased on the evaluations (45) of the derivatives of the material functions, we can now specify the derivatives occurring in

the algorithmic tangent operator A, see Eq. (34)


dRS

dS¼ � 1

2lþ Dcffiffi

Jp

!Iþ Dcffiffi

Jp n� n

dRS

dBi¼ Dcffiffi

Jp I� Dcffiffi

Jp n� n

dRBi

dS¼ �Hiri

dRS

dBi

� �þ 2g0iJDcLi � Cþ DcLi � g0iC

dRBi

dBj¼ � dRBi

dS

� �þ dij 1þ gi

DcjBij

� �I� giDcjBij

Li � Li þ DcLi � g0iBj

ð47Þ

valid for both, the rate independent as well as the rate dependent case. In (47)4 dij indicates the Kronecker-Symbol withdij ¼ 1 if i ¼ j and dij ¼ 0 if i – j, for i; j ¼ 1; . . . ;nB.

3.6.3. Derivatives for the iteration matricesBased on (45) and additionally for the rate dependent case (46), we evaluate the derivatives occurring in the iteration

matrices ½F�; ½H� and ½cDc� according to Eq. (35) as follows

dRS

dDc¼ �n

dRBi

dDc¼ �ri Hi þ H0iDcDc

� �nþ gi þ g0iDcDc

� �Li

ð48Þ

valid for both, the rate independent as well as the rate dependent case. Additionally we have only for the rate independentcase

dRDc

dS¼ n;

dRDc

dBi¼ � dRDc

dS

� �;

dRDc

dDc¼ 0 ð49Þ

and alternatively for the rate dependent case

dRDc

dS¼ �2f 0J DtC;

dRDc

dBi¼ � dRDc

dS

� �;

dRDc

dDc¼ 1 ð50Þ

3.6.4. Derivatives for the consistent tangent moduliFinally, we specify the partial derivatives [P] and [T] occurring in the expression (44) in connection with (43) for the iso-

choric part of the consistent tangent moduli Ciso as follows

½P� ¼ @R@�

� �¼ P 0 0 . . . 0½ �T and ½T � ¼ @RDc

@�

� �¼ ½0� ð51Þ

in terms of the fourth order deviatoric projection tensor P ¼ I� 1=31� 1.

4. Identification of the model parameters

Above, we presented all details required for e.g. a Finite Element implementation of the proposed model. In the following, weprovide a description of the parameter identification process as the last technicality needed for a successful application of themodel by any reader of the present paper. The identification of the material parameters is a difficult task in material modeling,especially for models with a broad application range. We will restrict ourselves to the description of the necessary steps in-volved in the identification procedure. Based on these explanations and if a sufficient set of test data is available it should alwaysbe possible to obtain a valid set of parameters for the proposed model. In view of an efficient test program we propose to performat each temperature where the material behavior shall be characterized the following 4 types of experiments (additionally tothe tests required for the determination of the elastic constants): 1. creep tests up to rupture at several stress levels (see Sec-tion 4.1), 2. a monotonic tensile test, 3. two special complex cyclic tests (see Section 4.2.3), 4. classical cyclic (e.g. low cycle fa-tigue) tests at several load levels allowing to specify a cyclic stress–strain relation (see Section 4.2.4).

Most of the material parameters introduced in our model occur in the material functions, see Box 2. A summary of allmaterial parameters, separated into two parameter sets is given in Table 1 for i ¼ 1; . . . ;nB backstress components Bi. Notethat we use the same hardening exponent m ¼ mi in each hardening law gi of the evolution Eq. (17) for the backstresscomponents.

4.1. Identification of parameters from classical tests

First, the identification of the parameters collected in set A is performed based on classical tests for the full temperaturerange to be covered. The parameters of set A are identified independent of the ones collected in set B and without consid-

Table 1Set of material parameters.

Parameter set A Parameter set B

Bulk modulus j Inelastic limit stress KShear modulus l Backstress evolution H0i; ri; mSecondary creep Asc

1 ; nsc1 ; Asc

2 ; nsc2 Hardening/ Softening ai; v1; bi; v2

Tertiary creep A0; A1; A2; A3; C Primary/Tertiary creep Apc ; Cpc ; npc ; ntc


ering the entire model. The bulk and shear modulus j and l are determined dynamically or from monotonic tension andshear tests. The secondary creep constants Asc

1 ; nsc1 ; Asc

2 and nsc2 are determined by fitting relation (13)2 to the secondary- also

denoted as stationary-creep rates obtained from uniaxial monotonic creep tests. Herein, it is important to cover experimen-tally both, low and high stress level creep in order to capture the previously addressed power-law breakdown correctly. Thetertiary creep constants A0; A1; A2; A3 and C are determined in a similar fashion by fitting the Larson-Miller relation (15) inconnection with (16) to the creep rupture times tr obtained from the same tests at failure.

4.2. Identification of parameters from complex cycle tests

The identification of the parameters collected in set B is much more complicated since the parameters are largely inter-dependent. Before identifying the parameters of set B, tertiary creep and the second part of the cyclic softening relation (20)related to bi and v2 are deactivated.

4.2.1. Parameter mFirst, the value of m will be specified. The influence of the choice of m is visualized in Fig. 3a where the uniaxial ratcheting

strain �rat is defined as the remaining total mean strain �rat ¼ ð�max þ �minÞ=2 after each cycle during cyclic stress controlledloading. With decreasing m the predicted ratcheting strain largely increases. Since the available experimental data for IN718reveals no significant ratcheting or mean stress relaxation at low temperatures we choose m ¼ 70. Higher values of m inducea very abrupt saturation of the backstress components and thus sometimes lead to numerical instabilities. A major drawbackof the Ohno-Wang formulation (17) is the multilinearity of the stress–strain response for high values of m. Consequently asufficiently high number nB of backstress components Bi has to be used in order to produce reasonably accurate approxima-tions of the stress–strain hysteresis loops. As a compromise between accuracy and computational costs we choose nB ¼ 4. Anexample of the resulting stress–strain response is given in Fig. 3b along with the corresponding experimental record.

4.2.2. Parameters v1 and v2

The remaining parameters of set B will now be determined based on complex cyclic tests. Following the concept outlinedin Renauld et al. (2002), page 23 ff., and the ideas underlying the tests on pages 194–197 in Shenoy (2006), these tests aredesigned such that they reveal the major characteristics of the material response to be covered by the constitutive materialmodel. Within the subsequent parameter identification process we employ at each temperature used for characterizing thematerial response two types of such complex cyclic tests. In the first type, in the following denoted also as constant rate tests(CRT), the specimen are subjected to uniaxial strain controlled loading with varying mean strain and varying strain ampli-tude. In the second type, in the following denoted also as varying rate tests (VRT), additionally the strain rate is varied anddwell times are included. The input strain signal and the resulting stress–strain response of an exemplary VRT at elevatedtemperature is visualized in Fig. 4.

a b

Fig. 3. Identification of parameters of set B. (a) Influence of exponent m on ratcheting strain �rat . (b) Exemplary monotonic and cyclic stress–strain relationsof IN718.


As can be seen exemplarily in Fig. 4b, IN718 shows a very quick change from relatively abrupt yield behavior upon reach-ing the inelastic limit surface for the first time to a very smooth response during subsequent cycling. We model this behaviorby cyclic softening which occurs at a high rate throughout these initial cycles. Therefore, we employ the first part of (20)related to ai and v1 with v1 ¼ 50 in order to model the high softening rate. The second part of (20) is used to model the rel-atively slow softening response, e.g. during cycling after the first few initial inelastic cycles up to specimen failure in a lowcycle fatigue test. Hence, we fix v2 to a sufficiently small value v2 ¼ 0:1. The resulting softening response requires, throughthe choice of v2, the accumulation of many cycles and thus has almost no impact on the CRT or VRT simulations.

4.2.3. Parameters K; H0i; ri; ai; Apc; Cpc; npc

After fixing m; v1; v2 and excluding bi¼1;...;4 and ntc (i.e. setting bi¼1;...;4 ¼ 1 and ntc ¼ 0 for now) the 16 parametersK; H0i¼1;...;4; ri¼1;...;4; ai¼1;...;4 and Apc

; Cpc; npc remain to be determined. In order to restrict the total amount of materialparameters to a minimum, we can assume their temperature dependency for IN718 as summarized in Table 2. Hence, ofall the parameters in set B only two, the limit stress K and the primary creep parameter Apc , have to be determined for eachtemperature. For other materials also a temperature dependency of H0i; ri; ai or bi might have to be assumed. Carefully note,that, as discussed in Section 2.3, this requires the inclusion of a temperature rate term within the backstress evolution equa-tion in order to avoid unreasonable translations of the stress–strain hysteresis loops under non-isothermal conditions. Now,in order to determine the remaining material parameters, we first restrict ourselves to room temperature. Hence, we canneglect Apc

; Cpc; npc for now. Before we start our optimization process we use the monotonic stress–strain curve to get aset of starting values for H0i; ri and K according to the relations derived by Jiang and Sehitoglu (1996) for the determinationof the parameters of the Ohno-Wang model. Due to the significant initial softening (see Section 4.2.2) these values can how-ever not be used as final parameters and have to be modified in combination with ai by fitting the CRT and the VRT through ageneral parameter optimization algorithm. Here we mainly rely on optimization algorithms (evolution strategies in combi-nation with other gradient free approaches) widely discussed in the literature. Among others, the optimization algorithm ofHugo Pförtner (see http://www.enginemonitoring.com/opti/fminsi.zip) has been employed in the present work. For furtherdiscussions in this respect we refer to the literature. Next, we determine, through further application of the parameter opti-mization algorithm at fixed H0i; ri and ai, a set of Apc and K at each temperature as well as values for Cpc and npc common toall temperatures. Then we finish the optimization procedure by consecutively optimizing subsets of the parametersK; H0i¼1;...;4; ri¼1;...;4; ai¼1;...;4 and Apc

; Cpc; npc.

4.2.4. Parameters ntc and bi

Finally the parameter values for ntc and bi¼1;...;nB are determined. ntc is determined through fitting monotonic creep tests upto rupture while keeping all other parameters constant. The values of bi¼1;...;nB are determined from so called cyclic stress–strain relations where typically the peak stresses at midlife are collected from various strain driven low cycle fatigue testsinto one stress–strain curve. The counterpart to this curve is the so called monotonic stress–strain curve which is simplydetermined from a uniaxial tension test, see Fig. 3b for a visualization at an intermediate temperature. The constitutive mod-el outlined above will predict the cyclic or stabilized stress–strain curve for large values of �c, i.e. after accumulation of a suf-ficient amount of inelasticity. Hence, after determining all parameters of set A and B except bi we identify bi by simply fittingthe simulation of a monotonic strain driven tension test with �c ¼ 1 at t ¼ 0 to the cyclic stress–strain curve. Note that thecyclic stress strain relation implicitly contains significant stress relaxation at elevated temperatures. Thus during the simu-lation it is important to choose an appropriate but artificial and very slow strain rate in order to reproduce the correspondingtime frame up to midlife in the test. In any case, this will be a quite crude choice since in the real cyclic tests the stress levelsvary as well as the time up to midlife. Despite these rough estimates, this method gives satisfactory results and has the ben-efit of being very effective.

a b

Fig. 4. Identification of parameters of set B. Characteristics of exemplary complex cyclic test with variable strain rates and dwell times. (a) Input strainsignal. (b) Stress–strain response.

http://www.enginemonitoring.com/opti/fminsi.zip

Table 2Temperature dependency of material parameters.

Parameter set A Parameter set B

Temperature dependent j; l; Asc1 ; nsc

1 ; Asc2 ; nsc

2 K; Apc

Temperature independent A0; A1; A2; A3; C H0i; ri; m; ai; v1; bi; v2; Cpc ; npc ; ntc


4.3. Temperature interpolation of the model parameters

No explicit assumptions for the temperature dependency of the material behavior have been made in the model pre-sented above, except for the one for tertiary creep included in the Larson-Miller relation. To determine the material param-eters for IN718 test data has been available at 9 temperatures ranging from 20�C to 750�C. For an intermediate temperatureT, the temperature dependent parameters P (see Table 2) are interpolated between T1 and T2 according to

P ¼ 10log10 ½PT1�þ

log10 ½PT2=PT1

�

log10 ½T2=T1 ��log10 ½T=T1 � ð52Þ

in terms of the parameter values PT1 at T1 and PT2 at T2. Alternatively, the temperature dependency could be included bymultiplying the parameters Asc

1 ; Asc2 and Apc with a classical Arrhenius term in the respective material functions. This would,

however, largely increase the complexity of the optimization process while decreasing the achievable accuracy. Further-more, for IN718 the limit stress K does not obey the Arrhenius relation and follows the well known trend for Ni-base super-alloys where it first increases with the temperature before it finally decreases again.

4.4. Local strain and stress driver

All the tests used for the identification of the material parameters are uniaxial stress or strain controlled tests. In order touse the fully three dimensional algorithmic implementation as presented in Section 3 for the simulation of these tests with-out any changes, we employ the strain and stress driver algorithms presented in Boxes 5 and 6. In Box 5 �� and �r indicate thetransverse strains and stresses, i.e. all components of the total strain/stress tensor except the prescribed axial strain �prescr andthe resulting axial stress raxial. The transverse strains �� are determined on basis of the necessary condition that all the trans-verse stresses �r have to vanish.

Box 5: Driver algorithm for strain driven uniaxial tension/compression tests.

1. Given at tnþ1 : �prescr ; T; at tn : �n;�inn ; �cn;Bi;n;D

tcn

2. Initialization �� ¼ ��n

3. Compute total strains � ¼ �prescre1 � e1 þ ��4. Compute stresses and tangent moduli (see Box 3)

r ¼ r̂ð�;�inn ; �cn;Bi;n;D

tcn Þ; C ¼ d

d�r

5. Partition stresses and moduli r ¼ raxial�1 � �1 þ �r; �C ¼ dd��

�r6. Update transverse strains ��( �� C�1 �r7. Check convergence IF j�rj > tol GO TO 3.

Box 6: Driver algorithm for stress driven uniaxial tension/compression tests.

1. Given at tnþ1 : rprescr; T; at tn : �n;�inn ; �cn;Bi;n;D

tcn

2. Initialization � ¼ �n

3. Compute stresses and tangent moduli (see Box 3)

r ¼ r̂ð�;�inn ; �cn;Bi;n;D

tcn Þ; C ¼ d

d�r

4. Update strains �( �� C�1 r� rprescre1 � e1ð Þ5. Check convergence IF jC�1 r� rprescre1 � e1ð Þj > tol GO TO 3.


5. Application to IN718-Numerical examples

In this section we will report on the performance of the proposed model and its capability to fit as well as to predict thereal stress–strain response of the Ni-base superalloy IN718. The model has been validated for many more isothermal as wellas nonisothermal tests than the ones presented in the following, however, we can not report on all of them here.

5.1. Fitting results

First, we will present exemplarily some fitting results at a low temperature and at a high temperature obtainedthrough the parameter identification process outlined in Section 4. Fig. 5 visualizes the fitting results for a constant rate(CRT) and a variable rate (VRT) test at low temperature. The input strain signal for a VRT was visualized in Fig. 4a. Thecorresponding input strain signal for a CRT test is quite similar, except that it contains no hold times and it maintains aconstant strain rate throughout the whole test. Fig. 5a and c depict the uniaxial stress–strain (r� �) response for theCRT and the VRT and Fig. 5b and c the respective stress-time (r� t) responses. In both cases, the model reproducesthe material response of IN718 obtained in the experiment quite well. The largest deviations result from the multilinearapproximation of the stress–strain response through the Ohno-Wang relation for the backstress evolution. This can beimproved by employing a higher number nB of backstress components Bi. As can be seen in Fig. 6, the model also fitsthe experimental measurements at high temperature quite well. Here, significant stress relaxation and strain rate effectscan be observed which largely increase for longer dwell times (covered by classical creep tests but not by the VRT) orvery high temperatures (>700�C).

Fig. 7 shows exemplarily the creep response obtained through the proposed model in comparison to the experimentalresults for four tests at the same high temperature as in Fig. 6. The three regimes, primary, secondary and tertiary creepare clearly observable. Among the data of many other creep tests the secondary creep rate and the rupture time of eachof the tests depicted in Fig. 7 has been used in order to optimize the material parameters. The primary creep responsevisible in Fig. 7 is a pure prediction since the corresponding parameters Apc

; Cpc and npc are all determined on basis ofthe VRT. Keeping in mind that creep tests usually show a large amount of scatter, the model represents the real materialbehavior quite well in the secondary and tertiary creep regime (see Fig. 7a and b) and also the primary creep regime(see Fig. 7b).

a

c

b

d

Fig. 5. Fitting results for IN718 at low temperature. Comparison of experimental and model r� � and r� t-response for (a) and (b) constant rate and (c)and (d) variable rate test.

a b

c d

Fig. 6. Fitting results for IN718 at high temperature. Comparison of experimental and model r� � and r� t-response for (a) and (b) constant rate and (c)and (d) variable rate test.

a b

Fig. 7. Creep response for IN718 at high temperature. (a) Visualization of entire creep test with r1 ¼ r2 þ 30 MPa ¼ r3 þ 80 MPa ¼ r4 þ 130 MPa. (b)enlargement of graph (a) to detail primary and secondary creep regime.


5.2. Predictions for isothermal and non-isothermal cyclic tests

Next, we will discuss the predictive capabilities of the proposed model within cyclic tests. Before we turn to the complexcase of cyclic nonisothermal tests with dwell times we will exemplarily look at the cyclic response during two classical iso-thermal low cycle fatigue (LCF) tests in Fig. 8 (Note that in the context of the present paper we are only interested in thestress response of the undamaged specimen and not in the specific fatigue lifetime or the material response during the lastcycles before final fracture). The tests in Fig. 8 are performed at an intermediate temperature between the ones of Figs. 5 and6. Hence, we expect some relaxation effects to take place but still will see some characteristics of the low temperature re-sponse. Since the strain controlled tests are performed with R� ¼ �min=�max P 0 an initial positive mean stress

a b

c d

0 0.005 0.01 0.015 0.02 0.025

−0.001 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0

0

10000 20000 30000 40000

0 10000 20000 30000 40000

50000

Fig. 8. Predictions for isothermal LCF tests at intermediate temperature. Strain controlled test with (a), (b) �min=�max ¼ 0%=0:7% and (c), (d)�min=�max ¼ 1:5%=2:5%. In (b) and (d) rmax and rmin indicate the max-/minimum stress of each cycle.


rmean ¼ ðrmax þ rminÞ=2 > 0 will occur. Due to the moderate temperature a slight mean stress relaxation is observable inFig. 8b and d. This mean stress relaxation can be solely attributed to the rate dependent material response. At low temper-atures no mean stress relaxation occurs which is also very obvious from the CRT in Fig. 5 where even for R� 0 no relaxationis observable in contrast to Fig. 6.

After having validated the model for isothermal cyclic tests, we will now look at the capability of the model to predict thereal material behavior under nonisothermal cyclic tests, so called thermo mechanical fatigue (TMF) tests. In these tests, addi-tionally to the strains the temperature is altered cyclically. Therein, any phase shift between the strain cycle and the tem-perature cycle can be chosen. In the present context we concentrate on Out-of-Phase TMF tests where the temperature ismaximal at minimum strain and minimal at the maximum strain level. The tests are performed with thermal strain compen-sation, i.e. the mechanical strain �mec cycle is realized by prescribing the total strain �tot such that the thermal strain �th iscompensated and the desired mechanical strain �mec ¼ �tot � �th is obtained. For further details about general aspects ofthe TMF testing procedure refer to the literature, e.g. Hähner et al. (2006). In Fig. 9, we report on the results of two TMF testsand the corresponding model predictions. Both tests are performed with the same minimum temperature, a 1% mechanicalstrain range and a 30 s dwell period at maximum temperature, i.e. minimum strain. In the first test (Fig. 9a and b) the tem-perature range is 300�C while in the second test (Fig. 9c and d) it is 360�C. The two test results are captured by the modelwith reasonable accuracy in both, the overall stress–strain response, as well as the evolution of the stress response alongwith the cycle accumulation.

Finally, we want to qualitatively investigate the cyclic creep behavior, the ratcheting behavior and the strain rate depen-dency predicted by the proposed model based on the selected material parameters. Fig. 10a visualizes the simulation resultfor cycles N ¼ 1;500;1000;1500;3000 and N ¼ 5000 of a stress controlled cyclic test with a stress amplitude ofra ¼ 700 MPa and a mean stress rm ¼ 200 MPa at elevated temperature. Due to the elevated temperature and the non-van-ishing mean stress, continuous creep occurs and the hysteresis loop is shifted towards increasing total strains. Hence, thetotal mean strain �m remaining after each cycle increases with the number of accumulated cycles due to cyclic creep. Ifthe stress amplitude ra (see Fig. 10b) or the mean stress rm is decreased the internal force driving creep is reduced andthe rate at which the mean strain �m increases is decreasing as well. If a negative mean stress is applied, cyclic creep evolvestowards increasingly compressive strains, consequently resulting in a behavior inverse to the one presented in Fig. 10a and b,see e.g. Park et al. (2007) for some experimental results performed however on a slightly weaker IN718 alloy configuration.The mean strain shift reported in Fig. 10a and b generally results from both, the underlying ratcheting behavior of the

a b

c d

Fig. 9. Out-of-Phase TMF tests with strain range D�mec ¼ 1%;30 s dwell at Tmax , same minimum temperature Tmin and temperature range (a), (b) DT = 300�Cand (c), (d) DT ¼ 360C.


material and the cyclic creep response. But, as discussed in Sections 2.3 and 4.2.1, at low temperatures IN718 shows no rat-cheting or mean stress relaxation and hence any cyclic mean strain or stress evolution at elevated temperatures can beattributed to cyclic creep alone and should not be confused with ratcheting. Such a differentiation between cyclic creepand ratcheting is however generally not pursued in the literature. As can be seen in Fig. 10c, the rate of the cyclic mean strainevolution predicted by the model on basis of the parameters for IN718 significantly decreases with decreasing temperature.At low temperature basically no creep occurs, hence no creep induced mean strain evolution takes place and since we chosethe exponent m in the Ohno-Wang model very large (m ¼ 70) also no strain ratcheting is predicted. Thus we have, after thefirst cycle, _�rat ¼ _�m � 0 at low temperature.

Looking finally at Fig. 10d, we want to assess the capability of the model to describe the strain rate dependency typicallyobserved during monotonic strain controlled tensile loading at different strain rates. As shown, the model predicts a signif-icant strain rate sensitivity for strain rates _� below 0.1 1/s induced by the creep relations (10) and (12). For higher strainrates, the model response is governed by the evolution equations of rate independent inelasticity alone and a further in-crease of the applied strain rate has no impact on the predicted behavior.

5.3. Validation of the implementation into 3 Finite Element packages

The presented model has been implemented into the three Finite Element (FE) packages ABAQUS, CalculiX and MARC as auser material subroutine (UMAT/ Hypela2). In order to validate the implementation, first a single 20-node quadratic brickelement with reduced integration has been subjected to a displacement signal resulting in the strain history shown inFig. 4 at several temperatures. The stress responses of all three FE packages have been proven to exactly agree with eachother, see Fig. 11a, and also with the one dimensional strain driver result. Therein, for very few steps it is essential to re-course to the available automated time stepping schemes in order to achieve convergence also for large initial time/displace-ment increments even in significantly inelastic loading steps. A smaller time step is invoked if locally at an integration pointthe algorithm of Section 3 experiences convergence difficulties, i.e. if after 10 local iterations the convergence criteria ofBox 4, step 3, are not fulfilled. Otherwise the time increment is increased. The resulting convergence rates are usually qua-dratic. The CPU times required by each FE package employing the user material subroutine (UM) for the computation of theentire strain profile of Fig. 4 are reported in Fig. 11b. Also, the CPU times required by ABAQUS based on a linear elastic (LE)

a b

c d

Fig. 10. Cyclic creep and ratcheting behavior predicted by the model at elevated temperature ((a)–(c)) Strain rate dependency during monotonic loading athigh temperature (d).

a b

Fig. 11. Comparison of r� � response and CPU times (LE-linear elastic, CH-combined hardening, UM-user material).


analysis or employing the ABAQUS/Standard combined hardening (CH) model are presented to give an idea about the addi-tional computational costs entailed by an application of the proposed model. The stress–strain response predicted by thesetwo models is also shown in Fig. 11a. Obviously, the CH model largely overpredicts the ratcheting response. Furthermore, aneighth of the specimen used during real complex cycle testing of IN718 has been discretized with 20-node quadratic brickelements and subjected to a load profile according to the full strain history in Fig. 4. The resulting model contains 9095 de-grees of freedom (DOF). All three FE-implementations performed absolutely stable and gave identical results. The respectiveCPU times are again reported in Fig. 11b. Clearly, despite the efficiency of the algorithm, for large FE-models and many cyclesthe acceptable CPU-time will still be a limiting factor. A possible remedy can be a submodeling technique.

6. Conclusion

Driven by the need to describe the real material behavior for a wide temperature range under monotonic and cyclic load-ing as exactly as possible while still in an efficient manner, we proposed a new constitutive model for the unified description


of rate dependent and rate independent inelasticity. As a key to the unified description we introduced a limit surface conceptand derived within this concept consistent evolution equations for the inelastic strain. Further, we included an appropriatebackstress formulation and hardening/softening behavior into our model and proposed a new creep formulation covering allthree creep stages. Then we developed a new, fully implicit integration algorithm which provides a unified treatment of therate dependent and the rate independent case, consistent with the continuous formulation. Here, as a key feature, we pro-posed an extension of the classical return mapping scheme through a rate dependent inelastic predictor step. As an example,we determined the material parameters for IN718. We showed that the model and the algorithmic implementation are suf-ficiently accurate, stable and computationally not overly expensive. Hence, the model is ready for application in structuraldesign, e.g. to analyze creep or plasticity induced deformations or to assess the fatigue lifetime.

For further investigations, three key fields can be identified. First, it will be interesting to see if the postulated constitutiverelations are also valid for other materials. Secondly, the material behavior usually shows a significant amount of scatterwhich plays a key role in the development of safe designs. Hence, the introduction and validation of appropriate stochasticconcepts applied to the relevant model parameters is of special importance. Finally, as a third point, methods need to bedeveloped to quickly determine the quasi-stabilized material response under cyclic loading without simulation of each sin-gle loading cycle. This is the key to an efficient application of the proposed constitutive model in the context of a lifetimeassessment of any structural component.

Acknowledgment

Helpful discussions with David M. Nissley from Pratt and Whitney (Hartford) and Bernard Fedelich and Vitaliy Kindra-chuk of the BAM (Berlin) are gratefully acknowledged.

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A constitutive model for rate dependent and rate independent inelasticity. Application to IN718