A.F. Saleeb

University of Akron, Akron, Ohio

S.M. Arnold

Glenn Research Center, Cleveland, Ohio

Specific Hardening Function Definition and

Characterization of a Multimechanism

Generalized Potential-Based

Viscoelastoplasticity Model

NASA/TM—2003-212219

February 2003

https://ntrs.nasa.gov/search.jsp?R=20030032927 2018-05-15T15:57:43+00:00Z
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A.F. Saleeb

University of Akron, Akron, Ohio

S.M. Arnold

Glenn Research Center, Cleveland, Ohio

Specific Hardening Function Definition and

Characterization of a Multimechanism

Generalized Potential-Based

Viscoelastoplasticity Model

NASA/TM—2003-212219

February 2003

National Aeronautics and

Space Administration

Glenn Research Center
Acknowledgments

Professor Saleeb would like to acknowledge the financial support provided by NASA Glenn Research Center under

Grant (NCC3–808) to the University of Akron to conduct this work.

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Trade names or manufacturers’ names are used in this report for

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endorsement, either expressed or implied, by the National

Aeronautics and Space Administration.
Specific Hardening Function Definition and

Characterization of A Multimechanism Generalized

Potential-Based Viscoelastoplasticity Model

A. F. Saleeb

Department Civil Engineering

University of Akron

Akron, OH 44325

S. M. Arnold

National Aeronautics and Space Administration

Glenn Research Center

Cleveland OH, 44135

Abstract

Given the previous complete-potential structure framework (see Auqrog dqg Sdohhe

[1994]), together with the notion of strain- and stress- partitioning in terms of separate contributions of several submechanisms (viscoelastic and viscoplastic) to the thermodynamic

functions (stored energy and dissipation), see Sdohhe hw do. [2001], a detailed viscoelastoplastic multimechanism characterization of a specific hardening functional form of the model

is presented and discussed. TIMETAL 21S is the material of choice as a comprehensive test

matrix, including creep, relaxation, constant strain-rate tension tests, etc. are available

at various temperatures. Discussion of these correlations tests, together with comparisons

to several other experimental results, are given to assess the performance and predictive

capabilities of the present model particularly with regard to the notion of hardening saturation as well as the interaction of multiplicity of dissipative (reversible/irreversible)

mechanisms.

Nomenclature

Invariants

−; −R; −IR complementary dissipation potential; R- reversiable, IR

-irreversible

©; ©R; ©IR Gibb’s complementary potential; R- reversiable, IR -

irreversible

F Bingham-Prager threshold function

G(b) normalized second invariant function

hb invariant material function

NASA/TM—2003-212219 1
Stresses

3⁄4ij Applied Cauchy stress tensor

(3⁄4s)ij viscoelastic equilibrium stress tensor

qij viscoelastic, non-equilibrium stress

3⁄4ij ¡ ®ij viscoplastic equilibrium stress tensor

®ij viscoplastic non-equilibrium stress tensor

∙ drag stress

∙b either a normalizing stress or hardening threshold stress

per mechanism

Strains

2ij ; 2ve

ij ; 2vp

ij ; 2th

ij total, reversable (viscoelastic), irreversible (viscoplastic)

and thermal strain tensors, respectively

pij internal reversible strain tensor (displacement-like)

°ij internal irreversible strain tensor

Material Parameters

Eijkl; M(a)

ijkl;  ́(a)

ij reversible elastic stiffness, viscoelastic stiffness and viscosity coefficient tensor for each ath mechanism, respectively

H(b); R(b) hardening and thermal recovery material parameters per

mechanism

1 material parameter associated with the irreversible viscosity of the material

Es; E(a) m denotes the elastic stiffness and maxwell spring stiffness

per ath mechanism, respectively

n;  ̄; m material exponents

r((G)(b)

); h(G(b)); f(F) material functions

1⁄2a relaxation time for the ath mechanism; = E(a)

η =E(a) m

o Possion ratio

Miscellaneous

Mijkl Devatoric operator (isotropic or anisotropic)

Nijkl Isotropic directionality tensor

±ij Kronecker delta function

h i Macauley bracket

Hv[] Heaviside unit function

(.

) time derivative (or rate) notation

M number of reversible mechanisms

N number of irreversible mechanisms

NASA/TM—2003-212219 2
1 Introduction

Extensive research efforts have been made over the years on the phenomenological representations of material behavior. Significant progress has been made in the development

of theories of viscoelasticity and viscoplasticity for the modeling of inelastic properties. In

particular, the present state-of-the-art in mathematical modeling of viscoplastic deformation

of metals is very well developed; i.e., based on the so-called internal variable formalism

in thermodynamics of irreversible processes (Crohpdq&Gxuwlq [1967], Lxeolqhu [1973],

Lhpdlwuh&Ckderfkh[1990], Ckderfkh[1989], Auqrog&Sdohhe [1994], Auqrog hw. do.

[1994a,1995], Fuhhg&Wdonhu[1993]).A larger number of specialized forms of unified viscoplastic models (e.g., isotropic or anisotropic, fully associative or nonassociative, etc.) have

been successfully applied in recent years to different high-temperature metals, alloys, and

composite material systems (Lhpdlwuh&Ckderfkh[1990], Auqrog hw. do. [1994a,1995],

Fuhhg&Wdonhu[1993], Kudxv}&Kudxv}[1996], Hduw [1976], Hxl [1985], Kxpdu hw do.

[1980] and Sdohhe&Wlow [1993]). In the present paper, we continue along our previous efforts, and utilize the fully associative generalized viscoelastoplastic potential framework1

(Auqrog&Sdohhe [1994], Auqrog hw. do. [1994a,b;1995], Sdohhe&Wlow [1993] and Sdohhe

hw do. [2001]) to represent a rather comprehensive set of experimental results performed on

the titanium alloy TIMETAL 21S. TIMETAL 21S is a  ̄-titanium alloy with a nominal composition of Ti-15Mo-3NB-3Al-0.2Si (% weight). Depending on the stress/deformations levels, this

material was found to exhibit both reversible and irreversible behavior with marked rate

dependency in each regime.

The objective of the present paper is twofold: the first being an evaluation of a specific form

for the material hardening function that has a true saturation value (or limit state) associated

with it, and the second is the characterization and evaluation of model representations with

multiple mechanism present in both the reversible and irreversible portions of the model, given

the above mentioned material system. The coupled multimechanism viscoelastoplastic model

(see Sdohhe hw do. [2001]) employed has been formulated to allow for sufficient generality in

its potential functions (Gibb’s and dissipation) through a systematic introduction of several

mechanisms (internal state variables of the tensorial type) for both viscoelastic and viscoplastic

response components. With the viscoelastic part, utilizing the concept of an equilibrium

stress, leading to rate dependency upon instantaneous loading, as well as to a unique limiting

state of elastic deformation at infinite times. The viscoplasticity formulation accounts for

both nonlinear kinematic hardening and static-recovery mechanisms. We have previously

reported (Sdohhe hw do. [2001]) on the implicit numerical-integration aspects, as well as the

computational aspects concerning the automation of material-parameter-estimation procedures

using the software called COMPARE (COnstitutive Material PARameter Estimator). We

refer the reader to the above paper (and other references therein) for further details.

This paper begins in section 2, with a brief outline of the general fully associative framework

for several interacting mechanisms of the viscoelastic and viscoplastic types, leading to the

governing flow and evolutionary rate equations. The specific functional forms describing the

hardening and thermal recovery of the material are then described. In section 3, the general

1This framework falls into the category of phenomenological (or macroscopic) material representation based

on continuum mechanics and internal state variables and has been labeled by us as belonging to the GVIPS

(Generalized VIscoelastoplastic with Potential Structure) class of material models.

NASA/TM—2003-212219 3
characterization strategy is discussed with the specific values for the various material parameters, given a single, two and three mechanism viscoplastic (irreversible) representation. The

six mechanism viscoelastic (reversible) portion of the model (taken from previous work) is also

repeated within for convenience. This is then followed by section 4 which describes in detail

the results of the comparison of saturating versus non-saturating material hardening functional

forms of the model and the correlative and predictive capability of multimechanism representations (wherein multiple viscoplastic internal material clocks are accounted for) verses a single

viscoplastic representation (that only admits a single internal clock).

2 Complete Potential Structure

We begin by summarizing the basic equations governing the behavior of a material element.

The discussion is limited to the case of small deformations and isothermal conditions, in which

the initial (virgin) state is assumed to be stress free (as well as, all initial values of internal

state parameters are assumed zero) throughout. Here, and for the remainder of the paper, a

Cartesian reference frame is utilized, along with index notation (wherein summation is implied

for repeated subscripts). However, for convenience, we also utilize superscript letters placed

between parentheses as indices to identify sets of internal state parameters and when needed

the summation over these will be indicated explicitly by the summation symbol.

All equations are written for the general 3D (three-dimensional) case; but the forms are

directly applicable in subspaces (e.g., 2D plane-stress or plane-strain, etc.); see (Sdohhe&Wlow

[1993], Sdohhe hw do. [1998;1999]). For conciseness, we define the total number of dissipative

viscoelastic state variables (each of the second-order tensor type) as M; i.e.; the corresponding

non-equilibrium stress tensors (Sdohhe&Auqrog[1997] and Auqrog hw. do. [1997]) are

q(a)

ij (a = 1; 2;:::M) and their associated conjugate (strain-like) tensors are p(a)ij . In addition,

we define the equilibrium stress tensor (σs)ij and its strain conjugate as εve

ij . Similarly, for

the viscoplastic mechanisms, we use the notation α(b)

ij (b = 1; 2;:::;N) for the back stresses

(kinematic hardening) and γij (b) for their conjugate or dual (strain-like) variables, for a total

of N viscoplastic (second-order) tensorial state variables.

2.1 Potentials and State Equations

In accordance with the basic hypothesis of the additive decomposition of total strain tensor,

εij , into three components, that is a reversible (i.e., elastic/viscoelastic), εve

ij ; an irreversible

(i.e., viscoplastic), εvp

ij ; and a thermal strain, εth

ij , component; we have:

εij = εve

ij + εvp

ij + εth

ij (1)

Similarly, an additive decompositon for the two fundamental potentials, that is, the Gibbs,

complementary function, ©; and dissipation function, −, are assumed:

©(σij ; α(b)

ij ; q(a)ij )=©R(σij ; q(a)ij )+©IR(α(b)ij ) (2)

−(σij ; α(b)

ij ; q(a)ij ) = −R(q(a)ij ) + −IR((σij¡αij ); α(b)ij ) (3)

NASA/TM—2003-212219 4
where the functional dependencies on the external ( σij ) and internal ( q(a)

ij and α(b)ij ) state

variables, with (a = 1; 2;:::;M) and (b = 1; 2;:::;N), and the conjugate variables εvp

ij (also

εij ), εve

ij (also p(a)

ij ), and γ(b)ij , respectively, are given by the following equations of state:

εij ¡ εijvp = @©R

@(σs)ij

(4)

εve

ij ¡p(a)

ij = @©R

@q(a)

ij

(5)

γij (b) = @©IR

@α(b)

ij

(6)

The corresponding rate forms are then

ε ̇ij ¡ ε ̇ijvp = d

dt μ @©R@(σs)ij ¶

= @2©R

@(σs)ij@(σs)kl

(σ ̇ s)kl (7)

ε ̇ijve¡p ̇

(a)

ij = d

dt Ã

@©R

@q(a)

ij !

= @2©R

@q(a)

ij @q(a)

kl

q ̇

(a)

kl (8)

γ ̇ ij (b) = d

dt Ã

@©

@α(b)

ij !

= @2©IR

@α(b)

ij @α(b)

kl

α ̇ (b)

kl ; or α ̇ (b)

ij =

"

@2©IR

@α(b)

ij @α(b)

kl

#−1

γ ̇ kl(b) (9)

where the above equation is the internal constitutive rate equations for the internal (nonequilibrium) state variable for each hardening mechanism.

The corresponding flow and evolution (rate) equations are similarly obtained from the dissipation function and are as follows:

ε ̇ijvp = @−IR

@σij

(10)

γ ̇ ij (b) = ¡@−IR

@α(b)

ij

; b = 1; 2;:::;N (11)

p ̇ ij (a) = @−R

@q(a)

ij

; a = 1; 2;:::;M (12)

Note in Eqn. (5) only a single viscoelastic strain component is associated with all viscoelastic

stress components. Also for the viscoelastic response, the total stress is decomposed into an

equilibrium stress, (σs)ij ; and a non-equilibrium (dissipative) reversible stress, qij ; i.e.,

(σs)ij = σij ¡ qij ; where qij = XM

a=1

q(a)

ij (13)

and for the viscoplastic response, the total stress is decomposed again into an equilibrium,(σij¡αij );

and non-equilibrium, αij ; component; where

NASA/TM—2003-212219 5
αij = XN

b=1

α(b)

ij (14)

The above GVIPS framework attempts to capture the underlying physical processes associated with microscopic defects (e.g. dislocations, grain boundaries, voids etc. in metals) and

their complicated interactions which span an entire spectrum of length scales (i.e. from the

“atomistic (nano) — microscale”, to the “mesoscale” to the final “macroscale”) by introducing

the notion of multiplicity of mechanisms in the mathematical description. These mechanisms, reflecting the vastly different interactions, give rise to the introduction of an aggregate

of individual internal state variables (i.e., q(a)

ij and α(b)ij ); e.g. many tensors each accounting

for interactions at different length scales, that is, “short” range between dislocations and other

dislocations, and “long” range between a well developed dislocation cell or subgrain and other

defects and interfaces. In addition to the differences in “microstructural” length scales, the time

rates of change governing their dynamics are also known to be vastly different, hence the notion

of multiple (or spectrum of) characteristic relaxation times in the GVIPS formulation. i.e.,

M(a)

ijkl and η(a)ijkl of Eqns.(15) and (17) . Furthermore, with energy measure providing the major

consideration in dislocation dynamic theories (e.g., in studying thermal activation, mass and vacancies diffusion, energy barrier such as jog formation energy, rate-dependent and multi-species

(distributions of finite-strength obstacles, etc.)) and the intricate interaction of such “unit”

processes, with their vastly different energy contents and rate-limiting values, the partitioning

of the overall supplied work into energy storage (e.g. hardening) and energy dissipation

(e.g. recovery and inelastic flows) components that proceed with varying degrees of competition

during the deformation of the material became a motivating factor, see Eqns. (2) and (3) - and

more specifically Eqns. (15) through (18). Hence the emphasis on complete2-potential “energy” structure which leads to a “hierarchy” of representations, from the Gibb’s and dissipation

functions (Grandparent), to the kinematic decomposition of the deformation and microstructure state equations (Parent), to the kinetic and evolution equations (Children). This structure

then guarantees i) that the overall response is always bounded in terms of the total and/or rate

quantities (e.g. transient to steady state creep, or from a state of nonlinear hardening to a

state of saturation in hardening) and ii) the availability of symmetric tangent stiffness matrices

which greatly enhance computational robustness. Finally, the use of multimechanisms (embedding the effects of many time/length scales) in the GVIPS class of formulation enables the

specialization of this general model into simpler (more restricted in scope) formulations, e.g.

purely elastic, linear viscoelastic, classical rate-independent elastoplastic, as well as more elaborate forms of hereditary descriptions (involving viscous effects, nonlinear hardening, dynamic

recovery, thermal/static recovery, relaxation, ratchetting or shakedown phenomena under load

cycles) to name a few.

2.2 Specific Functional Forms:

For concreteness, we summarize here the specified functional forms that have been used in our

previous work (Auqrog hw. do. [1994a,b;1995], Sdohhe&Wlow[1993], Sdohhe&Auqrog[1997],

2By complete we mean a formulation, as described by Eqns. (2) -(12), whereby the flow and evolutionary

equations are fully-associative (are normal to the same dissipation potential surface).

NASA/TM—2003-212219 6
Auqrog hw. do. [1997] and Sdohhe hw. do. [2001]) and will be used for the most part here.

These forms are quadratic for the viscoelastic contribution and essentially the same ”powertype” nonlinear forms previously used for the viscoplasticity functions H1(b) , −1, and −(b)

2 in

terms of the invariants of their respective arguments α(b)

ij , (σij¡αij ) and α(b)ij ; see eqns. 16,

18 and 20 below. However, a new functional form, h(G(b)

); for the hardening function will be

introduced and investigated as well.

The specific forms for the reversible and irreversible potentials and driving functions are as

follows:

©R = ©R(σij ; q(a)

ij ) = 1

2

(σs)ij E−1

ijkl(σs)kl +

1

2

XM

a=1

q(a)

ij [M(a)

ijkl]

−1

q(a)

kl +XM

a=1

q(a)

ij p(a)ij (15)

©IR = ©IR(σij ; α(b)

ij ) = σijεvp

ij +XN

b=1

H1(b)(G(b)

) (16)

and

−R = 1

2

XM

a=1

q(a)

ij [η(a)

ijkl]

−1

q(a)

kl (17)

−IR = −1(F) +XN

b=1

−(b)

2 (G(b)

) (18)

where

F = 1

2∙2 (σij ¡ αij )Mijkl(σkl ¡ αkl) ¡ 1 (19)

G(b) = 1

2∙2

(b)

(α(b)

ij Mijkl α(b)

kl ) (20)

and the required partial derivatives are:

1

∙2

@−1

@F = f(F) = F n21

1

∙2

(b)

@−(b)

2

@G(b) = r(G(b)

)

h(G(b))

1

∙2

(b)

@H1(b)

@G(b) = 1h(G(b)) (21)

Note in the above, Eijkl and Mijkl are fourth-order tensors of viscoelastic stiffness moduli (with

E−1

ijkl and [M(a)

ijkl]

−1 being the corresponding compliance tensors); η(a)

ijkl (a = 1; 2;:::;M) are

the fourth-order tensorial viscosity coefficients associated with the athdissipative viscoelastic

mechanism; the −1, and −(b)

2 are the inelastic dissipations due to plastic hardening and static

(thermal) recovery, respectively; and each H1(b) is taken to be a nonlinear function of the

corresponding viscoplastic internal state tensor α(b)

ij . To emphasize the functional dependency

in the potentials, e.g., ©R; ©IR and H1(b) , we have shown the corresponding arguments in

parentheses.

The only remaining functions that need to be assumed are those defining the competitive

processes within a material, that is the hardening and recovery terms. The function that drives

the thermal recovery is taken as previously to be a ”power-type” function, that is,

NASA/TM—2003-212219 7
r(G(b)

) = R(b)[G(b)]

m(b) (22)

whereas two related but distinct forms for the hardening function, h(G(b)

), will be assumed and

investigated herein. The first nonsaturating hardening form, which has been used extensively

in the past (e.g., Rrelqvrq [1978] and Auqrog hw. do. [1994a,b and 1995] and Sdohhe

hw.do. [2001]),

hnonsat(G(b)

) =

H(b)

[G(b)]

β(b)

(23)

is also of the power type. This form, although nonlinear in nature, is however, deficient in

two ways; i) it is unable to ever reach a true limit state (unless the constant  ̄ is unrealistically

high) in which hardening would cease, i.e., h(G)=0, and ii) no physical interpretation can be

associated with the ”normalizing” stress, ∙(b); for each individual mechanism. Consequently,

an alternative form which explicitly posses a limit state, irrespective of the exponent  ̄; can be

introduced, that is:

hsat(G(b)

) = H(b)

D

1 ¡

p

G(b)

Eβ(b)

(24)

Note that with this form the function h(G) will now always go to zero, for a given mechanism,

whenever the now explicit hardening threshold, ∙(b), is exceeded. Thus this new form can

simulate saturation of a tensile curve (particularly difficult at lower temperatures where thermal

recovery is naturally supressed) while still accurately representing other loading histories, e.g.,

creep and relaxation. A comparison of the respective capabilities of these forms will be discussed

in more detail in section 5.1. Note also that, though different in their behaviors, both forms

in Eqns. 23 and 24 involve the same number of material constants, thus no additional price is

paid for introducing saturation.

Finally, in the above, ∙; 1; n represent viscoplastic flow material constants, whereas the

H(b),  ̄(b) are hardening material constants; R(b); m(b) are recovery material constants; and

constants ∙(b) are either ”normalizing” or hardening threshold stresses, for the individual mechanism (b), where b = 1; 2;:::N; depending upon the form of the hardening function, h(G(b)

);

assumed.

2.3 Physical, Micromechanical Motivation

There are a number of observations, both from the viewpoints of physical micromechanical

origins as well as the macroscopic manifestations in experiments, which provide strong motivation for adopting expressions such as in Eqn. (24). For example, dislocations act as the

principle carriers of plastic deformations in metals. According to modern theories of dislocation dynamics [Hluwk&Lrwkh [1982]], there are numerous primary and secondary interactions

affecting dislocation mobility (e.g., among dislocations with other dislocations, or between dislocations and other obstacles/defects/ or internal and free-boundaries, etc.). The tendency of

this motion is towards more of a modulated substructure, i.e., dislocation cells and subgrains,

as well as heterogeneous patterns associated with slip bands, microshear bands, persistent slip

bands and dislocation tangles. To define these dislocation arrangements, there is a need to use

NASA/TM—2003-212219 8
a large number of parameters (e.g., in addition to dislocation densities, more irregular statistical distributions in the form of higher-order correlation functions are also required). However,

despite the inherent complexity in describing all the details of relevant elementary mechanisms

and denominate unit processes operating at the microscale, most studies [Zele hw do.[2002],

Auvhqolv&Pdun[2002] and Swdlqlhu hw do. [2002]] on the factors affecting the rate of

dislocation motion support the notion of a competing nature. That is, one distinguishes

between those mechanisms that hinder the motion (e.g., dislocations generation in pile-ups)

and those which facilitate the motion (such as the increased mobility of existing dislocations

and/or annihilation of dislocations by climb), wherein the macroscopic forces driving these

competing unit processes are the applied stresses, the loading or deformation rates, and the

temperature. For example, the above generation-annihilation formulation for a discrete set

of eighteen dislocation densities were utilized as internal state parameters in Auvhqolv&Pdun

[2002] to describe the state of crystallographic hardening in crystal plasticity.

On the other hand, recent three-dimensional simulations based on dislocation dynamics [Zele

hw do, 2002] have clearly demonstrated that when the dislocation sources start operating, the

dislocation density increases sharply at the beginning , but then the rate of increase diminishes

gradually afterwards, until a saturation value is finally reached. At this latter condition of

saturation, the portion of the dislocation segments generated by the sources are equalized by

the dislocation segments disappearing on interaction with the free surfaces and by dislocation

-dislocation annihilation. This saturation state was consistently observed at all deformation

rates considered; the differences being the attainment of higher dislocation densities at the

higher strain rates. Similarly, recent investigations on the micromechanical basis/multiscale

modeling of deformation in bcc crystals [Swdlqlhu hw do. [2002]] have also resulted in observing

saturation states of hardening at sufficiently large strains, with the marked dependency

of the saturation state of stress on temperature and strain rates. The analysis considered

an elaborate set of microscopic processes; i.e., double kink formation and thermally-activated

motion of kinks; short-range interactions between primary and forest dislocation species (with

different obstacle strength); dislocation multiplication due to breeding by double cross-slip as

well as fixed Frank-Read sources; and dislocations pair annihilation.

Alternatively, macroscopic scale experiments on metals, such as described in the later sections, exhibit saturation stress states in the fully developed plastic region. This often occurs

under a wide variety of (constant) strain rates, with a higher saturation stress resulting from the

increased loading rates. In summary, both microscopically as well as macroscopically, there

is ample evidence for the existence of states of hardening saturation. This is the main

motivation for adopting such a functional form as that proposed in Eqn. (24).

Finally, there are important differences in the present treatment of the saturation phenomena, as compared to existing alternatives in the literature (e.g. Lhpdlwuh dqg Ckderfkh

[1990], Fuhhg [1993], Kuhpsoh [1996] and Hr dqg Kuhpsoh [2002]). More specifically, we

do not introduce such notions as explicit bounding surfaces to limit the stress magnitudes,or

the idea of overstress/equilibrium versus hardening stress, or the separation of the material’s inelasticity into creep plus perfectly-plastic strains. In addition, our treatment indicates

that hardening-stauration is always possible, irrespective of the rates of continued loadings,

thus emphasizing the physical evidence that the internal structure will ultimately reach a stable/modulated microstructural configuration (of course this is true without the intervention of

other ”unstable” phenomena, such as damage and failures, which are outside the scope of this

paper).

NASA/TM—2003-212219 9
2.3.1 Resulting Multiaxial Flow and Evolution Equations

With the selection of the particular functional forms above, the present viscoelastoplastic

model is now complete within the context of a fully associative potential structure. The following expressions constitute the governing associated flow and evolution equations, assuming

multiaxial anisotropic behavior.

σ ̇ ij = Eijkl(ε ̇ kl ¡ ε ̇ klvp) +XM

a=1

q ̇

(a)

ij (25)

q ̇

(a)

ij = Mijkl(ε ̇ kl ¡ ε ̇ klvp) + Mijklη−1

klrsq(a) rs (26)

ε ̇

vp

ij = f(F) Γij if F  ̧ 0 (27)

ε ̇

vp

ij = 0 otherwise

α ̇ (b)

ij = Q(b)

ijkl "

ε ̇

vp

kl ¡ r(G(b)

)

h(G(b))

π(b)

kl #

if π(b)

kl ¢ (σkl ¡ ®kl)  ̧ 0 (28)

where

Q(b)

ijkl =

" @2

_

H(b)

@α(b)

ij @α(b)

kl

#−1

= h(G(b))

2

4Zm +

h0

(G(b))

h(G(b))[1 ¡ h0(G(b))

h(G(b)) (2∙2

(b)

G(b))]

α(b)

ij α(b)

kl

3

5 (29)

Γij=Mijkl(σkl¡αkl); π(b)

kl =Mklijα(b)

ij (30)

qij = XM

a=1

q(a)

ij ; αij = XN

b=1

α(b)

ij (31)

and

h0

(¢) = 1

∙2

(b)

@h(¢)

@G(b)

wherein h(¢) becomes either hold or hnew depending upon the specific form in view. Note Zm

is the ”generalized” inverse of Mijkl ; see Sdohhe&Wlow[1993] for further elaboration on this.

Details regarding the numerical solution of these equations have been discussed previously, see

Sdohhe&Wlow[1993], Saleeb et al. [2001]. Furthermore, note that the flow and evolutionary

loading conditions (Eqns. 27 and 28) are slightly modified as compared to this previous work,

due to our recent consideration of multiaxial ratchetting behavior. These differences will be

further elaborated upon in a future publication.

NASA/TM—2003-212219 10
3 Model Characterization

3.1 Outline of the General Strategy

The most important, and often times most difficult, aspect of modeling the behavior of a

given material at elevated temperature is obtaining the required material functions, e.g., f(F)

and g(G), for viscoplasticity and associated material parameters. The difficulty associated

with this process typically stems from not only the variety in mathematical forms for the

material functions (e.g., power law, exponential, hyperbolic sine, etc.), but also the fact that

given the material functions there is no unique set of material parameters for any given load

path. Therefore, numerous iterations and difficult compromises are required before a final set

of material parameters (for the assumed material functions) can be obtained.

Traditionally, characterization has been accomplished through basic trial and error procedures (i.e., graphical and/or mechanistic) which attempt to fit the predicted response from the

constitutive model to that response exhibited by the experimental test data. These approaches,

however, are rather limited, difficult, and many times less than fruitful for more general and

sophisticated constitutive models. This is particularly true when dealing with models that possess a very large number3 of material constants that; (i) are often lacking in their direct physical

interpretation (not the case in the present model), (ii) may have vastly different magnitudes,

and (iii) are highly interactive with each other. Further complications also arise when a large

number of experimental tests of various types (i.e., stress-, strain-, or mixed-control) under

transient and/or steady-state conditions, are expected to be simulated.

To overcome these difficulties, software tool (i.e., COMPARE) has been developed to enable a design engineer to easily and efficiently bridge this gap between constitutive theory and

experimental test data, in which optimum material parameters are determined by minimizing

the errors between experimental data and the correlated responses (Sdohhe hw do. [2001]

and [2002]). Within COMPARE the estimation of material parameters is cast as a minimumerror, weighted least-squares, multi-objective optimization problem; wherein, this optimization

problem is solved using the Sequential Quadratic Programing Technique. COMPARE is sufficiently general to handle a comprehensive set of test data, under arbitrary load-control variables,

multiaxial stress/strain state, and transient as well as steady-state response measurements.

3.2 Characterization of Material Parameters

Building upon our previous work [Auqrog hw. do. [1994a,b], Sdohhe&Auqrog[1997],

and Auqrog hw do. [1997] ], a primary objective in this paper is to specify material functions

and characterize the corresponding material parameters for the current multimechanism, viscoelastoplastic model of the GVIPS class with kinematic hardening; given TIMETAL 21S4,

an advanced titanium-based matrix commonly used in TMCs. Previous work (Auqrog hw. do.

[1994a,b: 1997, 2000], Sdohhe hw do [2001] ) indicated that six viscoelastic mechanisms (i.e.,

M=6) were sufficient to adequately span the entire reversible load spectrum with the proper rate

sensitivity at 650oC and that a single viscoplastic transient mechanism (i.e., N=1) appeared

insufficient to accurately (quantitatively) represent the irreversible domain over a wide stress

3 In the specific model presented, the total number of material parameters required are 5 +5N + 2M where N

defines the number of viscoplastic mechanisms and M the number of viscoelastic.

4TIMETAL 21S is a registered trademark of TIMET, Titanium Metals Corporation.

NASA/TM—2003-212219 11
range. Here we will expand upon this previous work by 1) comparing the simulation capabilities

of the two viscoplastic hardening forms given in Section 3.2 (given a single viscoplastic mechanism), and 2) upon proper down selection of the material hardening function, establish the

number of required viscoplastic submechanisms required to accurately simulate the irreversible

deformation behavior of TIMETAL 21S at 650oC. In all cases considered herein, the number

of viscoelastic mechanisms are held fixed at six, with the 14 reversible material constants being those determined previously [see Arnold et al. [2001]] and given in Table 1. Note a key

assumption is that Poisson ratio’s is identical in all three matrices Eijkl;M(a)

ijkl and η(a)ijkl; thus

making the forth-order tensor moduli coaxial and given the assumption of material isotropy,

these moduli take on the following form: Eijkl=EsNijkl; M(a)

ijkl =E(a) m Nijkl and η(a)ijkl = 1⁄2aM(a)ijkl;

where

Nijkl = f o

(1 + o)(1 ¡ 2o)

±ij±kl +

1

(1 + o)

(±ik±jl + ±il±jk)g (32)

With respect to the irreversible domain 3+5M constants are required in total; where three (

i.e., ∙; 1; and n) are associated with the flow law which is driven by the equilibrium (effective)

stress, and the remaining 5 per submechanism (three with the nonlinear hardening operator;

i.e., Hb,  ̄b and ∙b, the hardening threshold, and two with the thermal recovery term; i.e., Rb

and mb) are associated with the evolution of the nonequilibrium (back or internal) stress.

All tests addressed are uniaxial, isothermal, experiments conducted at 650oC, thus implying

that the multiaxial material constants are typically generalized from their uniaxial counterparts.

This need for generalization is precisely why a consistent multiaxial theory, such as that

developed here from a potential formulation, is imperative. The available data is quite extensive,

wherein five tests within the reversible domain (i.e., one creep with recovery upon complete

unloading, one relaxation and 3 rate dependent effective/instantaneous moduli ), and eleven

experiments conducted within the irreversible domain (i.e., three relaxation, four conventional

creep, one multi-step creep, and three tensile curves at different total strain rates) have been

used for characterization; while two multiple-step relaxation, one cyclic and one plasticity/creep

interaction tests were reserved for validating the predictive behavior of the model.

It is important to realize from the outset, that the resulting set of material parameters

is non-unique (due to the nonlinear nature of the problem), however if a sufficient amount

of ”data content”5 is provided to COMPARE then it is believed that the final obtained

parameters should be relatively independent of the initial guess and bounds provided. It is

beyond the scope and intent of this paper, however, to define what constitutes this sufficient

data content. For the purpose of the present exercise, the actual characterization process for

the multimechanism viscoplastic portion of the model was divided into three stages given the

fact that the viscoelastic constants were known from previous work, see Auqrog hw do. [1997,

2000], that is: 1) determine the flow and hardening parameters ( 1; n; Hb,  ̄b and ∙b) using the

three tensile curves at different total strain rates, 2) determine the thermal recovery parameters

(Rb and mb) given the three relaxation, four conventional creep, and one multi-step creep tests

5Data content is meant to imply the variety of experimental data provided not just quantity. In other words,

if one only provided tensile data to COMPARE one should not expect the model to be able to accurately

predict both creep and relaxation response (since the time duration of these types of tests far exceeds that of a

tensile test). Similarly, if one provided only creep or relaxation data to COMPARE, one should not expect

the model to accurately represent the typically shorter time domain tensile behavior of the material.

NASA/TM—2003-212219 12
Table 1: Final Viscoelastic material constants for TIMETAL 21S at 650oC

Constants Units Value

Es GPa 21.753

o - 0.365

Mechanisms a = 1 a = 2 a = 3 a = 4 a = 5 a = 6

M(a) GPa 41.37 6.895 21.120 6.523 3.965 3.434

1⁄2(a) sec 0.5 50.0 974. 9693. 14,460.0 28,218.0

Table 2: Final Viscoplastic material constants for TIMETAL 21S at 650oC given the

saturating hardening form, hnew

Constants Units Characterized Value

∙ MPa 4.25 4.25 4.25

n - 1.016 1.016 1.01646

1 GPa-sec 289,303.2 289,303.2 221,264.0

Mechanisms b = 1 b = 1 b = 2 b = 1 b = 2 b = 3

∙b MPa 104.8 98. 34.5 57. 56.9 43.5

Hb GPa 598. 12.1 8939.3 19,000.7 3.11 1.42

 ̄b - 5.72639 2.785 1.0 1. 6.64696 4.50

Rb 1/sec 6.868827E-02 2.2996E-03 9.8679E-01 1.5114 1.5483E-07 1.7506E-02

mb - 2.38539 1. 1. 1. 4.02489 7.21429

while keeping the flow and hardening parameter estimates from step 1 fixed and 3) fit all

eleven tests at one time to obtain final parameter estimates - this is accomplished by allowing

all viscoplastic parameters to be active (using the material parameters found in steps 1 and

2 as initial starting values) but with tight upper and lower bounds around each parameter.

COMPARE was utilized exclusively in all stages of the characterization process, with the

resulting set of viscoplastic parameters being given in Table 2. These three steps were then

repeated for the cases when two and three viscoplastic mechanisms were active as well, the

resulting parameter sets are also included in Table 2. For specific details regarding the required

input to COMPARE, see the COMPARE User’s guide, version 2.0, Sdohhe hw do. [2001].

In the case of multiple viscoplastic mechanisms, it is important to remember that the bounds

are selected so as to keep each mechanism separate from the other (as each mechanism should

possess a unique internal clock, or effective time domain). Note that the initial starting value

and both upper and lower bounds on the viscoplastic parameters were selected based on experience and physical meaning. For example, the maximum internal stress achievable (the hardening threshold, ∙b), which will strongly influence when a tensile stress-strain response saturates

(given hsat ), is equal to PN

b=1

∙b; where N is the total number of viscoplastic mechanisms. This

NASA/TM—2003-212219 13
value (PN

b=1

∙b = (3⁄4sat ¡ p3∙)=

p3)) can be estimated directly from the available constant strainrate tensile (stress-strain curves) tests, provided they have saturated. For TIMETAL 21S it

appears that this total amount is in the range of 13 to 32 ksi for the different strain rates used

in the experiments. Note, in the above definition, 3⁄4sat is defined to be the measured maximum

stress, at the fully developed plastic/saturation state. Consequently, for the case of two and

three mechanisms, the initial starting values for ∙b are taken to be equal, i.e., ∙single=3 and then

allowed to move from there. The lower bound for p3 PN

b=1

∙b being equal to the highest creep

stress examined. Also, one is reminded that the threshold parameter ∙ (representing that value

of stress below which only reversible (albeit time-dependent or time-independent) behavior is

present, i.e., no measurable inelastic behavior occurs) delineates the reversible and irreversible

regimes. This is the reason why below ∙ only viscoelastic behavior need be considered; alternatively when the stress state exceeds ∙ both viscoelastic and viscoplastic behavior are fully

active. Finally, it is highly recommended that the values of the exponents (i.e., n;  ̄b, and mb)

be always taken greater than one so as to ensure proper behavior of the various derivatives.

Values less than one can at times produce excellent correlations while at the same time provide

peculiar response histories for other loading configurations.

4 Results:

4.1 Comparison of Saturating and Nonsaturating Hardening Forms

In this section, we will examine the tensile response of TIMETAL 21S over three orders of

magnitude in total strain rate (i.e., "_ = 8:33x10−4; 8:33x10−5; 8:33x10−6per second) as well as

its cyclic response in order to demonstrate the difference between the previously widely used

nonsaturating hardening form [see Auqrog&Sdohhe [1994], Auqrog hw. do. [1994a,1995],

and Sdohhe hw do [2001]] and the newly introduced saturating form. The single mechanism

representation in Table 2 will only be considered so that a consistent comparison can be made

to our previous work (see Sdohhe hw do [2001]). Figure 1, illustrates the experimental stress

strain response histories (indicated by symbols) and the corresponding correlated response via

the saturated form (solid lines) and the nonsaturating form (dashed lines, see Sdohhe hw do

[2001]). If we confine ourselves to the early portion (< 1% total strain) of the response curves,

see Fig. 1a, it is clear that both forms do an adequate job of capturing the experimental data

over the range of strain rates. This however, is only true because we purposely limited the

strain range to be less than or equal to 1%. If we enlarge the strain range to that of 6% one

can immediately see the significant difference between the two hardening functional forms, the

saturating form being the more accurate representation overall. Similarly, in the case of cyclic

loading (see Fig. 2) the saturating form does a significantly better job of representing the real

material behavior over the entire strain range. Finally, simulations of other classes of loading

(creep, relaxation, etc.) will not be adversely affected by the utilization of this new saturating

hardening functional form, as demonstrated in the next section.

NASA/TM—2003-212219 14
Table 3: The Viscoplastic material constants for the previous, see Saleeb et al [2001],

nonsaturating hardening function, hnonsat(G(b)

)

Viscoplastic Units Value

∙ MPa 4.24732

n - 0.961

1 MPa-sec 4.19216e08

 ̄(1) - 0.333

H(1) MPa 74341.89

m(1) - 0.374

R(1) 1/sec 0.983e-04

Figure 1: Comparison of specific hardening functions and their ability to represent the stressstrain behavior TIMETAL 21S over three different strain rates (i.e., 8.33E-04, 8.33E05 and 8.33E-06/sec). a) strain range 0-1%; b strain range 0-6%.

NASA/TM—2003-212219 15
Figure 2: Comparsion of specific hardening forms under cyclic loading conditions.

4.2 Correlation Capability of Single versus Multiple Mechanisms

Here we examine the influence of including multiple viscoplastic mechanisms on the ability of

the presented GVIPS class viscoplastic model to accurately simulate TIMETAL 21S. To date,

only multiple mechanisms in the viscoelastic (reversible) portion of the model have been utilized,

see Auqrog hw do. [1997]. In the following results, six viscoelastic mechanisms are included

along with one, two or three viscoplastic mechanisms. In Figs. 3-6 the viscoelastoplastic

correlations for, i) the tensile response of TIMETAL 21S over three orders of magnitude in

total strain rate (i.e., "_ = 8:33x10−4; 8:33x10−5; 8:33x10−6= sec) , ii) relaxation responses given

three initial strain levels (i.e., " = 139; 400 and 602 1" loading, respectively, to the following

”peak” stress levels, 3⁄4 = 103; 238; and 345 MPa, all produced given a total strain rate of

"_ = 5:0x10−4= sec), and iii) short term creep at four stress levels (i.e., 3⁄4 = 14; 72; 110; and 128

MPa) and one multiple step creep test are shown, respectively. In the graphs, the symbols

denote experimental data and the lines the model correlations, with the solid line denoting a

single mechanism; dotted line - two, and the dashed line - three viscoplastic mechanisms. In

the authors’ opinions, these correlations (irrespective of the number of mechanisms utilized)

are qualitatively very good given the very wide variety of loading conditions examined. Overall

the maximum quantitative correlation errors are between 1% to 50% depending upon the load

history imposed in a given test. This is well within typical experimental scatter, particularly

when dealing with creep measurements which can show upwards of 200-300% variation.

Examining the tensile responses, shown in Fig. 3, it is clear that the model with a single

(or multiple) irreversible dissipative mechanism is able to qualitatively capture the ratedependency of the material. Note how only the 8:33x10−5= sec strain rate test is quantitatively

accurate with the lower and higher strain rates being quantitatively in greater error. This is

expected, since the single irreversible dissipative mechanism (although stress dependent) must

compromise over the entire time spectrum of all the experiments being correlated. Clearly,

NASA/TM—2003-212219 16
given the current data set, this mechanism will be biased toward the longer time domain –

as eight of the eleven experiments are long term. When three mechanisms are considered the

model appears to be better able to accurately capture (both qualitatively and quantitatively)

the actual strain rate dependency of the material.

Qualitatively the relaxation behavior exhibited by the material, shown in Fig. 4, is very well

captured by the model again irrespective of the number of viscoplastic mechanism used. This is

particularly true with regard to the primary relaxation feature, that is, the initially high stress

reduction rate and overall large stress-drops at all stress levels. Note that the three starting

relaxation stress levels in Fig. 4 represent stress levels6 that are: i) below the apparent yield,

ii) within the knee (e.g. 115 to 330 MPa), and iii) near the ultimate stress level of the material

(approximately 345 MPa), see Figs. 3 and inserts of Fig. 4. In order to accurately capture

(quantitatively) the entire relaxation history it appears to require at least two mechanisms be

incorporated within the model (three being slightly better).

Similarly, when examining the creep response correlations given in Fig. 5, the model again,

irrespective of the number of viscoplastic mechanism used, is able to capture qualitatively the

stress dependence over the entire stress range. However, clearly, incorporating additional viscoplastic mechanisms (three being overall the most accurate) enhances the present model’s

ability to accurately (quantitatively) simulate both short term and longer term creep behavior.

Note, how the secondary creep rate is significantly more accurate when three mechanisms are

utilized. Figure 6 shows the correlation of the multiple step creep test, where the resulting

inelastic strain versus time (at constant stress) is shown. Here the simulation using multiple

viscoplastic mechanisms (two - dotted and three - dashed lines), as compared with the experimental data (denoted by the symbols), is seen to accurately represent the first two steps but

under-predict the accumulated inelastic strain of the third step. This is not surprising, however,

given the fact that this test was performed under constant load and not constant stress as was

the simulation. As a consequence of the constant load controlled creep test, the experimental

results have both additional geometric effects and possibly creep damage (thereby leading to

tertiary creep) included in them. Neither of these factors have been included in the numerical

simulation. The influence of these factors is particularly evident during the third step, where

the inelastic strain rate during the experiment is significantly greater than that simulated (cf.

3⁄4 = 128 MPa in Fig. 5d).

It is important to note that if one were to focus on only one type of loading condition, superior

correlations could of course be achieved for that class of loading. However, predictions of other

classes of loading may severely suffer. For example, during a separate characterization process

involving only the three tensile tests, excellent correlation was obtained, but at the expense of

poor creep and relaxation behavior for a given set of material parameters. Conversely, when the

model was calibrated for creep responses, poorer tensile and relaxation behavior were similarly

predicted. Thus when judging the ”goodness” of a given model one must keep in mind the

appropriate required ”data content” for accurately capturing the full range of material behavior

of interest. Similarly when comparing the current correlation with that done previously (see

Sdohhe hw do. [2001]) it should be remembered that here all exponents were constrained to be

always greater than or equal to one, even though superior correlations in some cases could be

obtained if one used exponents (in particular  ̄) that are less than one. This latter condition

6Given a 5.0 x 10−4/ sec total strain rate ramp-up history, see Fig. 4.

NASA/TM—2003-212219 17
Figure 3: Illustrates the correlation capability of present model when one, two or three viscoplastic mechanisms are employed given constant strain rate, tensile histories; i.e.,

a) 8.33E-04, b)8.33E-05 and c) 8.333E-06 m/m/sec

NASA/TM—2003-212219 18
Figure 4: Illustrates the correlation capability of the present model when one, two or three

viscoplastic mechanisms are employed given relaxation histories. Peak stress levels

are: a) 103 MPa, b) 238 MPa and c) 345 MPa

NASA/TM—2003-212219 19
Figure 5: Illustrates the correlation capability of the present model when one, two or three

viscoplastic mechanisms are employed in the context of creep (constant stress) loading.

The applied stress is: a) 14, b) 72, c) 110 and d) 128 MPa

NASA/TM—2003-212219 20
Figure 6: Illustrates the correlation capability of the present model when one, two and three

mechanism are used, given a multistep load history, i.e., 72,110 and 128 MPa steps.

was implemented since some anomalies in certain response histories were observed due to higher

order derivatives of the hardening and recovery functions employed in the implicit integration

algorithm. This derivative can become problematic when exponent values less than one are

used.

4.2.1 Back Stress Partitioning within Irreversible Domain

To gain insight into and help explain the differences between the use of a single irreversible

mechanism and that of multiple mechanisms, regarding the evolution of the effective stress

(σij¡αij ) which drives the inelastic response of the material, we have graphed the uniaxial

external stress (σ); individual internal (back) stresses, α(b)

; for each mechanism, as well as the

sum of the back stresses, ®; for each class of loading examined, that is, tensile, creep, relaxation

and cyclic; see Figs. 7 through 10.

Figure 7 illustrates the stress versus time response corresponding to the tensile response

(given a total strain rate of "_ = 8:33x10−5= sec) shown in Fig. 3b. It is clearly evident from Fig.

7 that as the externally applied stress increases so too does the internal (or non-equilibrium)

stress, albeit at a slower rate, until the maximum value of the back stress is reached. This

maximum (uniaxial) value being related to the obtained (estimated) shear threshold value ∙b;

i.e., p3∙b. The determined shear threshold value for this material is 104.7 MPa in the case of

the single viscoplastic mechanism and 158 MPa (for the sum) in the case of three viscoplastic

mechanisms (see Table 2). The fact that the non-equilibrium stress (internal stress)does not go

to zero after an ”infinite” amount of time, but rather reaches a constant value is a distinguishing

feature between the reversible and irreversible time dependent modeling.

Figure 7b demonstrates clearly the existence of different internal viscoplastic clocks, since

the rate of growth of the first mechanism (depicted by the solid line) is relatively short as

compared to mechanisms two and three (as well as the case of a single mechanism, shown in

NASA/TM—2003-212219 21
Figure 7: Illustrates the evolution in time of both the external and internal (for each mechanism)

stress fields during the tensile response shown in Fig. 3b.

NASA/TM—2003-212219 22
Fig 7a). Specifically, in the single mechanism case the internal stress saturates after approximately 300 seconds whereas the first mechanism in the three mechanism model saturates after

approximately 100 secs. Further, one can see that the maximum value of internal stress reached

by mechanism-one (see Fig. 7b) is very near (i.e., 93.4 MPa) its threshold value of 98.7 (or

57.0 in shear) MPa. This is in sharp contrast to the single mechanism case wherein the internal stress essentially saturates at 103 MPa which is substantially below its threshold value

of 189.6 (or 104.7 in shear) MPa. The reason for the ”apparent” premature saturation stems

from the larger hardening exponent,  ̄ = 5.726, used in the case of a single mechanism whereas

 ̄ = 1 was employed for the first mechanism in the three mechanism model. The other two

constant strain rate ("_ = 8:33x10−4 and 8:33x10−6 /sec) tensile tests exhibit qualitatively the

same (stress-time) behavior with only the actual magnitude of both internal and external stress

being different. The ordering of the final resulting equilibrium (effective) stress (i.e., (σ ¡ α))

at saturation of the externally applied stress (for all three tensile tests) is consistent with the

applied total strain rate, i.e., the faster the rate the higher the effective stress. This should

not be surprising, since from eqn. 25 it is immediately apparent that after sufficient time has

elapsed (i.e., q ̇ =0) and the change in external stress has stopped (i.e., σ ̇ = 0; the tensile curve

has flattened), the inelastic strain rate ( ε ̇ vp ) is equal to the applied total strain rate (ε ̇) and

is consequently proportional to the equilibrium (effective) stress.

Figure 8 illustrates the stress versus time response corresponding to the creep response (given

an applied stress of 3⁄4 = 72:0 MPa) shown in Fig. 5b. The single mechanism representation

shown in Fig 8a is very similar to the tensile case shown in Fig. 7a, in that initially one has

a rapid increase in internal stress followed by a saturation of the internal stress. The nonlinear nature of the response is apparent when one compares the ”steady-state” response of

Figs. 7a and 8a, that is (σ ¡ α 1⁄4151:7 MPa, "_ = 8:33x10−5= sec) and (σ ¡ α 1⁄417:9 MPa,

"_ = 1:8x10−7= sec): The behavior of the multimechanism model (see Fig. 8b) is in sharp contrast, however, to that of the single mechanism model; in that, although the overall internal

stress does indeed still rise in time (albeit at a much slower rate) the first mechanism actually

decreases (or relaxes). Also, examining the time corresponding to ”constant” internal stress (i.e.

steady-state), it is clear from Fig. 8 that the single mechanism arrives at this condition in approximately 1500 secounds whereas the three mechanism model requires more than 15000 secs,

thus indicating once again a significant difference in the individual internal irreversible material

time clocks. Figure 5b clearly illustrates that the three mechanism response is significantly

more accurate than the single mechanism with the ”steady-state” strain rate of approximately

3:4x10−07 /sec being very well represented given an effective stress of approximately 20.6 MPa.

Similar behavior to that described above is observed for relaxation and cyclic loading cases,

see Figs. 9 and 10. It is interesting to note that in the case of relaxation (Fig. 9), the relaxation

of the internal stress follows that of the external stress almost exactly, especially for the single

mechanism representation, except early on where the internal stress reached a maximum value

of 42.7 MPa after 593 seconds. Alternatively, in the three mechanism case the maximum

internal stress in the first mechanism (which is the dominate mechanism) is approximately 51.0

MPa after 3.9 seconds and subsequently relaxes to 4.5 MPa after 80,000 secs (the sum being 8.4

MPa). In the case of cyclic loading (Fig. 10) it again is clear that the single mechanism and

first mechanism of the three mechanism model have very similar internal clocks as the evolution

of internal stress is almost identical over the entire load history. Furthermore, it is clear that

the first mechanism dominates this load case.

NASA/TM—2003-212219 23
Figure 8: Illustrates the evolution in time of both the external and internal (for each mechanism)

stress fields during the creep response shown in Fig. 5b.

NASA/TM—2003-212219 24
Figure 9: Illustrates the evolution in time of both the external and internal (for each mechanism)

stress fields during the relaxation response shown in Fig. 4a.

NASA/TM—2003-212219 25
Figure 10: Illustrates the evolution in time of both the external and internal (for each mechanism) stress fields during the cyclic response of Figs. 2 and 11.

NASA/TM—2003-212219 26
Figure 11: Illustrates the predictive capability of a single and three viscoplastic mechanism

model subjected to a cyclic history, given a constant strain rate of 8:33x10−5 /sec:

4.3 Predictive Ability of Single versus Multiple Mechanisms

The present viscoelastoplastic multimechanism GVIPS model, characterized to represent

the behavior of TIMETAL 21S at 650oC, is now exercised and its predictive capability assessed

relative to the number of viscoplastic mechanisms employed. The assessment begins by considering a cyclic test, followed by multi-step relaxation tests, creep reversal or recovery tests and

finally a plasticity/creep interaction test.

Figure 11 illustrates the cyclic behavior under strain control given a total strain rate of

8.33 x 10−5= sec, wherein the symbols represent the experimental results and the solid (single

mechanism) and dotted (three mechanism) line the GVIPS prediction. Here we see that the

agreement is very good with less than a 0.5% over-prediction of the maximum stress at +0:01

and 20% at ¡0:01 strain. This is in sharp contrast to our previous (see Sdohhe hw do [2001])

predictive results employing the non-saturating hardening form, see Fig. 2, where a 17% overprediction of the maximum stress at +0:01 and 45% at ¡0:01 strain was obtained. Clearly, when

using the current saturating hardening form, either, a single or multiple mechanism formulation

provides the same level of accuracy.

Multistep relaxation tests were performed to document the influence of prior history on

the time-dependent response of TIMETAL 21S and validate the predictive capability of the

present model(see Sdohhe hw do [2001]). The two-step load sequence was comprised of a 345

MPa stress level, relaxed for 24 hours followed by an initial 103 MPa stress level relaxed for

approximately one hour. The experimental stress versus time response and GVIPS simulations

given multiple mechanisms are shown in Fig. 12. The first step of this two step relaxation test

was used for correlation purposes (see Fig. 4c), with the second step shown in Fig. 12 being

a measure of the predictive capability of the model. Clearly, in this case the two mechanism

characterization is more accurate than the three mechanism case. Similarly, in the case of

the three step relaxation load sequence (shown in Fig. 13, wherein step 1 began at a stress

NASA/TM—2003-212219 27
Figure 12: Illustrates the predictive capability of a single viscoplastic and multimechanism models subjected to a two-step relaxation history. Here only the response corresponding

to the second step is shown. The starting stress level being 103 MPa.

level of 103 MPa, step 2 at 238 MPa and step 3 once again at 103 MPa with each constant

total strain hold period lasting approximately 24 hours) the two mechanism model appears once

again to slightly out perform the three mechanism model. Obviously, these results demonstrate

that a single mechanism viscoplastic GVIPS model is not capable of accurately capturing the

influence of multistep load histories when compared to multimechanism representations.

Lastly, a classic plasticity-creep interaction experiment, which illustrates the need for unified

viscoplastic models, was performed and simulated. In this type of experiment the material is

subjected to an overload prior to performing a creep test at a lower stress value, see Fig. 14.

Typically, as is found here, the initial creep response following the overload (denoted by the

open circle symbols) is significantly reduced when compared to that produced from a pure

creep test (denoted by the open triangles) at this lower stress amplitude, see Fig. 14b. Once

again comparing the numerical simulations (for the one, two and three mechanism GVIPS

models) to that of the experimental results one observes very good overall agreement for the

multimechanism simulations; whereas, for the single mechanism case approximately a 50% error

is incurred. For this type of loading history, it appears that the three mechanism model slightly

out performs the two mechanism model.

5 Conclusions

Hereditary behavior encompasses such important features as nonlinearity, loading-rate sensitivity, creep, relaxation, creep-plasticity interaction effects, as well as memory effects for both

the short-term (transient) and long-term (steady states of limit equilibrium) material responses.

To date a general, potential-based, viscoelastoplastic multimechanism deformation model has

been formulated and implemented. In this study, two specific material hardening functions

NASA/TM—2003-212219 28
Figure 13: Illustrates the predictive capability of a single viscoplastic and multimechanism models subjected to a three-step relaxation history. The starting stress levels for each

step being, 103, 238 and 103 MPa, respectively.

have been evaluated as well as the influence of employing multiple viscoplastic mechanisms

as opposed to only a single mechanism. Material characterization of a titanium based alloy,

TIMETAL 21S, was accomplished using the recently developed COMPARE software, a fullyautomated procedure for material parameter estimation, and a fairly comprehensive set of test

data under different loading conditions. In addition to these correlation tests, several other

experimental results were reserved to assess the predictive capability of the model.

The overall performance of the multimechanism model was found to be very good, clearly

demonstrating the great potential, effectiveness, as well as practical utility of the proposed

model. In particular, the utilization of a material hardening function (h(G)) that saturates

when the internal stress reaches a threshold value was found to be superior to the previously

used non-saturating form when relatively rapid rates of loading (i.e. on the order of seconds and

minutes as opposed to miniutes and hours) are imposed, as typical in strain controlled tensile

tests. Furthermore, it was clearly demonstrated in this study that the introduction of this

saturating hardening functional form also demands the use of more than a single viscoplastic

sub-mechanism if one desires to accurately represent both relaxation and creep behavior as

well as tensile behavior. Finally, the existence of multiple internal clocks which account for

the wide range in observed response history is illustrated by plotting the evolution of internal

(non-equilibrium) stress within the multimechanism model. The coupled nature of the model

is clearly seen in the external and internal stress plots (see Figs. 7 through 10) wherein for

example one internal stress may rise/fall while the other(s) fall/rise. Accurate results, both

qualitatively and quantitatively, over the entire load-time spectrum, can be achieved with

the use of additional sub-mechanisms within the irreversible domain. This conclusion is

consistent with our previous findings within the reversible domain (the need for 6 viscoelastic

mechanisms, to capture both time dependency and rate-sensitivity) wherein multiple mechanism

were required to accurately represent the deformation behavior over a wide time spectrum, see

Auqrog hw. do. [1997].

NASA/TM—2003-212219 29
Figure 14: Illustrates the predictive capability of a single and multimechanism model subjected

to a plasticity/creep interaction history: a) stress versus total strain response and

b) creep strain versus time response.

NASA/TM—2003-212219 30
References

[1967] Crohpdq., B.D. dqg Gxuwlq, M.E.: ”Thermodynamics with Internal ”State Variables”, J. Chem. Phys., Vol. 47, No.2, pp. 597-613.

[1973] Lxeolqhu, J.: ”On the Structure of Rate Equations of Materials with Internal Variables, ” Acta Mechanica, Vol. 17, pp. 109-119.

[1976] Hduw, E.W.: ”Constitutive Relations for Nonelastic Deformation of Metals”, J. Eng.

Mater. Technol., Vol. 98, pp. 193-202.

[1980] Kxpdu, V., Mrumduld, M. dqg Mxnkhumhh, S.: Numerical Integration of Some

Stiff Constitutive Models of Inelastic Deformation”, J. Eng. Mater. Technol., Vol. 102,

pp.92-96.

[1982] Hluwk, J.P. dqg Lrwkh, J.; Theory of Dislocations, Wiley, N.Y..

[1985] Hxl, C.Y. : ”A Uniformly Valid Asymptotic Solution of Hart’s Equations for Constant

Inelastic Extensional Strain Rate”, Int. J. Solids and Struct., Vol. 21, pp. 411-421.

[1987] Rrelqvrq, D.N. : ”A Unified Creep-Plasticity Model for Structural Metals at High

Temperatures”, ORNL TM-5969.

[1989] Ckderfkh, J.L.: ”Constitutive Equations for Cyclic Viscoplasticity”, Int. J. Plasticity, Vol. 5, pp. 247-302.

[1990] Lhpdlwuh, J.; dqg Ckderfkh, J.L.: Mechanics of Solid Materials, Cambridge

University. Press, New York.

[1993] Avkedxjk, N.E., dqg Kkredle, M.: Unpublished Data, University of Dayton Research Institute, Dayton, Ohio.

[1993] Fuhhg, A.D. dqg Wdonhu, K.P.: ”Viscoplasticity with Creep and Plasticity Bounds,

Int. J. Plasticity, Vol. 9, pp. 213-242.

[1993] Sdohhe, A.F. dqg Wlow, T.E.: ”Analysis of the Anisotropic Viscoplastic-Damage

Response of Composite Laminates-Continuum Basis and Computational Algorithms,”

Int. Jnl. Num. Meth. Engng., Vol. 36, pp. 1629-1660.

NASA/TM—2003-212219 31

[1994] Auqrog, S.M.; dqg Sdohhe, A.F. : ”On the Thermodynamic Framework of Generalized Coupled Thermoelastic Viscoplastic - Damage Modeling”, Int. J. Plasticity, Vol.

10, No. 3, pp. 263-278.

[1994a] Auqrog, S. M., Sdohhe, A.F., dqg Cdvwhool M.G.: ”A Fully Associative, Nonlinear Kinematic, Unified Viscoplastic Model for Titanium-based Matrices”, Life Prediction Methodology for Titanium Matrix Composite, ASTM STP 1253, W.S. Johnson,

J.M. Larsen, and B.N. Cox, Eds. American Society for Testing and Materials, Philadelphia, pp. 231-256, or NASA TM-106609.
[1995] Auqrog, S.M., Sdohhe, A.F, dqg Wlow, T.E.: ”A Modeling Investigation of Thermal and Strain Induced Recovery and Nonlinear Hardening in Potential Based Viscoplasticity,” J. Engng. Mater. Tech. ASME, Vol. 117, pp. 157-167.

[1996] Ckderfkh, J.L. dqg Cdloohwdxg, G.: ”Integration Methods for Complex Plastic

Constitutive Equations”, Comput. Meth. Appl. Mech. Eng., Vol. 133, pp. 125-155.

[1996] Kuhpso, E. : ”A Small Strain Viscoplasticity Theory Based on Overstress”, In: Unified Constitutive Laws of Plastic Deformation, Krausz, A and Krausz, K. (Eds.), Academic, San Diego, pp. 281-318.

[1996] Kudxv}, A.S. dqg Kudxv}, K.(eds.): ”Unified Constitutive Laws of Plastic Deformation”, Academic Press, New York.

[1997] Sdohhe, A.F. dqg Auqrog, S.M.: ”A General Reversible Hereditary Constitutive

Model: Part I: Theoretical Developments”, NASA TM-107493; also see Jnl. Eng. Mat.

Tech., Vol. 123, pp. 51-64, 2001.

[1997] Auqrog, S.M.,Sdohhe, A.F. dqg Cdvwhool, M.G.: ”A General Reversible Hereditary Constitutive Model: Part II: Application to A Titanium Alloy”, NASA TM 107494;

also see Jnl. Eng. Mat. Tech., Vol. 123, pp. 65-73, 2001.

[1998] Sdohhe,A.F., Wlow, T.E., dqg Ll, W.: ”An Implicit Integration Scheme for Generalized Viscoplasticity with Dynamic Recovery”, Comp. Mech., Vol. 21, No. 6, pp.

429-440.

[1998] Sdohhe, A.F., Ghqg|, A.S., Wlow, T.E., dqg Turzeulgjh, D.A.[1998]: ”COMPARE - Constitutive Material Parameter Estimation, User’s Guide - Version 1.0”,

Technical Report, Dept. of Civil Engineering, University of Akron, Akron, Ohio.

[1998b] Sdohhe,A.F., Wlow, T.E., dqg Ll, W. : ”Robust Integration Schemes for Generalized Viscoplasticity with Internal-State Variables”, Comp. Mech., Vol. 21, No. 6, pp.

429-440.

NASA/TM—2003-212219 32

[1994b] Auqrog, S.M.; Sdohhe, A.F., dqg Cdvwhool, M.G.: A Fully Associative,Nonisothermal, Nonlinear Kinematic, Unified Viscoplastic Model for Titanium

Based Matrices, Thermomechanical Fatigue Behavior of Materials: Second Volume,

ASTM STP 1263, M.J. Verrilli and M.G. Castelli, Eds., Philadelphia,or NASA TM106926.

[2001] Sdohhe, A.F., Auqrog, S.M., Cdvwhool, M.G , Wlow, T.E., dqg Gudi, W.E.:

“A General Hereditary Multimechanism-Based Deformation Model With Application

to The Viscoelastoplastic Response of Titanium Alloys, Int. Jnl. of Plasticity, Vol. 17,

No. 10, pp. 1305-1350.

[2002] Auvhqolv, A. dqg Pdunv, D.M.; ”Modeling the Evolution of Crystallographic Dislocation Density in Crystal Plasticity”, Jnl. Mech. Physics of Solids, Vol. 50, pp. 1979-

2009.
[2002] Hr, K. dqg Kuhpsoh, E. : ”Extension of the Viscoplasticity Theory Based on Overstress (VBO) to Capture Non-Standard Rate Dependence in Solids”, Int. Jnl. of Plastcity, Vol. 18, pp. 851-872.

[2002] Sdohhe, A.F., T.E. Wlow, D.A. Turzeulgjh, dqg Ghqg|, A.S.; “Effective Strategy for Automated Characterization in Complex Viscoelastoplastic and Damage Modeling for Isotropic/Anisotropic Aerospace Materials”, Jnl of Aerospace Engineering,

Vol. 15, No. 3, pp.84-96.

[2002] Swdlqlhu, L., Cxlwlqr, A. M. dqg Ouwl}, M.; ”A Micromechanical Model of Hardening, Rate-Sensitivity and Thermal Softening in BCC Single Crystals”, J. Mech. Phys.

Solids, Vol. 50 , pp. 1511-1545.

[2002] Zele, H.M. dqg Rxeld, T.D.; ”A Multiscale Model of Plasticity”, Int. Jnl. of Plasticity, Vol. 18, pp. 1133-1163.

NASA/TM—2003-212219 33
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39

Specific Hardening Function Definition and Characterization of a

Multimechanism Generalized Potential-Based Viscoelastoplasticity Model

A.F. Saleeb and S.M. Arnold

Viscoplasticity; Viscoelasticity; Nonlinear hardening; Isothermal; Deformation; Internal

state variable; Correlations; Predictions; Multiaxial

Unclassified -Unlimited

Subject Category: 39 Distribution: Nonstandard

A.F. Saleeb, University of Akron, Akron, Ohio 44325; and S.M. Arnold, NASA Glenn Research Center. Responsible

person, S.M. Arnold, organization code 5920, 216–433–3334.

Given the previous complete-potential structure framework together with the notion of strain- and stress-partitioning in

terms of separate contributions of several submechanisms (viscoelastic and viscoplastic) to the thermodynamic functions

(stored energy and dissipation) a detailed viscoelastoplastic multimechanism characterization of a specific hardening

functional form of the model is presented and discussed. TIMETAL 21S is the material of choice as a comprehensive

test matrix, including creep, relaxation, constant strain-rate tension tests, etc. are available at various temperatures.

Discussion of these correlations tests, together with comparisons to several other experimental results, are given to

assess the performance and predictive capabilities of the present model particularly with regard to the notion of hardening saturation as well as the interaction of multiplicity of dissipative (reversible/irreversible) mechanisms.