Mechanics Research Communications 35 (2008) 589–594

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Mechanics Research Communications

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

Formulation of a new single crystal law for modeling the cyclic softening

P. Evrard *, V. Aubin, S. Degallaix, D. KondoEcole Centrale de Lille, Laboratoire de Mécanique de Lille (UMR CNRS 8107), 59651 Villeneuve d’Ascq Cedex, France

a r t i c l e i n f o

Article history:Received 5 February 2008Received in revised form 2 June 2008Available online 11 June 2008

Keywords:Low-cycle fatigueCyclic softeningSingle crystal lawComposite modelingSelf-consistent scheme

0093-6413/$ - see front matter � 2008 Elsevier Ltddoi:10.1016/j.mechrescom.2008.06.001

* Corresponding author. Tel.: +33 3 20 33 60 23;E-mail address: pierre.evrard@ec-lille.fr (P. Evrar

a b s t r a c t

The present paper proposes to modify the single crystal law [Hoc, T., Forest, S., 2001. Poly-crystal modelling of IF-TI steel under complex loading path. Int. J. Plast. 27, 65–85] used inpolycrystalline model developed by [Pilvin, P., 1990. Approches multiéchelles pour la pré-vision du comportement anélastique des métaux. Ph.D. Thesis, Université Pierre et MarieCurie; Cailletaud, G., 1992. A micromechanical approach to inelastic behavior of metals.Int. J. Plast. 8, 55–73], in order to account for the cyclic softening observed in some metallicmaterials during fully reversed low-cycle fatigue tests under total strain control. Experi-mental observations [Mayama, T., Sasaki, K., 2006. Investigation of subsequent viscoplasticdeformation of austenitic stainless steel subjected to cyclic preloading. Int. J. Plast. 22, 374–390] show that the cyclic softening is partially due to a rearrangement of dislocations inhard and soft zones within the grains during cyclic straining. We propose thus to modifythe initial single crystal law by including a composite model of hard and soft zones on eachslip system within a grain. The softening is modeled through the evolution of soft zones.The model is applied to a FCC polycrystal. The self-consistent scheme is used in order toobtain the polycrystalline macroscopic behavior. The results show that the new model cor-rectly describes an initial cyclic hardening, a subsequent cyclic softening and finally a sta-bilized state.

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1. Introduction

In low-cycle fatigue, most metallic materials show three stages during the life time: an accommodation phase usuallycovering a short number of cycles, followed by a stabilization phase covering the major part of the life time, and finally acrack propagation phase leading to fracture. In low-cycle fatigue under fully reversed total strain control, the accommoda-tion phase corresponds to the first cycles during which the material adapts to the applied strain amplitude. It consists ofeither a hardening, or a softening, or a hardening followed by a softening, depending on the material, heat treatment andapplied strain amplitude.

The polycrystalline model proposed by Hoc and Forest (2001) uses a single crystal constitutive law with a unique dislo-cation density on each slip system as internal variable. This model correctly accounts for the cyclic hardening, but not for asubsequent cyclic softening. Consequently, the stabilized stress level is often overestimated. In fact, the hardening/softeningbehavior is the result of simultaneous hardening and softening mechanisms, which compete together and finally lead to astabilized state. TEM observations (Mayama and Sasaki, 2006), performed on an austenitic stainless steel, show that thehardening is due to an increase of the number of dislocations, distributed homogeneously, while the softening is due to arearrangement of dislocations in hard and soft zones with high and low dislocation densities, respectively. Thus, the

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fax: +33 3 20 33 71 53.d).

Fig. 1. Schematic representation of physical mechanisms for hardening/softening accommodation behavior.

590 P. Evrard et al. / Mechanics Research Communications 35 (2008) 589–594

dislocation distribution becomes more and more heterogeneous and dislocation structures appear in the grains. Fig. 1 rep-resents schematically these microstructural evolutions.

In this paper, according to TEM observations, the single crystal law used in Hoc and Forest (2001) is modified in order totake into account the progressively heterogeneous dislocation distribution. This new single crystal law consists in introduc-ing hard and soft zones (named respectively walls and channels in this paper) on each slip system in a grain. For each slipsystem, two hardening laws are considered: one for the walls and the other for the channels. Both are based on the evolutionof dislocation densities and the forest hardening. The rearrangement of dislocations is taken into account through the vol-ume fraction of channels, which increases during the cyclic straining. The macroscopic behavior is obtained using the clas-sical self-consistent scheme and the Cailletaud-Pilvin interaction law (Pilvin, 1990; Cailletaud, 1992).

In the following, X;X;X and X define respectively scalar, vector, second-rank and four-rank tensors. Moreover, indexes ‘g’,‘s’, ‘w’ and ‘c’ refer to grain, slip system (s), wall and channel, respectively. Because we consider fatigue behavior of metallicmaterials, a small strain hypothesis is chosen. Furthermore, we consider the plastic behavior as a limit of the strain ratedependent behavior.

2. Initial single crystal law

The single crystal law used in Hoc and Forest (2001) supposes a homogeneous dislocation distribution and a unique dis-location density on each slip system. The viscoplastic flow law is expressed by:

_cðsÞ ¼ jsðsÞj � sðsÞl

K

* +n

signðsðsÞÞ if jsðsÞjP sðsÞl ð1Þ

and _cðsÞ ¼ 0 otherwise.The critical shear stress is obtained by:

sðsÞl ¼ lbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX

t

htðsÞqtr

þ s0 ð2Þ

where b;l; s0 and htðsÞ are respectively the Burgers vector, the shear modulus, the initial shear stress and the interaction ma-trix, that characterizes the interactions between slip systems. In this paper, a simple form of this matrix is chosen. Diagonalterms, which correspond to the self-hardening, are equal to 1, and non diagonal terms, which correspond to the latent hard-ening, are equal to a constant parameter h. Finally, the evolution of the dislocation density is computed by:

_qðsÞ ¼ 1b

1NðsÞ� 2ycqðsÞ

� �_cðsÞ ð3Þ

with qðsÞðcðsÞ ¼ 0Þ ¼ q0 where yc is a parameter proportional to the length characteristic for the annihilation process of dis-location dipoles and NðsÞ is the mean free path on the slip system (s). NðsÞ is expressed as

NðsÞ ¼ LffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPt 6¼ðsÞqt

q ð4Þ

where L is a material parameter controlling the dislocation production.

P. Evrard et al. / Mechanics Research Communications 35 (2008) 589–594 591

3. Constitutive equations of the proposed model

3.1. Viscoplastic flow law

According to heterogeneous dislocation structures observed in low-cycle fatigue, Mughrabi (1983) proposed a compositemodel where the grain contains simultaneously walls and channels, which are zones with high and low dislocation densities,respectively. In this paper, we propose to use this composite model at the slip system scale. The resolved shear stress, sðsÞ1, isgiven by:

1 ThemðsÞg ¼ 1

2

sðsÞ ¼ f ðsÞw sðsÞw þ f ðsÞc sðsÞc ð5Þ

where sðsÞw ; sðsÞc are the resolved shear stresses in walls and channels and f ðsÞw ; f ðsÞc are the volume fractions of walls and chan-nels, respectively.

As other authors (Estrin et al., 1998; Tóth, 2005), we suppose that the shear strain rate is the same in walls and channelsð _cðsÞw ¼ _cðsÞc ¼ _cðsÞÞ. Furthermore, although several authors like (Feaugas, 1999) consider that dislocation structures are alsoresponsible for intragranular back stress, the same formalism as the initial crystal law is used, where only an isotropic hard-ening is taken into account. Considering that signðsðsÞw Þ ¼ signðsðsÞc Þ ¼ signðsðsÞÞ, the viscoplastic flow laws for the walls andchannels are thus written as

_cðsÞ ¼jsðsÞw j � sðsÞlw

Kw

!nw

signðsðsÞÞ ¼jsðsÞc j � sðsÞlc

Kc

!nc

signðsðsÞÞ ð6Þ

where Kw;nw and Kc;nc are the viscosity parameters in walls and channels, respectively. sðsÞlw; sðsÞlc are the critical shear stres-ses in walls and channels, respectively. Including Eq. (6) in Eq. (5), the resolved shear stress is:

jsðsÞj ¼ f ðsÞw Kw _c1

nw þ f ðsÞw sðsÞlw þ f ðsÞc Kc _c1

nc þ f ðsÞc sðsÞlc ð7Þ

Considering the same viscosity parameters in walls and channels (Kw ¼ Kc and nw ¼ nc), the activation of a slip system fol-lows the viscoplastic flow law:

_cðsÞ ¼ jsðsÞj � ðf ðsÞw sðsÞlw þ f ðsÞc sðsÞlcÞK

* +n

signðsðsÞÞ if jsðsÞjP f ðsÞw sðsÞlw þ f ðsÞc sðsÞlc ð8Þ

and _cðsÞ ¼ 0 otherwise.Consequently, in the single crystal law proposed, the critical shear stress on slip system (s), sðsÞl , includes the critical shear

stresses of walls and channels via a mixture law:

sðsÞl ¼ f ðsÞw sðsÞlw þ f ðsÞc sðsÞlc ð9Þ

3.2. Hardening laws in walls and channels

The dislocation densities in walls and channels evolve following two different laws. The initial critical shear stress, s0, isassumed to be the same in walls and channels.

During the plastic flow, the critical shear stress in the walls, which are the dislocation rich zones, is defined by Franciosiand Zaoui (1982):

sðsÞlw ¼ lbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX

t

htðsÞqtw

rþ s0 ð10Þ

where b;l and htðsÞ have the same meanings as in the initial single crystal law. Finally, the evolution of the dislocation den-sity in walls is given by Essmann and Mughrabi (1979):

_qðsÞw ¼1b

1NðsÞw

� 2ycqðsÞw

!j _cðsÞj with qðsÞw ðcðsÞ ¼ 0Þ ¼ q0 ð11Þ

where yc is a parameter proportional to the length characteristic for the annihilation process of dislocation dipoles and NðsÞw isthe mean free path in walls on the slip system (s). NðsÞw is expressed as

NðsÞw ¼LwffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPt 6¼ðsÞqt

w

q ð12Þ

where Lw is a material parameter controlling the dislocation production in walls.

total resolved shear stress is calculated from the granular stress rg such as sðsÞ ¼ rg : mðsÞg , where mðsÞg is the orientation tensor:ðnðsÞ � bðsÞ þ bðsÞ � nðsÞÞ with bðsÞ and nðsÞ the slip direction and the normal direction to slip plane, respectively.

592 P. Evrard et al. / Mechanics Research Communications 35 (2008) 589–594

In the channels, which are dislocation poor zones, the interactions between slip systems can be neglected, thus noneinteraction matrix is used. The critical shear stress is thus defined by

2 Thi

sðsÞlc ¼ lbffiffiffiffiffiffiffiffiqðsÞc

qþ s0 ð13Þ

where qðsÞc is the dislocation density in the channels. The evolutions of dislocation densities in walls and channels are quasisimilar, with a term of dislocation production and a term of dislocation annihilation. In the channels, neglecting the inter-actions between slip systems, a simple form of the mean free path is used. The evolution of the dislocation density in thechannels is given by:

_qðsÞc ¼1b

ffiffiffiffiffiffiffiffiqðsÞc

qLc� 2ysqðsÞc

0@

1Aj _cðsÞj with qðsÞc ðcðsÞ ¼ 0Þ ¼ q0 ð14Þ

where Lc is a material parameter controlling dislocation production and ys is a parameter proportional to the length charac-teristic for the annihilation process of screw dislocations (Bíly, 1993).

3.3. Evolution of wall and channel structure

We suppose that the wall and channel structure evolution on the slip system (s) is described by the channel volume frac-tion. Using the relation f ðsÞw þ f ðsÞc ¼ 1, the relation (5) is modified as

sðsÞl ¼ ð1� f ðsÞc ÞsðsÞlw þ f ðsÞc sðsÞlc ð15Þ

According to (Estrin et al., 1998), the expansion of channel volume fraction on each slip system can be defined by a phenom-enological law which follows the differential equation:

f_ðsÞ

c ¼ 1� 1f1

f ðsÞc

� �j _cðsÞj ð16Þ

where f1 is the asymptotic volume fraction of channels.

3.4. Self-consistent scheme and interaction law

The macroscopic behavior is obtained by the classical self-consistent scheme, which considers each grain surrounded bythe homogeneous equivalent medium (HEM). The Cailletaud-Pilvin2 interaction law is used in order to link the macroscopicstress R and the microscopic granular stress rg:

rg ¼ Rþ lðB� bgÞ with B ¼XNg

g

f gbg ð17Þ

where f g and Ng are the grain volume fraction and the number of grains, respectively. The evolution law of bg is written viaan Armstrong-Frederick kinematic hardening (Armstrong and Frederick, 1966):

_bg ¼ _epg � Dðbg � depgÞk_epgk ð18Þ

where D and d are material parameters, which have to be identified from experimental data-base. The microscopic plasticstrain rate _epg is obtained as _epg ¼

PðsÞm

ðsÞg _cðsÞ.

The total macroscopic strain rate is obtained by: _E ¼ _Ee þ _E

pwith _E

p ¼PNg

g¼1f g _epg . The elastic term _Ee

is given by Hooke’slaw for an isotropic medium.

4. Results

For the simulation, a FCC polycrystal (12 slip systems {111} h110i) is considered, composed of 40-grains with an isotropicdistribution of crystallographic orientations. The parameters of the initial polycrystalline model ðE; m;K;n; b; s0;h;q0; Lw;

yc;D; dÞ have been obtained in a previous study for an austenitic phase (FCC structure) within an austenitic-ferritic stainlesssteel (Evrard et al., 2008), giving thus an order of magnitude of these parameters. Table 1 gives the values of modelparameters.

Concerning the modified single crystal law, two parameters have been introduced for modeling the channel behavior. Theparameter proportional to the length characteristic for the annihilation process of screw dislocations can be computed by:ys ¼ lb

2ps0 (Essmann and Mughrabi, 1979), giving ys ¼ 30 nm. The parameter controlling dislocation production Lc is identifiedfrom literature for an austenitic stainless steel (Gaudin, 2002), giving Lc ¼ 2:5.

s interaction law supposes that the elastic behavior is homogeneous and isotropic, and the grains are spherical.

Fig. 2. Hardening/Softening curves for a uniaxial low-cycle fatigue test with a total strain amplitude of ±0.5% ð _E11 ¼ 6:6� and10�4 s�1Þ. Initial-Single CrystalLaw (I-SCL) and New-Single Crystal Law (N-SCL). Influence of the asymptotic volume fraction of channels, f1 ¼ 50%;70%;90%.

Table 1Set of material parameters for a FCC material, obtained in a previous study Evrard et al. (2008)

E (MPa) m K (MPa s1/n) n b (m)

190,000 0.3 20 10 3.16 � 10�10

s0 (MPa) h q0 (m�2) Lw yc (nm)100 0.6 1012 33 3.2D d204 0.05

P. Evrard et al. / Mechanics Research Communications 35 (2008) 589–594 593

A fully reversed low-cycle fatigue test under total strain control at ±0.5% is simulated. The hardening/softening curves forthe initial and new single crystal laws for different values of channel asymptotic volume fraction f1 are given in Fig. 2. Theresults show that the model with the new single crystal law is able to account for an accommodation phase characterized bya hardening followed by a softening, and for the subsequent stabilized phase. Furthermore, when the asymptotic volumefraction of channels increases, the softening rate increases. This result is coherent with the hypothesis that considers thatthe expansion of soft zones controls the cyclic softening.

However, we can remark that the peak stress is higher than that of a purely austenitic stainless steel (of the order of310 MPa for a AISI 304L (Mayama and Sasaki, 2006)) for the same total strain amplitude. In order to obtain more realisticpeak stress level, the identification procedure should be performed on a purely austenitic stainless steel.

5. Conclusion

This work aimed at modeling the cyclic hardening/softening behavior observed in some polycrystalline materials in low-cycle fatigue. TEM observations show that the softening following a hardening during the accommodation phase can beattributed to a rearrangement of dislocations in hard and soft zones on the active slip systems. A new single crystal lawis proposed, which takes into account this dislocation structure evolution.

In spite of some assumptions, the proposed model offers advantages: its formulation is simple, it gives a good descriptionof the cyclic hardening/softening curves observed experimentally, and consequently a good description of the stabilizedstress state. Moreover, when the volume fraction of channels increases, the softening rate increases, in coherence withthe softening effect of the dislocation pattern rearrangement.

In a future work, this model will be used for modeling the low-cycle fatigue behavior of a bi-phased steel.

References

Armstrong, P.J., Frederick, C.O., 1966. A mathematical representation of the multiaxial Bauschinger effect. CEGB Report RD/B/N731, Berkeley Nucl.Bíly, 1993. Cyclic deformation and fatigue of metals. Elsevier.Cailletaud, G., 1992. A micromechanical approach to inelastic behavior of metals. Int. J. Plast. 8, 55–73.Essmann, U., Mughrabi, H., 1979. Annihilation of dislocations during tensile and cyclic deformation and limits of dislocations densities. Phil. Mag. 40, 731–

756.Estrin, Y., Tóth, L.S., Molinari, A., Bréchet, Y., 1998. A dislocation-based model for all hardening stages in large strain deformation. Acta Mater. 46, 5509–

5522.Evrard, P., Pilvin, Ph., Aubin, V., Degallaix, S., Kondo, D., 2008. Implementation and validation of a polycrystalline model for a bi-phased steel under non-

proportional loading path. Mech. Res. Commun. 35, 336–343.Feaugas, X., 1999. On the origin of the tensile flow stress in the stainless steel AISI 316L at 330K. Back stress and effective stress. Acta Mater. 47, 3617–3632.Franciosi, F., Zaoui, A., 1982. Multislip in f.c.c single crystals: a theoretical approach compared with experimental data. Acta Metall. 30–38, 1627–1637.

594 P. Evrard et al. / Mechanics Research Communications 35 (2008) 589–594

Gaudin, C., 2002. Etude des mécanismes associés au rochet cyclique d’un acier austénitique. Ph.D. Thesis, Université technologique de Compiégne.Hoc, T., Forest, S., 2001. Polycrystal modelling of IF-TI steel under complex loading path. Int. J. Plast. 27, 65–85.Mayama, T., Sasaki, K., 2006. Investigation of subsequent viscoplastic deformation of austenitic stainless steel subjected to cyclic preloading. Int. J. Plast. 22,

374–390.Mughrabi, H., 1983. Dislocation wall and cell structures and long range internal stresses in deformed metals crystals. Acta Mater. 31, 1367–1379.Pilvin, P., 1990. Approches multiéchelles pour la prévision du comportement anélastique des métaux. Ph.D. Thesis, Université Pierre et Marie Curie.Tóth, L.S., 2005. Modelling of strain hardening and microstructural evolution in equal channel angular extrusion. Comput. Mater. Sci. 32, 568–576.