Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

51, 1, pp. 63-73, Warsaw 2013

CONSTITUTIVE MODEL FOR TIME-DEPENDENT RATCHETTING

OF SS304 STAINLESS STEEL: SIMULATION AND ITS FINITE

ELEMENT ANALYSIS

Xiangjun Jiang

Xi’an Jiaotong University, Key Laboratory of Education Ministry for Modern Design and Rotor-Bearing System, Xi’an,

PR China; e-mail: jxj97011311@yahoo.com.cn

Yongsheng Zhu

Xi’an Jiaotong University, Theory of Lubrication and Bearing Institute, Xi’an, PR China; e-mail: yszhu@mail.xjtu.edu.cn

Jun Hong

Xi’an Jiaotong University, State Key Laboratory for Manufacturing System, Xi’an, PR China

e-mail: jhong@mail.xjtu.edu.cn

Youyun Zhang

Xi’an Jiaotong University, Theory of Lubrication and Bearing Institute, Xi’an, PR China

e-mail: yyzhang1@mail.xjtu.edu.cn

Qianhua Kan

Southwest Jiaotong University, School of Mechanics and Engineering, Chengdu, PR China

e-mail: qianhuakan@yahoo.com.cn

Time-dependent ratchettingbehaviourof SS304stainless steelwasexperimentally conducted
at room temperature and 973K. The material shows distinct time-dependent deformation.
However, under cyclic stressingwith a certain peak/valley stress hold and at 973K,more si-
gnificant time-dependent inelastic behaviourwas observed.Basedon theAbdel-Karim-Ohno
nonlinear kinematic hardening rule with the static recovery term, a time-dependent harde-
ning rule incorporating an internal variable in the dynamic recovery term of the back stress
is proposed to reasonably describe the evolution behaviour of time-dependent ratchetting
with a certain peak/valley stress hold and at high temperature. Simultaneously, the propo-
sed model is implemented into the ANSYS finite element package by User Programmable
Features (UPFs). It is shown that the customizedANSYSmodel exhibits better performan-
ce than the reference model, especially under cyclic stressing with the certain peak/valley
stress hold and at high temperature.

Key words: ratchetting, time-dependence, constitutive model, finite element method

1. Introduction

Fatigue failure is thefinal result of a complexmicroscopic phenomenonwhich occursunder cyclic
loading. In structure components of a nuclear power system, the viscosity of thematerial and the
time-dependence of ratchetting at high temperature, such as at 973K, become as very remarka-
ble, and should be addressed in detail. The time-independent and time-dependent deformation,
caused by the slight opening of the hysteresis loop, can be reasonably described by some revised
versions of the Armstrong-Fredrick non-linear kinematic hardeningmodel (Abdel-Karim, 2005,
2010;Chaboche, 1991;ChenandJia, 2004; JiangandSehitoglu, 1994;KangandGao, 2004;Kang
et al., 2002; McDowell, 1995; Ohno and Abdel-Karim, 2000; Yaguchi and Takahashi, 2005a,b).
However, regarding the stress relaxation behaviour with a certain peak/valley stress hold under
cyclic loading, few studies have been conducted so far. According to the creep-ratchetting expe-
riments of SS304 stainless steel at room temperature and 973K, the effects of stressing rate,



64 X. Jiang et al.

peak-stress hold and stress ratio on the ratchetting have been discussed by employing a unified
visco-plasticmodel and a plasticity-creep superpositionmodel (Mayama andSasaki, 2006; Taleb
and Cailletaud, 2011). A combined nonlinear kinematic hardeningmodel with a static recovery
termwas firstly proposedbyChaboche (1977)] and extended byKan et al. (2007) to describe the
time-dependent ratchetting behaviour. It was shown that the unified visco-plastic model even
employing the static recovery term cannot describe reasonably the creep-ratchetting behaviour
especially at low stress ratios and high temperature. In order to provide an accurate simulation
on the time-dependent ratchetting behaviour of the material more reasonably, it is necessary
to develop a constitutive model to describe such remarkable time-dependent behaviour. The
application of an advanced constitutive model into the finite elementmethod is also indispensa-
ble to achieve an accurate stress-strain analysis for engineering structures. It was demonstrated
that the ability to simulate structural ratchetting could be improved by implementing a more
advanced constitutive model into ANSYS material library (ANSYS, 1995; Hassan etal, 1998;
Kang, 2004, 2006; Kobayashi and Ohno, 2002; Rahman et al., 2008).
In this work, a visco-plastic constitutive model with the static recovery term and the dy-

namic recovery term is proposed to describe the time-dependent ratchetting behaviour and to
improve the capability of themodels to simulate the interaction between creep and ratchetting.
The capability of the proposedmodel is discussed by comparing with the corresponding experi-
ments of the SS304 stainless steel material at room temperature and 973K (Kang et al., 2006).
Simultaneously, the proposedmodel is successfully implemented into the finite element package
ANSYS (1995). The corresponding consistent tangent operator is derived afresh by considering
the influence of McCauley’s bracket term.

2. Description of the constitutive models

In this work, the unified visco-plastic nonlinear kinematic hardening rule developed by Abdel-
Karim and Ohno (simplified as UVPmodel hereafter) is employed as

αn+1 =
M
∑

k=1

r(k)b
(k)
n+1 (2.1)

where α is the total back stress which is divided into M components denoted as b(k) multi-
ply r(k) (k=1,2, . . . ,M). The critical state of dynamic recovery is reflected by a surface

f(k) =α(k)2−r(k)2 =0 (2.2)

where α(k) =
√

3
2
α(k) :α(k) is the equivalent back stress and r(k) is the radius of the critical

surface.
Evaluation of the UVPmodel byKan et al. (2007), Kang andKan (2007) has demonstrated

that thismodelwasnot robust enough to simulate the time-dependent ratchetting of thematerial
at room and high temperature. Same as the evolution rule of the back stress using in the
UVPmodel, a visco-plasticity-creep superposition constitutive model by employing a kinematic
hardening rule with a static recovery term (simplified as MUVP I model) was proposed for
simulation of creep or cyclic creep deformation. It could be found that the simulations were
in good agreement with experiments (Kang and Kan, 2007), however, it should be noted that
the superposition model is difficult to be directly implemented into finite element software. In
this work, a modified Abdel-Karim and Ohno model (simplified as MUVP II model) with the
additional dynamic recovery term of the back stress is introduced into the non-linear kinematic
hardening term. Then the back stress evolution equation is expressed as

ḃ
(k) =

2

3
ξ(k)ε̇in−Yṗ−ξ(k)

[

µ(k)ṗ+H(f(k))
〈

ε̇
in :
α
(k)

r(k)
−µ(k)ṗ

〉]

b
(k)−χ(α(k))(m−1)b(k) (2.3)



Constitutive model for time-dependent ratchetting... 65

where ξ(k) and r(k) are temperature-dependent material parameters; (:) indicates the inner
productbetween second-ranktensors; H(f(k)) is theHeaviside function: if f(k) ­ 0,H(f(k))= 1;

f(k) < 0, H(f(k)) = 0, p =
√

2
3
εin : εin stands for accumulated inelastic strain, µ(k) is a

parameter allowingEq. (2.3) to control thehysteresis loopwith slight opening, if 0<µ(k) ¬ 1. In
this analysis, the ratchetting parameter µ(k) is adopted identically for all the parts of back stress
as µ. The static recovery term of the back stress, −χ(α(k))(m−1), is used to represent the time-
dependent deformation with peak/valley stress hold, where χ and m are material constants. It
can be determined by the trials-and-errors method from the uniaxial experimental ratchetting
results. The second-rank tensor Y is the internal variable that was proposed and developed in
Yaguchi et al. (2002a,b), Zhan and Tong (2007) to describe the evolutionary behaviour of the
back stress with the dynamic recovery property. In the present study, the driving force of the
variable Y is assumed to be time-dependent, and the evolution equation can be expressed as
follows

Ẏ=−α′{Yst sgn(α
(k))+Y}(α(k))(m−1) (2.4)

Here α′ and Yst are material constants, α
′ describes the evolutionary rate of the variable Y,

and Yst denotes the saturated value of Y. The sgn function is defined as sgn(x) = 1 if x> 0
and sgn(x) = −1 if x < 0. The initial value of Y is taken to be zero, and the constants α′

and Yst are assumed to be positive and can be determined by the trails-and-errors method at
certain temperature.

3. Finite element model

Thefinite elementmodel should represent the central part of the specimenwhere stresses can be
assumed to be uniform.A 3D eight-noded solid185 brick element is employed for calculating the
uniaxial time-dependent ratchetting behaviour of thematerial. Additionally, an axi-symmetrical
notched-bar subjected to the uniaxial cyclic stressing with non-zero mean stress is discussed. A
2D axi-symmetrical four-noded plane182 element mesh is employed to set up the finite element
model for the notched-bar (Kan et al., 2007). Detailed formulation for the implementation of
thematerial model is presented in AppendixA. Thematerial parameters are the same as those
used in constitutive simulation and are listed in Tables 1 and 2.

Table 1.Material constants for all models at two temperatures

M =8, room temperature

ξ(1) =3341 ξ(2) =1833 ξ(3) =765.6 ξ(4) =210.4

ξ(5) =69.92 ξ(6) =35.91 ξ(7) =23.04 ξ(8) =13.0

r(1) =37.85 r(2) =34.56 r(3) =18.89 r(4) =11.92

r(5) =8.38 r(6) =6.74 r(7) =12.41 r(8) =70.33
E =192GPa ν =0.33 K =72 n=15 Q0 =90

M =8, at 973K

ξ(1) =3306 ξ(2) =1703 ξ(3) =726.7 ξ(4) =208.5

ξ(5) =69.35 ξ(6) =36.15 ξ(7) =22.94 ξ(8) =13

r(1) =12.16 r(2) =14.14 r(3) =13.19 r(4) =3.76

r(5) =7.86 r(6) =16.08 r(7) =7.91 r(8) =24.01
E =125GPa ν =0.33 K =35 n=9 Q0 =48



66 X. Jiang et al.

Table 2.Correlation coefficients for the parameters of the models

At room temperature At 973K
UVP MUVP I MUVP II UVP MUVP I MUVP II

µ 0.04 0.03 0.05 0.035 0.01 0.01

χ 0 5E-13 0 0 1.7E-9 2.43E-9

m 0 5 0 0 4.5 4.54

Yst 0 0 30 0 0 8

α′ 0 0 3E-16 0 0 6.55E-13

4. Simulations and results

Theprevious experimental results firstly conducted byKan et al. (2007) are used to evaluate the
performance of non-linear kinematic hardening ruleswith finite element application in thiswork.
In addition, temperature-dependent material parameters for the proposed model are obtained
from the uniaxial tensile response. The material constants E, ν, K, n, ξ(k), r(k), Q0 can be
determinedby theproceduredescribed in thepreviouswork (Kang et al., 2006) shown inTables 1
and 2. The cnstants χ, m and α′ can be determined from the cyclic stress-strain curve with
peak/valley stress hold by the trials-and-errors method at certain temperature. The uniaxial
ratchetting behaviour of SS304 stainless steel with constant or variable stress rates are simulated
numerically by the visco-plastic constitutive model mentioned in this paper. The simulation
results are shown in Figs. 1, 2, 3b,c, 4, 5b,c for different loading conditions, respectively. In
these simulations, a constant m=8 is applied for all components of the back stress.

Fig. 1. (a) Simulation by the UVPmodel for different rates and at room temperature; (b) simulation by
MUVP II (χ=0)model for different rates and at room temperature; (c) simulation by MUVP II
(χ=0)model with or without peak/valley stress hold and at room temperature and stressing rate

of 2.6MPa/s



Constitutive model for time-dependent ratchetting... 67

Fig. 2. Results of uniaxial ratchetting strain vs. cyclic number for different rates and at 973K:
(a) simulations by UVPmodel; (b) simulations by MUVP Imodel; (c) simulations byMUVP II model

Fig. 3. Results of uniaxial ratchetting (40±100MPa) at 973K and stressing rate of 10MPa/s:
(a) experiment; (b) simulation by ANSYS with theMUVP II model; (c) simulation of ratchetting for

the notched-bar by theMUVP II model



68 X. Jiang et al.

Fig. 4. Results of uniaxial ratchetting strain vs. cyclic number with or without peak/valley stress hold
at 973K and stressing rate of 10MPa/s: (a) simulations by the UVPmodel; (b) simulations by the

MUVP Imodel; (c) simulations by theMUVP II model

Fig. 5. Results of ratchetting (40±100MPa) with peak/valley stress hold at 973K and stressing rate
of 10MPa/s: (a) experiment; (b) simulation by ANSYS with theMUVP II model; (c) simulation of

ratchetting for the notched-bar by theMUVP II model



Constitutive model for time-dependent ratchetting... 69

From the experimental and simulated ratchetting results at room temperature shown in
Figs. 1, it can be concluded that: (1) theUVPmodel can provide good simulations for the unia-
xial ratchetting of thematerial at moderate stressing rates, such as at 65MPa/s and 13MPa/s,
however, at a lower stressing rate, i.e., 2.6MPa/s, the simulation by this model shows a larger
divergence from the corresponding experiments; (2) the UVPmodel has the capability of simu-
lating the variation of ratchetting strain caused by the varied hold-times at peak/valley stress
points reasonably, though it cannot well simulate the time-dependent ratchetting behaviour of
the material at lower stressing rate well; (3) though the improvement at varied stressing rates
andvariedhold-times at peak/valley stress points could beobtainedbyusing theMUVPImodel
proposed in Kan et al. (2007), the MUVP II model with the dynamic recovery term only pro-
videsmore reasonable simulations for the time-dependent ratchetting behaviour of thematerial
compared with the UVP model for varied hold-times at peak/valley stress points.
Theexperiments andsimulations at 973Kare shown inFigs. 2, 3, 4 and5. It canbeconcluded

that: (1) since the viscosity of thematerial is very remarkable at 973K, the UVPmodel cannot
provide good simulations for the time-dependent ratchetting of the material, even at moderate
stressing rates, such as at 40MPa/s and 10MPa/s, and for those with various hold-times at
peak/valley stress points, as shown in Figs. 2a and 4a; (2) although the MUVP I model can
provide good simulations for the time-dependent ratchetting behaviour of thematerial presented
at varied stressing rates and with different hold-times at room temperature, it cannot simulate
the time-dependent ratchetting at lower stressing rates as shown in Fig. 2b. Moreover, the
MUVP Imodel cannot describe the ratchetting strain by various hold-times precisely as can be
seen in Fig. 3b, since the static recovery term only adopted in this model cannot reasonably
take this effect into account at lower stressing rates; (3) comparedwith the simulation results by
theMUVP I model, theMUVP II model can provide good simulations for the time-dependent
ratchetting behaviour of thematerial presented at varied stressing rates andhold-times at 973K,
as shown in Figs. 2c and 4c; (4) similar to that at room temperature, the customized ANSYS
model based on theMUVP IImodel can provide reasonable simulations for the time-dependent
ratchetting behaviour of the material presented at 973K, as shown in Figs. 3b,c and 5b,c.
Nevertheless, it should be noted that at 973 K, owning to the simulations by this modelmainly
focused on the value of time-dependent ratchetting strain, the predicted shape of stress-strain
hysteresis loop apparently deviates from the experimental one even with the MUVP II model,
which should be improved in the further work.

5. Conclusions

Based on the time-dependent deformation characteristics observed from experimental results
obtained byKang et al. (2006), three types of constitutive models are employed to simulate the
uniaxial time-dependent ratchetting of SS304 stainless steel at roomtemperature and973K. It is
shown fromtheabovediscussions that theproposedvisco-plastic constitutivemodel is reasonably
formulated by the unified visco-plastic constitutive model rule with the static and dynamic
recovery term.The simulation results found through the proposedmodel are in good consistence
with the experimental results of the uniaxial time-dependent ratchetting of SS304 stainless
steel at room temperature and 973K. Finally, the customized ANSYSmodel has demonstrated
that it is capable of simulating the time-dependent ratchetting deformation accurately for some
structures at room and even high temperatures with a certain peak stress hold.

A. Appendix

The plasticitymodel discussed in Section 3 is implemented into the general purposeFEpackage
ANSYS through the user-defined subroutine USERMAT. A new explicit stress update algori-



70 X. Jiang et al.

thm and the corresponding consistent tangent operator are derived based on backward Euler
algorithmand radial returnmethod.The algorithm reduces the plasticitymodel into a nonlinear
equation that can be solved by Newton’s method.
The discretized evolution rules of kinematic hardening are given by

αn+1 =
M
∑

k=1

r(k)b
(k)
n+1 (A.1)

— forMUVP I

ḃ(k) =
2

3
ξ(k)ε̇in− ξ(k)

[

µ(k)ṗ+H(f(k))
〈

ε̇
in :
α
(k)

r(k)
−µ(k)ṗ

〉]

b(k)−χ(α
(k)
n+1)

(m−1)b
(k)
n+1 (A.2)

— forMUVP II

ḃ
(k) =

2

3
ξ(k)ε̇in−Yṗ− ξ(k)

[

µ(k)ṗ+H(f(k))
〈

ε̇
in :
α
(k)

r(k)
−µ(k)ṗ

〉]

b
(k)−χ(α

(k)
n+1)

(m−1)
b
(k)
n+1

(A.3)

and

b
(k)
n+1 = θ

(k)
n+1b

∗(k)
n+1 (A.4)

where, if ∆εinn+1 : b
(k)
n+1−µ

(k)∆pn+1 > 0 and H(f
(k))= 1 then

θ
(k)
n+1 =

1

1+µ(k)ξ(k)∆pn+1+ξ
(k)(∆εinn+1 :b

(k)
n+1−µ

(k)∆pn+1)+χ(r
(k)b
(k)
n+1)

(m−1)b
(k)
n+1

(A.5)

otherwise

θ
(k)
n+1 =

1

1+µ(k)ξ(k)∆pn+1+χ(r(k)b
(k)
n+1)

(m−1)b
(k)
n+1

(A.6)

and

b
∗(k)
n+1 =











b
(k)
n +

2

3
ξ(k)∆εinn+1 for MUVP I

(A.7)b
(k)
n +

2

3
ξ(k)∆εinn+1−Yn+1∆pn+1 for MUVP II

(A.7)

as donebyANSYS (1995), Yaguchi et al. (2002b), Zhan andTong (2007), a newnonlinear scalar
equation of stress integration for the proposedmodel is obtained as

Yn+1−Y
∗

n+1+

(

3G+
m
∑

i=1

θ
(k)
n+1ξ

(k)r(k)
)

〈Yn+1−Q0
K

〉n

∆tn+1 =0 (A.8)

where

Yn+1 =

√

3

2
‖sn+1−αn+1‖ Y

∗

n+1 =

√

3

2

∥

∥

∥

∥

s
∗

n+1−
m
∑

i=1

r(k)θ
(k)
n+1b

(k)
n

∥

∥

∥

∥

(A.9)

where s∗n+1 is the elastic predictor of deviatoric stress sn+1, and b
∗(k)
n+1 is the predictor of ḃ

(k)
n+1.

The stress and internal variables are determined in the iteration process, while the last sub-
section is convergent. If the equilibrium for the global FE equations is satisfied, the solution



Constitutive model for time-dependent ratchetting... 71

is validated. Then, a new global equilibrium iteration is required. The consistent tangential
modulus d∆σn+1/d∆εn+1 should be determined to ensure quadric convergence. This consistent
tangential matrix can be derived for the proposed constitutive model as

d∆σn+1
d∆εn+1

=D−4G2(L−1n+1 :J
0
n+1) : Id (A.10)

where D is the fourth-order elasticity tensor.
This is similar to that obtained by ANSYS (1995), Yaguchi et al. (2002b), Zhan and Tong

(2007) in form, but

L
(k)
n+1 = I+J

0
n+1 : (2GI+b

(k)
n+1⊗b

(k)
n+1) (A.11)

where

J0n+1 =
3

2
A(n0n+1⊗nn+1)+

√

3

2
∆pn+1Jn+1

n
0
n+1 =nn+1+∆pn+1Jn+1 :b

′

n+1

Jn+1 =
1

‖sn+1−αn+1‖
(I−nn+1⊗nn+1)

d∆pn+1 =A

√

3

2
nn+1 :

[

2GId : d∆εn+1−

(

2GI+
m
∑

i=1

H
(k)
n+1

)

: d∆εinn+1

]

A=
n(∆tn+1/K)〈Fy(n+1)/K〉

n−1

1+n(∆tn+1/K)〈Fy(n+1)/K〉
n−1(nn+1 :b

′

n+1)

(A.12)

where, if ∆εinn+1 :b
(k)
n+1−µ

(k)∆pn+1 > 0 and H(f
(k))= 1 then

b′n+1 =0 H
(k)
n+1 = θ

(k)
n+1ξ

(k)r(k)M
(k)−1

n+1 :
(2

3
I−b

(k)
n+1⊗b

(k)
n+1

)

M
(k)
n+1 = I+θ

(k)
n+1ξ

(k)b
(k)
n+1⊗∆ε

in
n+1+(m−1)θ

(k)
n+1χr

(k)(m−1)b
(k)(m−3)
n+1 b

(k)
n+1⊗b

(k)
n+1

(A.13)

otherwise

b
′

n+1 =
M
∑

i=1

(−1)θ
(k)
n+1ξ

(k)r(k)µ(k)M
(k)−1

n+1 :b
(k)
n+1

H
(k)
n+1 = θ

(k)
n+1ξ

(k)r(k)M
(k)−1

n+1 :
(2

3
I−b

(k)
n+1⊗b

(k)
n+1

)

M
(k)
n+1 = I+(m−1)θ

(k)
n+1χr

(k)(m−1)b
(k)(m−3)
n+1 b

(k)
n+1⊗b

(k)
n+1

(A.14)

the symbol ⊗ represents the dyadic tensor product, (:) stands for tensor contraction, and I is

the fourth order unit tensor, Id =
(

I− 1
3
1⊗1

)

represents the deviatoric operation of a tensor,

1 is a second-order unit tensor.

Acknowledgments

Wewould like to thank for the financial support of this workby theNationalBasicResearchProgram

ofChina (GrantNo. 2011CB706601)and the keyproject ofNationalNatural ScienceFoundationofChina

(No. 50935006).

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Konstytutywny model zależnego od czasu zjawiska ratchetingu dla stali SS304 – symulacja

i analiza metodą elementów skończonych

Streszczenie

W pracy przedstawiono wyniki badań eksperymentalnych nad zależnym od czasu procesem zmę-
czeniowym typu ratcheting przeprowadzonych w temperaturze pokojowej oraz podwyższonej do 973K.
Materiał wykazał wyraźnie zależną od czasu funkcję deformacji. Podczas cyklicznego obciążania przy za-
danych wartościachmin/max naprężeń w temperaturze 973K zaobserwowano silnie nieliniowe i zależne
od czasu zachowanie się badanej stali. Dowyjaśnienia tego zjawiska, zwanego ratchetingiem zależnymod
czasu, wykorzystano model umocnienia materiału oparty na nieliniowej formule kinematycznego umoc-
nienia Abdela-Karima-Ohno ze statycznym członem odprężania.Model ten zmodyfikowano,wprowadza-
jąc wewnętrzną zmienną w dynamicznym członie odprężania przy obciążeniu powrotnym. Jednocześnie
zaproponowanymodel wdrożono do systemu ANSYS poprzez zastosowanie pakietu User Programmable
Features (UPFs).Wykazano, że takamodyfikacja systemuANSYS charakteryzuje się lepszymdziałaniem
w stosunku do standardowego oprogramowania. Jest to szczególnie zauważalne dla symulacji cyklicznego
obciążenia stali w podwyższonej temperaturze.

Manuscript received December 15, 2011; accepted for print March 15, 2012