The aim of this paper is the analysis of inelastic behavior under periodic cyclic loading regimes for materials showing recovery effects. Starting with the Armstrong–Frederick type model and applying the two-time-scale asymptotic technique, the constitutive equation for the mean inelastic strain rate and the evolution equation for the mean backstress variable are derived. The advantage of the presented technique is the closed analytical form of the solutions such that the influence of the loading profiles on the resulting slow cycle-by-cycle strain accumulation can be analyzed explicitly. To validate the derived equations, the original constitutive model is integrated numerically by applying the time-step procedure for various cyclic loading profiles. Numerical examples are presented to illustrate cyclic creep behavior of X20CrMoV12-1 heat resistant steel.
NagodeMLänglerFHackM. Damage operator based lifetime calculation under thermo-mechanical fatigue for application on Ni-resist D-5S turbine housing of turbocharger. Eng Fail Anal2011; 18: 1565–1575.
2.
Kostenko Y, Almstedt H, Naumenko K, et al. Robust methods for creep fatigue analysis of power plant components under cyclic transient thermal loading. In: ASME turbo expo 2013: turbine technical conference and exposition. USA: American Society of Mechanical Engineers, 2013, pp.V05BT25A040–V05BT25A040.
3.
HoldsworthSMazzaEBindaLet al.Development of thermal fatigue damage in 1CrMoV rotor steel. Nucl Eng Des2007; 237: 2292–2301.
4.
WangWBuhlPKlenkAet al.The effect of in-service steam temperature transients on the damage behavior of a steam turbine rotor. Int J Fatigue2016; 87: 471–483.
5.
ZhuXChenHXuanFet al.Cyclic plasticity behaviors of steam turbine rotor subjected to cyclic thermal and mechanical loads. Eur J Mech A Solids2017; 66: 243–255.
6.
AltenbachHNaumenkoK. Creep bending of thin-walled shells and plates by consideration of finite deflections. Comp Mech1997; 19: 490–495.
7.
AltenbachHKushnevskyVNaumenkoK. On the use of solid- and shell-type finite elements in creep-damage predictions of thinwalled structures. Arch Appl Mech2001; 71: 164–181.
8.
ChabocheJL. A review of some plasticity and viscoplasticity constitutive equations. Int J Plast2008; 24: 1642–1693.
9.
KremplECreep-plasticity interaction. In: AltenbachHSkrzypekJ (eds). Creep and damage in materials and structures (cISM Lecture Notes No. 399), Wien, New York: Springer, 1999, pp. 285–348.
10.
Miller AK (ed.) Unified constitutive equations for creep and plasticity. London, New York, Elsevier, 1987.
11.
OhnoN. Recent topics in constitutive modelling of cyclic plasticity and viscoplasticity. Appl Mech Rev1990; 43: 283–295.
12.
FrederickCOArmstrongPJ. A mathematical representation of the multiaxial Bauschinger effect. Mater High Temp2007; 24: 1–26.
13.
NaumenkoKAltenbachH. Modeling high temperature materials behavior for structural analysis. Part I: continuum mechanics foundations and constitutive models. Advanced structured materials2016; vol 28, Cham: Springer.
14.
OhnoN. Recent progress in constitutive modeling for ratchetting. J Soc Mater Sci Jpn1997; 46: 1–9.
15.
Abdel-KarimM. An evaluation for several kinematic hardening rules on prediction of multiaxial stress-controlled ratchetting. Int J Plast2010; 26: 711–730.
16.
ZhangS-LXuanF-Z. Interaction of cyclic softening and stress relaxation of 9–12% Cr steel under strain-controlled fatigue-creep condition: experimental and modeling. Int J Plast2017; 98: 45–64.
17.
ZhaoPXuanF-ZWuD-L. Cyclic softening behaviors of modified 9–12% Cr steel under different loading modes: role of loading levels. Int J Mech Sci2017; 131: 278–285.
18.
BelytschkoTLiuWKMoranBet al.Nonlinear finite elements for continua and structures, New York: John Wiley & Sons, 2013.
19.
de Souza NetoEAPericDOwenDR. Computational methods for plasticity: theory and applications, New York: John Wiley & Sons, 2011.
20.
JahanshahiM. Numerical assessment of a double step integration algorithm for J2 plasticity with Armstrong–Frederick evolution of back stress. Sci Iran2012; 19: 1168–1179.
21.
ShutovA. Efficient implicit integration for finite-strain viscoplasticity with a nested multiplicative split. Comput Meth Appl Mech Eng2016; 306: 151–174.
22.
ShutovAVLarichkinAYShutovVA. Modelling of cyclic creep in the finite strain range using a nested split of the deformation gradient. ZAMM2017; 97: 1083–1099.
23.
LinJDunneFHayhurstD. Approximate method for the analysis of components undergoing ratchetting and failure. J Strain Anal Eng Des1998; 33: 55–65.
24.
LabergereCSaanouniKSunZDet al.Prediction of low cycle fatigue life using cycles jumping integration scheme. Appl Mech Mater2015; 784: 308–308.
25.
SandersJAVerhulstF. Averaging methods in nonlinear dynamical systems, New York: Springer, 1985.
26.
BensoussanALionsJLPapanicolaouG. Asymptotic analysis for periodic structures, North-Holland: Amsterdam, 1978.
27.
NayfehAH. Introduction to perturbation methods, New York: John Wiley and Sons, 1993.
28.
MorachkovskiiOK. Nonlinear creep problems of bodies under the action of fast field oscillations. Int Appl Mech1992; 28: 489–495.
29.
AltenbachHBreslavskyDMorachkovskyOet al.Cyclic creep damage in thin-walled structures. J Strain Anal Eng Des2000; 35: 1–11.
30.
OskayCFishJ. Fatigue life prediction using 2-scale temporal asymptotic homogenization. Int J Numer Methods Eng2004; 61: 329–359.
31.
YuQFishJ. Temporal homogenization of viscoelastic and viscoplastic solids subjected to locally periodic loading. Comput Mech2002; 29: 199–211.
HaoualaSDoghriI. Modeling and algorithms for two-scale time homogenization of viscoelastic-viscoplastic solids under large numbers of cycles. Int J Plast2015; 70: 98–125.
34.
JosephDSChakrabortyPGhoshS. Wavelet transformation based multi-time scaling method for crystal plasticity FE simulations under cyclic loading. Comput Methods Appl Mech Eng2010; 199: 2177–2194.
35.
BesselingJF. A theory of elastic, plastic and creep deformation of an initially isotropic material showing anisotropic strain hardening, creep recovery and secondary creep. Trans ASME J Appl Mech1958; 25: 529–536.
36.
BesselingJFvan der GiessenE. Mathematical modelling of inelastic deformation, London: Chapman & Hall, 1994.
37.
NaumenkoKAltenbachHKutschkeA. A combined model for hardening, softening, and damage processes in advanced heat resistant steels at elevated temperature. Int J Damage Mech2011; 20: 578–597.
38.
NaumenkoKKutschkeAKostenkoYet al.Multi-axial thermo-mechanical analysis of power plant components from 9–12% Cr steels at high temperature. Eng Fract Mech2011; 78: 1657–1668.
39.
LänglerFNaumenkoKAltenbachHet al.A constitutive model for inelastic behavior of casting materials under thermo-mechanical loading. J Strain Anal Eng Des2014; 49: 421–428.
40.
NaumenkoKGariboldiE. A phase mixture model for anisotropic creep of forged Al–Cu–Mg–Si alloy. Mater Sci Eng A2014; 618: 368–376.
41.
LemaitreJChabocheJ-L. Mechanics of solid materials, Cambridge: Cambridge University Press, 1990.
42.
EisenträgerJNaumenkoKAltenbachH. Calibration of a phase mixture model for hardening and softening regimes in tempered martensitic steel over wide stress and temperature ranges. J Strain Anal Eng Des2018; 53: 156–177.
43.
BoyleJTSpenceJ. Stress analysis for creep, London: Butterworth, 1983.
