Modeling and Finite Element Simulation of Low Cycle Fatigue ... - Core

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Procedia Engineering 55 (2013) 774 – 779

6th International Conference on Cree ep, Fatigue and Creep-Fatigue Interaction [CF-6 6]

Modeling and Finite Elemeent Simulation of Low Cycle Fatig gue Behavviour of 316 SS J.Shit, S S.Dhar∗, S.Acharyya Department of Mechanical E Engineering, Jadavpur University, Kolkata, India

Abstract In this model the cyclic plastic behaviour in the trannsition cycle (from virgin state to saturated state ) has been stud died for the material SS316. The dynamic recovery terms iin Chaboche’s kinematic hardening law have been modified with a memory function to include cyclic hardening/softening behaviour during transition cycles. The memory function has been calibrated for this material to obtain a saturated vvalue at the stable hysteresis loop. The hardening rate ( incrrease in hardening stress per cycle) is also matched by propper calibration constant. All the material constants have been derived from uniaxial strain controlled low cycle fatigue (LC CF) test results from virgin state to saturated state at stable hy ysteresis loop. The material model developed with the kinem matic hardening rules has been plugged into a Finite Elemen nt (FE) program. The results obtained from FE simulations have been compared with the experimental results at differen nt strain amplitudes. Authors. by Published by Elsevier Ltd. Openand/o access CC BY-NC-ND license. ©© 2013 2013The Published Elsevier Ltd. Selection orunder peer-review under responsibility of the Indira Gandhi Cen ntre for Selection and peer-review under responsibility of the Indira Gandhi Centre for Atomic Research. Atomic Research

Keywords: cyclic plasticity; kinematic hardening;low cyclic fatigue; dilatation

Nomenclature a Cyclic hardening rate parameter bk Coefficients of dynamic recovery term in kinematic hardening rule ck Strain hardening coefficients in kinematic hardening rule s accumulated plastic strain S0 Memory range p(s) Memory function ps Saturated value of memory function Sij Deviatoric stress tensor Xij Back stress tensor

σij σf σ0 εij εpij ξ β K n

Cauchy’s stress tensor Flow stress Initial yield stress Strain tensor Plastic strain tensor Back stress in uniaxial tension compression loading Dilatation coefficient Isotropic hardening strength coefficient Isotropic hardening exponent

∗

Corresponding author: E-mail address: [email protected]

1877-7058 © 2013 The Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of the Indira Gandhi Centre for Atomic Research. doi:10.1016/j.proeng.2013.03.330

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1. Introduction Inelastic material behavior in closed loading path, repeated for a number of times is known as cyclic plasticity. The experimental observations show a number of related phenomena. Those are i.) Bauschinger effect, ii.) cyclic hardening/softening, iii.) mean stress relaxation and ratcheting . Some materials also exhibit SD ( strength differential) effect [1]. Above all, there is additional hardening due to non proportional loading path. Addressing all these phenomena in constitutive equations becomes complicated and difficult to implement numerically. In general, Armstrong – Frederick law and also its multi-segmented version due to Chaboche and Rousselier can not address cyclic hardening / softening phenomenon. It can model only saturated state but not the transition process from virgin state to saturated state. Chaboche et al [2] introduced a strain memory surface, which translate and expand also. Thus, controlling isotropic hardening rate cyclic hardning is explained. Ohno [3] modified the concept to achieve a realistic model. But this model can not address fading memory as the non hardening surface can expand only. In general, Von Mises model considers equal flow stress in tension and compression and the effect of hydrostatic stress on flow stress is neglected. But the effect of hydrostatic stress on flow stress is observed in some materials after a few cycles [4]. This produces higher flow stress in compression than that in tension ( SD effect ). 2. Mathematical formulation 2.1. Yield function The yield function used in the present model is a Von mises yield function modified with a dilatation term. This is as follows

Φ=

3 ( S ij − X ij )( S ij − X ij ) − σ 2f = 0 2

Here, flow stress,

σ f = σ 0 {1 − β ( β=

Dilatation coefficient,

σm )} σ eq

(1) (2)

ε vp ε eqp

(3)

Introduction of ȕ in yield function produces higher flow stress for negative σ m and reduces its value for positive σ m and hence strength differential effect introduced. The value of ȕ is calibrated to match the experimental LCF data. σ 0 = Yield strength of the material, S ij = deviatoric part of stress tensor, X ij = back stress tensor, also deviatoric in nature, σm = mean stress 2.2. The flow rule The Plastic strain rate,

εijp = λ

εijp , follows from the flow rules as

∂Φ ∂σ ij

Here, λ is a scalar multiplier.

(4)

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J. Shit et al. / Procedia Engineering 55 (2013) 774 – 779

2.3. Kinematic hardening rule The kinematic hardening rules used here are modified Chaboche’s kinematic hardening laws. The modification has been done in dynamic recovery terms with a memory functional. Thus the kinematic hardening rules are written as

X ijk = ck εijp − bk bˆ( p( s) )sX ijk

(5)

1

2 p p 2 where s = ( εij εij ) 3 The functional, bˆ( p ( s ) ) , has the following properties. i) bˆ( p( s)) ≥ 0 , for all values of p(s) ii) bˆ(0) = 0 for p(s)=0 and iii) bˆ ′( p(s) ) ≤ 0 for all possible values of

p(s). One of the simplest choices of bˆ( p ( s) ) is given in ref [5]. That is,

bˆ( p (s )) =

1 1 + ap (s )

, Here, ‘a’ is a material constant. This choice of

bˆ( p ) gives a kinematic hardening

law as

X ijk = c k εijp −

bk sX ijk 1 + ap (s )

(6)

The evolution of back stress will be complete with the evolution equation of memory function p ( s ) .For that the back stress from uniaxial tension compression test has been calculated. The evolution law of memory function p ( s ) is given as

p =

s S0

§ 2 · ¨ ξ − p¸ ¨ 3 ¸ © ¹ with p ( 0 ) = 0

(7)

Here ξ is the uniaxial back stress as calculated from uniaxial tension compression test data. S0 is the memory range. That is to say that beyond S0 the plastic strain memory is erased. From the above differential equation it is seen that

dp 2 ξ = p (s ) + S 0 ds 3

(8)

This shows clearly that the back stress depends on current memory p(s) and its history over a plastic arc of length S0. 3. The material constants Elastic Modulus, E and Poisson’s ratio Ȟ are obtained from tensile test. 3.1. Kinematic hardening variables Ck and bks are obtained from saturated cyclic stress- strain hysteresis curve [6]

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J. Shit et al. / Procedia Engineering 55 (2013) 774 – 779 s

Here, bk is the value of the coefficient of recall term as obtained from the saturation cycle of the hysteresis loop. In this model Ck, bks are three sets of coefficients which are derived from three different segments of the saturated LCF loop.

bks = bk bˆ( p s ) = Next,

bk 1 + ap s

(9)

Here, ps = saturated value of memory function a= cyclic hardening rate. If a=0, the effect of memory function on recall terms vanishes. bk= The coefficient of recall term as obtained from the 1st cycle of the hystersis loop. The saturated value of memory function, ps, is obtained from back stress Vs plastic strain curve of a saturated loop. This is as follows

ps =

1 Δε p

Δε p

³ ξ ( p s )ds

0

(10)

Here, ξ is equal to the back stress of a saturated loop in uniaxial tension compression loading and Δεp is the plastic strain range (i.e twice the plastic strain amplitude). The first trial value of ‘a’ is obtained from equation (9). Knowing the values of bks and bk and using the value of ps as given in equation (10) the value of ‘a’ can be determined from equation (9). Further adjustment in the value of ‘a’ may be required to obtain a better cyclic hardening rate. The memory range, S0, is obtained as the accumulated plastic strain value of a loading branch in a saturated

³

loop. Thus, S 0 = s.dt , over a loading branch in a saturated loop. 4. Mechanical properties of SS316 4.1. Table-1 tensile properties

Young’s Modulus (GPa)

Poisson’s Ratio

σYP (MPa)

σUTS (MPa)

% Elongation

210

0.3

267

637

78

4.2. Kinematic hardening coefficients (saturated values) C1=70000 MPa, C2=30000 MPa, C3=4000 MPa, b1s=1700, b2s =557 , b3s =0 4.3. Kinematic hardening coefficients ( as obtained from 1st cycle) C1=70000 MPa, C2=30000 MPa, C3=4000 MPa, b1=2200, b2=800.0 , b3=0 4.4. Cyclic hardening coefficients S0 = 0.04480

, a= 0.003308 MPa-1

σ=K(εp)n K

n

1150

0.286

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4.5. Table-2 calibration of yield stress with strain amplitude εP(%)

(σYP)cyc (MPa)

0.81

215

0.43

199

0.33

196

5. Finite element implementation of the model The FE simulation of the model is implemented on a round bar specimen under strain controlled tension compression loading. Low cycle fatigue loops are simulated with finite element simulation. The first attempt is to simulate the saturated hyteresis loop. The saturated values of Chaboche’s kinematic hardening coefficients are used. The material constant ‘a’ and ‘S0’ are set to zero value. Thus, the memory function calculation is bypassed and its effect on kinematic hardening is deactivated . The next attempt is to simulate cycle hardening in the transition cycle .This is to simulate the LCF loop from virgin state to saturation state. The values of material constants ‘a’ and ‘S0’ are assigned. Thus, the memory function calculations are activated. The recall terms of Chaboche’s kinematic hardening rules are calibrated with the memory function. The cyclic hardening is achieved. 6. Results and discussions Fig 1 shows the comparison of simulated saturated hysteresis loop for a strain amplitude ± 1.0% (plastic strain amplitude ± 0.85%) with the experimental one. It is found that ȕ=0.0 gives higher peak stress in tension and ȕ=0.01 gives higher peak stress in compression. For, ȕ=0.005 the matching is satisfactory with respect to peak stresses in tension and compression. Figs 2 &3 shows the peak stress Vs cycle curves for the material SS316. The results of first 20 cycles are compared with the experimental values. It is observed in Fig-2 that for small strain amplitudes (±0.6% ), the effect of ȕ is negligible. For higher strain amplitude (Fig3) of ±1.0% the effect of ȕ is noticeable. β=0.01 gives better matching with the experimental results for higher strain amplitudes.

(β=0.0) (β=0.01)

Fig. 1. Saturated loop for strain amplitude ± 1.0% ( Material : SS 316).

J. Shit et al. / Procedia Engineering 55 (2013) 774 – 779

Fig. 2. Peak stress Vs cycles for a strain amplitude ± 0.6%. Material SS 316.

Fig. 3. Peak stress Vs cycles for a strain amplitude ± 1.0%. Material SS 316.

7. Conclusions From the above results the following conclusions can be drawn. • Cyclic hardening rate is simulated for the material SS316 stainless steel. • The dilation effect in the material SS316 is negligible for low strain amplitudes ( below 1.0%).The effect is noticeable at higher strain amplitudes. • The saturated LCF loops are simulated satisfactorily for the materials SS316. References [1] [2] [3] [4] [5] [6]

W.A.Spitzig, R.J.Sober, O.Richmond, Pressure Dependence of Yielding and Associated Volume Expansion in Tempered Martensite, Acta Metall,23(1975), 885. J.L.Chaboche, Dang Van, G. Cordier, Modelization of the Strain Memory Effect on Cyclic Hardening of 316 Stainless Steel, Paper L 11/3 1979; 5th SMIRT, Berlin. N.Ohno, Y.Kachi, A Constitutive Model of Cyclic Plasticity for Nonlinear Hardening Material, J. Appl. Mech 53(1985) 395. Mahnken Rolf, Strength Difference in Tension and Compression and Pressure Dependence of Yielding, Comp. Methods and Appl. Mech. Engg. 190(2001) 5057. P. Haupt, M. Kamlah, Representation of Cyclic Hardening and Softening Properties using Continuous Variables, Int. J. Plasticity 11(1995) 267-291. S.Bari, T.Hassan, Anatomy to coupled constitutive models for ratcheting simulation. Int J. Plasticity 16(2000) 381-409.

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