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International Journal of Pressure Vessels and Piping

journal homepage: www.elsevier.com/locate/ijpvp

Comparative analysis of the sin-hyperbolic and Kachanov–Rabotnov creep-damage models

Mohammad Shafinul Haque∗, Calvin Maurice StewartUniversity of Texas El Paso, 500 West University Avenue, El Paso, TX, 79902, USA

A R T I C L E I N F O

Keywords:Creep deformationCreep-damageStress-ruptureMinimum creep strain rateSin-hyperbolic modelHigh-stress to low-stress transitionContinuum damage mechanics

A B S T R A C T

In this study, the Sin-Hyperbolic (Sinh) and the classic Kachanov-Rabotnov (KR) creep-damage models are fit tocreep data for 304 stainless steel at 600 and 700 °C. The ability of the models to predict minimum creep strainrate, creep deformation, damage evolution, and rupture are compared. It is shown that both models can predictthe creep deformation accurately; however, the minimum creep strain rate, damage evolution, and rupturepredictions of the KR model is deficient. Using the KR model, critical damage is often less than unity; a violationof continuum damage mechanics (CDM). The KR minimum creep strain rate and rupture predictions do notexhibit the sigmoidal bend on a log-log scale that is observed in experimental data. In the Sinh model, criticaldamage is always unity and the sigmoidal bend in the minimum creep strain rate and stress-rupture curves ispresent. Overall the Sinh model is more accurate.

1. Introduction

When subject to an elevated temperature and pressure environmentmost metallic materials are susceptible to creep deformation and stressrelaxation. Alloys such as 304 stainless steel, show strain rate sensitivityand creep deformation at room temperature [1]. This “unified elastic-viscoplastic” behavior occurs when the boundary between elastic andinelastic deformation is difficulty to define. It is critically important tostudy and model the minimum creep strain rate, creep deformation,damage, and rupture of alloys such as 304SS.

Creep deformation can be separated in three successive stages:primary, secondary, and tertiary as depicted in Fig. 1. The primarycreep regime dominants at low temperature and low stress. In high-temperature alloys, negligible primary creep strain will accumulate andit is often not observed in experiments. The secondary creep regimedominants at low-to-intermediate stress and temperature. This regimeis the most stable where a balance of strain-hardening and recoverymechanics make the prediction of creep deformation simple. At inter-mediate-to-high temperature and stress, the tertiary creep regimedominants. This regime is characterized by the non-linear accumulationof creep-damage that contributes to gross creep deformation and rup-ture [2]. In extreme environments, such as Advanced Super-CriticalCarbon Dioxide power cycles, the secondary and tertiary creep regimewill be dominant.

Creep-damage is the irreversible accumulation of defects that

contribute to rupture as depicted in Fig. 1. Cauvin and Testa char-acterized damage in three scales micro-, meso- and macro-scales.Atomic voids and dislocations appear on the micro-scale while visuallyor near-visually observable damage is on the macro-scale [3]. Themesoscale consists of a representative volume element that averages theeffects of micro-scale cracks, voids, and other distributed deteriora-tions. It is the mesoscale that should be accounted for when usingcontinuum damage mechanics (CDM). Marigo stated that damage takesplace when atomic bonds break at the microstructural level and damagerepresents surface discontinuities in the form of micro-cracks and vo-lume discontinuities in the form of cavities [4]. Lindborg explained that(a) localized stress concentrations mount up due to micro-cracks; (b)rapid plastic and/or viscoplastic flow can occur at the crack tips; and (c)sharp cracks are more critical for damage than round voids [5]. Ibijolaclassified damage as brittle when the crack initiates at the mesoscalewithout plastic strain and damage is ductile when damage occurs si-multaneously with plastic deformation [6]. Lemaitre added that cracksand voids inside materials are oriented randomly [7]. It is common toconsider these orientations as an intrinsic variable of damage [8,9].Ibijola proposed a model assuming that cracks and voids are equallydistributed in all directions inside the material and do not depend onorientation [6]. Later it was proposed that there are two com-plementary approaches to damage, one that focuses on actual damagemanifestations at mesoscale (the micromechanics approach) and theone that assumes homogeneous relation with damage (the

https://doi.org/10.1016/j.ijpvp.2019.02.001Received 26 May 2015; Received in revised form 17 April 2017; Accepted 1 February 2019

∗ Corresponding author.E-mail address: mhaque@miners.utep.edu (M.S. Haque).

International Journal of Pressure Vessels and Piping 171 (2019) 1–9

Available online 04 February 20190308-0161/ © 2019 Elsevier Ltd. All rights reserved.

T

phenomenological approach) which has been coined as “continuumdamage mechanics” [3,10]. Hill and Hashin have done significant workon the micromechanics approach and solved the boundary value pro-blem of a representative volume element (RVE) with distributed mi-crostructural features [11,12]. Kachanov originated the idea of a scalarphenomenological damage quantity for damage modeling based onCDM. Rabotnov derived damage as the “area of cavities” in the cross-sectional area of a uniaxially stressed specimen. Rabotnov developed adamage evolution equation that when coupled to a power-law creepstrain rate is known as the Kachanov-Rabotnov (KR) creep-damagemodel [13,14]. The KR model was extended by Penny to extrapolatelong-term creep-rupture behavior using short-term creep-rupture data[15–17]. Le May and Furtado studied the relationship between the KRdamage equation and physical void formation [18]. Analytical toolshave been developed to rapidly determine the material constants for theKR model given creep data [2].

Overall, the Kachanov-Rabotnov (KR) model has been found to ac-curately predict creep deformation; however, there are deficiencies inthe functional form of the equations that contribute to poor minimumcreep strain rate, damage evolution, and rupture predictions. The KRapproach is a local CDM approach where rupture is reached when thedamage variable within a representative volume element becomesunity. This is consistent with the idea that at rupture the “area ofcavities” is equal to the cross-sectional area of a specimen. When the KRmodel is applied, it is reported that the analytical damage at rupture iswell below unity, a result that is inconsistent with CDM [16,19]. Sev-eral investigators have observed the stress sensitivity and mesh de-pendency when implementing the KR model [20–22]. In finite element(FE) simulations of notched geometry, damage increases the stressconcentration faster than the plasticity driven stress-relaxation. Thedamage rate becomes near infinite contributing to mesh dependenceand numerical instability. Brittle-type, localized damage develops at thenotch tip. Upon mesh refinement, FE simulations do not converge to aunique solution [23–25]. Stewart hypothesized that these issues andothers could be overcome by introduce a set of coupled creep-damageequations that incorporate the Sine-hyperbolic (Sinh) function [26].Stewart developed a multistage Sinh creep-damage model that canmodel primary, secondary, and tertiary creep. The Sinh model mitigatesstress sensitivity and the damage rate remains finite even on the vergeof rupture [23,24]. When the Sinh model is applied, rupture only occurswhen damage equals unity.

In this study, the KR and Sinh creep-damage models are calibratedand compared to 304 Stainless Steel creep data. The creep data consists

of two isotherms and four stress levels that have been tested for re-peatability five times (a total of twenty individual tests) [27]. Analy-tical approaches to evaluate the material constants of each model arediscussed in detail. The minimum creep strain rate, creep deformation,damage evolution, and rupture of the models are compared to eachother. Additional minimum creep strain rate and creep-rupture datathat exhibits sigmoidal behavior is used to compare the predictioncapability of the models [26,28,29]. It is found that while both modelsaccurately predict creep deformation, the Sinh provides a more accu-rate prediction of creep-damage and rupture. Complimentary to thisstudy, the Sinh model has been compared to the Theta-projection andMPC Omega models [30,31].

2. Creep-damage constitutive models

2.1. Continuum damage mechanics

The initiation and growth of microstructural defects lead to gradualloss of material strength. The main purpose of Continuum DamageMechanics is to model and incorporate this process in the constitutiveequations for a given material. Continuum Damage Mechanics (CDM)based damage rate equations can provide accumulated damage, re-sidual life, and rupture prediction given the applied stress and tem-perature. It is assumed that damage happens in a continua (a homo-geneous representative volume) thereby the expression continuumdamage mechanics is coined.

Generally, creep-damage can be classified in two forms: trans-granular damage and inter-granular damage. Trans-granular damagearises when dislocations inside the crystal grains form under high stressand low temperature. Inter-granular damage arises when micro-cracking and cavitation takes place at grain boundaries and triple pointsunder high temperature and low stress [34].

The growth and accumulation of cavities under creep conditionslead to a reduction in the effective cross-section and thus an increase inthe effective stress. For the calculation of damage, three type of stresscan be considered [32]. The first is the true stress which accounts forthe geometric reduction in cross-section due to macroscopic deforma-tion

= = ⋅ +σ FA

σ ε(1 )Ti (1)

where σT is true stress, F is applied force, Ai is an initial cross-sectionalarea, σ is engineering stress and ε is engineering strain. The second isthe net stress which is a type of true stress that accounts for geometrichomogenization

⎜ ⎟= ⋅⎛⎝

⎞⎠

=−

σ σ AA

σΩ(1 )N T

i

N

T

(2)

where Ai is the current cross-sectional area, AN is the mean area of de-cohesion, and Ω is the fraction of area de-cohesion. The net stress ishigher than the true stress. Finally, there is the effective stress whichaccounts for local stress concentration, the interaction between defects,and mechanical behavior homogenization. The effective stress can bedefined as the stress required to attain macroscopic engineering strain εin an undamaged volume element subject to the engineering stress σ[32,33].

= ⋅ =

= −

−σ σ

ω

¯

1

AA

σω

EE

¯ (1 )

(3)

where A is an effective (reduced) cross-sectional area, ω is damage, andE and E are the effective and initial elastic modulus respectively. Asdamage accumulates to unity, the effective elastic modulus approacheszero.

Fig. 1. Creep deformation and damage evolution: The green, yellow, and redzones represent the primary, secondary, and tertiary stages respectively wherethe dash lines represent the transition between stages. The red dash-dot linesrepresent the irreversible meso-scale damage that accumulates at the onset oftertiary creep. (For interpretation of the references to colour in this figure le-gend, the reader is referred to the Web version of this article.)

M.S. Haque, C.M. Stewart International Journal of Pressure Vessels and Piping 171 (2019) 1–9

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2.2. Kachanov-Rabotnov model

In 1967, Kachanov observed a varying creep ductility but did notidentify the damage mechanism responsible for tertiary creep [13].Kachanov assumed creep displacement rates are much faster within theuncavitated regions than cavity growth regions and cavitation does notaffect the macroscopic creep deformation. Fracture occurs when cav-ities are linked up throughout whole transverse grain boundaries. Thisinitial model did not predict tertiary creep for constant stress condi-tions. Later, in 1969, Rabotnov assumed the creep displacement rates isslower at the uncavitated regions than those of cavity growth regions[14]. Thus, stress is loaded up to the uncavitated zones from fastgrowing cavitated zones. The redistributed stress increases the creeprate. Rabotnov coupled the damage with the creep rate through anstress increment associated with a load reduction in bearing area as thecavitations/damage develops. This coupled model leads to an ac-celerated/tertiary creep rate [36].

In the Kachanov-Rabotnov model, amplified stress due to a localreduction in cross-sectional area is the basis of the damage variable, ω.The reduction in cross-section area is due to the formation of micro-cracks, cavities, and voids. The effective stress [Eq. (3)] is applied. TheKachanov-Rabotnov coupled creep-damage constitutive equations areas follows

= = ⋅⎛⎝ −

⎞⎠

ε dεdt

A σω

˙1cr

crn

(4)

= = ⋅−

ω dωdt

M σω

˙(1 )

ψ

θ (5)

where A and n are the Norton power law constants, σ is equivalentstress and M , ψ and θ are the tertiary creep-damage constants [13,14].The ψ constant controls the magnitude of Mand can be set arbitrarily toany number greater than or equal to unity. Rupture time and criticaldamage can be predicted by integration of the damage evolution [Eq.(5)] and assuming the initial time t0 and initial damage ω0 are equal tozero [2,35].

= − − ⋅ + ⋅ ⋅+ −t ω θ M σ[1 (1 ) ] [( 1) ]θ ψ1 1 (6)

= − − + ⋅ ⋅ ⋅ +ω t θ M σ t( ) 1 [1 ( 1) ]ψ θ1

( 1) (7)

The tertiary creep constant M can be calculated by rearranging [Eq.(6)]

= − −+ ⋅ ⋅

+M ω

θ σ t1 (1 )

( 1)r

θ

ψr

1

(8)

At rupture time, assuming the critical damage =ω 1r gives

=+ ⋅ ⋅

Mθ σ t

1( 1) x

r (9)

One of the major drawbacks of the KR model is the difficulty de-termining the tertiary creep-damage constants. Two complementaryanalytical procedure was recently developed (Strain approach andDamage approach) to mitigate this issue [2]. In the strain approach(SA), the damage evolution equation is incorporated into the creepstrain rate equation such that the experimental creep strain can be di-rectly applied to find the constants. In the damage approach (DA), finitedifference is applied to obtain the creep strain rate and the creep strainrate plugged into the creep strain rate equation to calculate “analyticaldamage”. This analytical damage is then used to calibrate the damagerate equation. A brief discussion of these two approaches is follows.

2.2.1. Strain approachIn the Strain Approach (SA) the damage prediction [Eq. (7)], is

introduced into the creep strain rate [Eq. (4)] and integration is per-formed to get a creep strain equation. Then theM constraint [Eq. (8)], is

introduced and the equation simplified to produce the following creepstrain formula

⎜ ⎟=⋅ + ⋅⎡⎣

⋅ − ⋅ + ⋅ ⎤⎦+ −

= ⎛⎝

− ⎞⎠

+( ) ( )ε t

A θ t t t σ

θ nτ t

t( )

( 1)

1, 1cr

στ

nr

στ

nr

n

r

θ1

1

(10)

here, both the M and ψ tertiary creep-damage constants are eliminated.The ψ constant can be set arbitrarily to any number greater than orequal unity. The M constant should be calculated from [Eq. (8)]. Theconstant, θ can be determined from available creep strain data using thecreep strain [Eq. (10)] and regression software.

2.2.2. Damage approachIn the Damage Approach (DA), the creep strain rate [Eq. (4)], is

algebraically rearranged to get analytical damage at each time step

=−( )

( )ω ε

σ( ˙ )cr

εA

n

εA

n

˙ 1/

˙ 1/

cr

cr(11)

Thus the creep strain rate from experiments, εcr will produce ananalytic damage curve, ω ε( ˙ )cr . The introduction of M constraint [Eq.(8)] into the damage prediction [Eq. (7)] gives the following

⎜ ⎟= −⎡

⎣⎢⎢

⋅⎡⎣⎢

⎛⎝

− ⎞⎠

− ⎤

⎦⎥ +

⎤

⎦⎥⎥

+ +

ω t tt

ω( ) 1 1 1 1r

r

θ θ11

1

(12)

where the tertiary creep-damage constant, θ, is the only unknown. Theconstant θ can be determined by using the available creep strain data,the analytic damage, [Eq. (7)], the modified damage, [Eq. (12)], andregression analysis software. The ψ constant can be set arbitrarily toany number greater than or equal unity. The M constant should becalculated from [Eq. (8)].

2.3. Limitation of local approach

Kachanov-Rabotnov's (KR) local CDM approach shows stress sensi-tivity. In the KR approach damage is assumed to be an internal statevariable which evolves from zero to critical damage, ≤ <ω ω(0 )cr ,where critical damage is often assumed equal to unity =ω( 1)cr . Qi et al.and Lemaitre made it clear that the assumption that critical damage isequal to unity is not realistic [19,37]. The value of critical damage isless than unity and varies from 0.2 to 0.8 for most metals. Chaboche,explained that KR model uses local cross-sectional area reduction toaccount for effective stress amplification during damage, but in reality,microscopic damage gives little loss of effective area before crack in-itiation thus the damage variable, ω is a very small value until a largefraction of life has been exhausted [38]. He proposed that an im-provement can be obtained by introducing an additional damageparameter. The KR model tries to model both the creep-damage andfracture within a continuous function; however, the very high rate ofdamage that occurs during fracture cannot be accommodated, thus theKR model develops critical damage values equal to much less thanunity. Penny stated that the damage rates of KR model become ex-cessively high at about 90% of a lifetime such that the critical damageωcr cannot be unity [16]. This becomes clear when examining the da-mage function of KR [Eqs. (4) and (5)], as follows

⎜ ⎟= ⎛⎝ −

⎞⎠

ε fω

g σ T˙ 1(1 )

( , )cr n (13)

⎜ ⎟= ⎛⎝ −

⎞⎠

ω fω

g σ T˙ 1(1 )

( , )θ (14)

when damage approaches unity these functions becomes infinitely largesuggesting that these functions attempt to encapsulate both the con-tinuous damage of creep and the instant of fracture [39]. For further

M.S. Haque, C.M. Stewart International Journal of Pressure Vessels and Piping 171 (2019) 1–9

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proof, taking the variation of damage ∂ω t( ), [Eq. (12)], with an in-finitesimal variation of stress ∂σ t( ) produces

∂ =− +

∂

+

ω t ψ Mσ t t

θ Mσ t t

σ tσ t

( ) ( )

[1 ( 1) ( ) ]

( )( )

ψ

ψ θθ 1 (15)

Replacing the portion, Mσ t t( )ψ , by rearranging [Eq. (12)] and in-troducing it into the above equation provides

∂ = − −−

∂+ω t ψ ω

ωσ t

σ t( ) 1 (1 )

(1 )( )

( )

θ

θ

1

(16)

For an infinitesimal variation of stress, ∂σ t( ), when damage is cri-tical →ω( 1)cr , the damage variation ∂ω t( ) is near infinite as the de-nominator becomes zero. Damage cannot be equal to unity. To over-come these issues, a damage evolution equation that exhibits a finitevariation of damage ∂ω t( )for an infinitesimal variation of stress ∂σ t( ) isneeded.

Numerous investigators have written papers describe the limitationsof the local CDM approach [21,40–42]. These authors observed damagelocalization around the crack tip in FEM analysis and experienced meshdependency where the FE solutions upon mesh refinement do notconverge to a single solution. Extensive mesh dependence studies on theKR and Sinh models have been reported elsewhere [23,24].

2.4. Sinh model

In this study, a CDM-based Sinh creep-damage constitutive model isused. A detailed monograph on this model is available [26]. Classically,the simplest stage of creep to analysis is the secondary stage where abalance between hardening and recovery mechanisms leads to a steady-state/minimum creep strain rate, εmin . The Mcvetty minimum creepstrain rate equation

=ε B σ σ˙ sinh( / )c s (17)

can fit the non-linear bend observed in minimum creep strain rateversus stress on a log-log scale. The constants B (1/s) and σs(MPa) arethe creep coefficient and secondary creep mechanism-transition stressrespectively [26,43].

2.4.1. Creep strain rate and analytical damageThe CDM based damage variable has three functions; to model the

tertiary creep regime, track the evolution of creep-damage, and predictrupture. To achieve these goals the damage variable, ω, is coupled tothe secondary viscous function as follows

= ⋅ ⋅ε f σ g T h ω˙ ( ) ( ) ( )sc (18)

where the h ω( ) function describes how the current state of damageinfluences the strain rate. A detailed monograph of the damage evolu-tion has been derived [26].

Introducingh ω( ), into the total creep strain rateεcr , [Eq. (17)], andsolving for h ω( ) produces

=h ω t ε tB σ σ

[ ( )]˙ ( )

sinh( / )cr

s (19)

It is proposed that =h ω λω( ) exp( )p , where λ and p are unit-lessmaterial constants. Introducing it into above equation and solving fordamage provides

= ⎧⎨⎩

⎡⎣⎢

⎤⎦⎥

⎫⎬⎭

∗ω tλ

ε tB σ σ

( ) 1 ln˙ ( )

sinh( / )cr

s

p1/

(20)

where ∗ω t( ) is the analytical damage derived from the creep strain rate,ε t˙ ( )cr . Considering the time just before fracture ≃t tr , where =ω 1cr , thecreep strain rate, [Eq. (19)], becomes

= ≈

= =( )ε ε B σ σ λ

λ ε B σ σ

˙ ˙ sinh( / )exp( )

ln ; ˙ sinh( / )

cr final s

εε s˙˙ minfinal

min (21)

where the material constant λ can be determined directly from ex-perimental data. It is proposed that the best value for p is 3/2 unit-less.Thus, the creep strain rate of the Sinh model becomes

=ε B σ σ λω˙ sinh( / )exp( )cr s3/2 (22)

2.4.2. Damage modelUsing the concept of Liu and Murakami [39], that the problem of

stress-sensitivity and mesh-dependence can be mitigated by re-presenting damage as an exponential function within the creep rate anddamage evolution equations, the following damage evolution equationis proposed

⎜ ⎟=− − ⎛

⎝⎞⎠

ωQ ϕ

ϕσσ

ϕω˙ [1 exp( )]sinh exp( )

t

χ

(23)

where Q, ϕ, χ , and σt are material constants that must be greater thanzero. The portion − −ϕ ϕ[1 exp( )]/ is necessary to avoid an undefinederror when the damage evolution equation is integrated. Integration ofthe damage rate [Eq. (23)] with the assumption of the initial time, toand initial damage, ω0 equal to zero, gives the following damage andrupture prediction

⎜ ⎟= − ⎡⎣⎢

− − − ⎤⎦⎥

= ⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

−

ω tϕ

ϕ tt

t Q σσ

( ) 1 ln 1 [1 exp( )] ; sinhr

rt

χ 1

(24)

where the constants Q, χ , and σt can be found using stress-rupture data.The remaining constants ϕ, can be determined simultaneously byminimizing the error of the following equation [a combination of Eqs.(20) and (24)]

=

− ⎡⎣− − − ⎤⎦

= ⎡⎣

⎤⎦

∗

{ }ω t ω t

ϕ

( ) ( )

ln 1 [1 exp( )] lnϕtt λ

ε tB σ σ

p1 1 ˙ ( )sinh( / )

1/

rcr

s (25)

It can be observed from [Eq. (24)] that the damage trajectory de-pends on the material constant ϕ. Taking the variation of damage, ∂ω t( )with an infinitesimal variation of stress, ∂σ t( ) [Eq. (24)] and replacingthe portion Q σ σsinh( / )t

χ by rearranging [Eq. (24)] gives

∂ =− ∂ω t

ϕωϕ

σ tσ

( )[exp( ) 1] ( )

t (26)

where when damage is critical →ω( 1)r , the variation of damage, ∂ω t( )does not become infinite and remains finite throughout the damageevolution.

3. Result

For verification, the constitutive models are calibrated to experi-mental data for 304SS at four stress levels (160, 180, 300, 320MPa)and two isotherms (600–700 °C) [27]. Each test was repeated five times,thus a scatter band is observed.

3.1. Kachanov-Rabotnov model

A list of optimized constants for the KR model is presented inTable 1. The material constants were obtained by fitting to the repeattest nearest the middle of the scatter band. Values of constant A and nwere evaluated using Norton power law ( =ε Aσ˙ n

min ). The remainingconstants M, ψ, and θ were found using the Strain (SA) and Damage(DA) approaches. For the strain approach (SA) the experimental creepdata was fit to the strain [Eq. (10)] and optimized values of M, ψ, and θobtained, These constants were plugged into [Eq. (7)] to evaluate

M.S. Haque, C.M. Stewart International Journal of Pressure Vessels and Piping 171 (2019) 1–9

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damage evolution using SA. For the Damage Approach (DA), the ana-lytical damage is calculated using [Eq. (11)] then M, ψ, and θ constantswere optimized through comparing with the damage prediction [Eq.(12)]. These constants are used in [Eq. (10)] to evaluate the creep strainusing DA.

The creep deformation curves at 600 °C (300 and 320MPa) and700 °C (160 and 180MPa) are plotted in response to the KR model(both using the SA and DA approach) in Fig. 2a and Fig. 3a respectively.It is evident that the KR model fits through the middle of the scatterband.

Plots of the damage evolution to analytical damage data [Eq. (11)]are shown in Fig. 4a and Fig. 5a. There is minimal difference betweenthese two constant determination approaches. Overall, the KR modelproduces an accurate prediction of creep deformation and the analyticaldamage; however, critical damage is ≪ω 1.

3.2. Sinh model

The same experimental data is used to determine the material

constants of the Sinh model. The secondary creep constants B and σs aredetermined from minimum creep rate and model equation [Eq. (21)].The unit-less material constant λ is calculated for each data set using[Eq. (21)]. The value of ϕ is determined by minimizing the error of [Eq.(25)]. A list of the optimized constants for the Sinh is presented Table 2.

The constant λ is the natural logarithm of the quotient of final andminimum creep strain rate. As creep strain rate depends on stress andtemperature, the λ constant also evolves accordingly. From Table 2 it isobserved that with a decrease in stress the constant λ increases. At lowstress, machine components deform slowly towards failure causing alower εmin value, thus λ increases. Not enough experimental data isavailable about the final creep strain rate εfinal to generalize this state-ment across all levels of stress and temperature.

The constant ϕ controls the trajectory of the Sinh damage modelcurve. At →ϕ 0 the trajectory becomes linear. For a given temperature,the value of ϕ increases with decreasing stress. At low stress, machinecomponents have a comparatively higher life. The damage evolutionmust have a long trajectory to accommodate longer life. The constant ϕguides the damage trajectory.

Table 1Kachanov-Rabotnov material constants for 304SS

Temp, T(°C) Stress, σ A n Strain Approach Damage Approach

M ψ θ M ψ θ

(MPa) (MPa-n hr−1) (MPa−ψ hr−1) (MPa−ψ hr−1)

700 160 6.53E-31 12.78 1.01E-10 3 12 9.71E-11 3 12.5180 6.53E-31 12.78 1.00E-10 3 18 9.80E-11 3 18.5

600 300 1.56E-35 13.36 1.05E-11 3 27 7.56E-12 3 38.0320 1.56E-35 13.36 2.22E-11 3 24 1.97E-11 3 27.0

Fig. 2. Creep deformation simulations at 600 °C, (a) Kachanov-Rabotnov and (b) Sinh model.

Fig. 3. Creep deformation simulation at 700 °C, (a) Kachanov-Rabotnov and (b) Sinh model.

M.S. Haque, C.M. Stewart International Journal of Pressure Vessels and Piping 171 (2019) 1–9

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The creep deformation and damage evolution of the Sinh model tothe 304SS data is plotted in Figs. 2–5. Similar to the KR model, the Sinhmodel produces an accurate prediction of the creep deformation;however, notice that critical damage is to unity in every case.

4. Comparison

4.1. Minimum creep strain Rate

The KR model uses the classic Norton power law to model theminimum creep rate while the Sinh model uses the Mcvetty Sin-hy-perbolic law. The ability of the KR and Sinh models to predict theminimum creep strain is examined by calibrating the these laws to fiveisotherms of minimum creep strain rate data collected from literature[26]. The predictions at 566, 593, 649, 760, and 816 °C are shown inFig. 6. Examining the available experimental data, it is observed thatthe 593, 649, 760 °C isotherms exhibit a sigmoidal behavior while the566 and 816 °C are linear. The KR rupture predictions are linear on alog-log scale and thus are not able to accurately model the sigmoidalbehavior observed in the experimental data. The KR model can be re-calibrated to fit for either high stress or low stress but the model cannotmodel both regions simultaneously. The Sinh rupture predictions bendon a log-log scale and are able to accurately model the sigmoidal be-havior. In the Sinh model, the constant σs controls the bend. Overall, theSinh model better fits the experimental data when compared with theKR model over a wide stress range. The KR works best when the appliedstress range is limited.

4.2. Creep deformation and damage

The ability of the KR and Sinh models to model the full creep de-formation curve, including both the secondary and tertiary creep re-gimes, is plotted in Figs. 2 and 3. Both models are able to accuratelypredict the creep deformation of 304SS using the given experimentaldata. While the ductility and rupture time vary slightly from model tomodel, it is not enough to define one model as superior to the other.

The ability of the KR and Sinh models to model the analytical da-mage is plotted in Figs. 4 and 5. For KR, both the strain and damageapproach to material constant determination were applied. These ap-proaches produce virtually identical predictions of damage. When cal-culating the analytical damage using the KR or Sinh model, sometimesanomalous negative values of damage are calculated. For KR, the nu-merator of the analytical damage [Eq. (11)] is negative when the creepstrain rate is less than the minimum creep strain rate. This anomalousdamage appears during the primary and secondary creep regime whenextensometer fluctuation cause the creep strain rate to registered asslightly lower than the average minimum creep strain rate. This

Fig. 4. Damage evolution simulations at 600 °C, (a) Kachanov-Rabotnov and (b) Sinh model.

Fig. 5. Damage evolution simulation at 700 °C, (a) Kachanov-Rabotnov and (b) Sinh model.

Table 2Sinh material constant for 304SS

Test Temp, T Stress, σ B σs λ ϕ χ Q σt

(°C) (MPa) (hr−1) (MPa) (hr−1) (MPa)

1 700 160 1.48E-4 27.99 4.517 5.5 3 0.024 289.442 700 180 1.48E-4 27.99 4.24 3.8 3 0.028 289.443 600 300 1.45E-6 24.76 3.72 5.99 3 0.003 257.084 600 320 1.45E-6 24.76 1.93 2.62 3 0.004 257.08

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anomalous damage can be neglected.A similarly trend is observed for the Sinh analytical damage [Eq.

(20)]. If the creep strain rate is equal to or less than the minimum creeprate, the natural logarithm operator will generate zero or a negativevalue for the analytical damage. This anomalous damage is excluded.

The damage evolution prediction can be broken-down to threeparts: rupture time, critical damage, and damage trajectory (damagerate). Overall, the KR and Sinh models predict similar rupture times forthe given creep curves; however, the critical damage and damage tra-jectory are dissimilar. In the KR model, using the analytical damagecalculation [Eq. (11)], the critical damage varies between 0.2 and 0.3suggesting that rupture takes place when damage is well below unity.This contradicts the continuum damage mechanics (CDM) theory. As aresult, the damage trajectory is near-infinite near rupture. In the Sinhmodel, because of the material constants, λ, the critical damage cal-culated using the analytical damage [Eq. (20)] always resolves to unity.In Figs. 4b and 5b, Sinh damage consistently evolves from zero to unity.As a result, the damage trajectory is always finite near rupture.

4.3. Stress-rupture

The ability of the KR and Sinh models to predict stress-rupture isexamined by taking the rupture prediction [Eq. (6)] and [Eq. (24)]respectively and calibrating to three isotherms of stress-rupture datacollected from literature [28,29]. The stress-rupture predictions areshown in Fig. 7(a) at 593 °C, 843 °C and 954 °C respectively [29].Comparing the KR and Sinh model, it is observed that the Sinh betterfits the experimental data. The KR rupture prediction is linear on a log-log scale and thus is not able to accurately model the sigmoidal bend inthe stress-rupture data. Sinh model bends on a log-log scale and thus isable to accurately model the region. The physical realism of the KR andSinh rupture predictions can be further examined by extrapolatedrupture predictions as shown in Fig. 7b at 700 °C. The nominal yieldstrength and ultimate tensile strength of 304SS at 700 °C are plotted as atest for physical realism. In the Sinh rupture prediction [Eq. (24)], theconstant σt controls the location of the sigmoidal bend. The Sinh rup-ture prediction bends at the yield strength and approaches a value lessthan but near the nominal ultimate tensile strength of 304SS as rupturetime approaches zero (a conservative prediction in the high stress re-gion). The KR rupture prediction does not bend at the yield strengthand approaches a value 1.35x larger than the ultimate tensile strength

as rupture time approaches zero (a non-conservative prediction in thehigh stress region). Penny identified the ill-nature of the KR ruptureprediction as the “brittle curve” phenomena and modified the KR toaccommodate the high-stress to low-stress bend by introducing addi-tional factors and material constants while retaining the flaw that cri-tical damage is less than unity [16]. Gorash proposed different types ofcreep behavior for high-stress and low stress rupture. In the high-stressregion, power law creep is dominant. In the low-stress region, viscouscreep is dominant. As a result, two sets of constitutive equations (basedon KR) are applied to introduce the bend. Critical damage remains low(between 0.2 and 0.4) and the damage trajectory is near-infinite nearrupture [44]. The Sinh model, models the high-stress to low-stresstransition and regions without these additional complications andlimitations.

5. Conclusion

In this study, the KR and Sinh models are compared mathematicallyand with respective to the minimum creep strain rate, creep deforma-tion, damage, and rupture predictions. It is observed that.

• Mathematically, the damage variation, ∂ω t( ), for an infinitesimalvariation of stress, ∂σ t( ) in the KR model [Eq. (16)] will approachinfinity, while the same in the Sinh model [Eq. (26)] will remainfinite.

• The KR model does not model the bend in the minimum creep strainrate versus stress. The Sinh model is designed to accurately modelthe minimum creep strain rate bend.

• For the given creep deformation curves of 304SS, both the KR andSinh model produce accurate predictions of creep deformation.Creep deformation data collected over a wider range of stress andtemperature is needed to fully vet this observation.

• The damage evolution of the KR and Sinh are dissimilar. Damageevolution is broken-down to three parts: rupture time, critical da-mage, and damage trajectory (damage rate).o The rupture predictions of KR and Sinh using the limited creepdeformation data are identical; however, using several isothermsof stress-rupture data collected from literature, the KR model doesnot model the sigmoidal bend in stress-rupture accurately. TheSinh model is designed to accurately model the sigmoidal beha-vior.

Fig. 6. Minimum creep strain rate versus stress using the KR and Sinh models [26].

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o The critical damage of the KR model is always less than unity. TheSinh model is designed to always produce critical damage equal tounity.

o The damage trajectory of the KR model is near-infinite near rup-ture. The damage trajectory of the Sinh model remains finitethroughout life.

The Sinh model has several advantages that make it a compellingcreep deformation, damage, and rupture prediction tool. In the future,functional relationships between the KR and the Sinh will be exploitedto instantly convert established KR material constants to those for Sinh.

Acknowledgements

This material is based upon work supported by the Department ofEnergy, National Energy Technology Laboratory under Award Number(s) DE-FE0027581.

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