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Effects of strain state on the kinetics of strain-induced martensite in steels
\
Pergamon
J[ Mech[ Phys[ Solids\ Vol[ 35\ No[ 8\ pp[ 0502Ð0524\ 0887 Þ 0887 Elsevier Science Ltd[ All rights reserved Printed in Great Britain 9911Ð4985:87 ,*see front matter
PII ] S9911Ð4985"87#99990Ð4
EFFECTS OF STRAIN STATE ON THE KINETICS OF STRAIN!INDUCED MARTENSITE IN STEELS J[ M[ DIANI Centre de Recherche et d|Etude d|Arcueil\ 05 bis av[ Prieur de la co¼te d|or\ 83003 Arcueil\ France
D[ M[ PARKS Department of Mechanical Engineering\ Massachusetts Institute of Technology\ 66 Massachusetts Avenue\ Cambridge MA 91028\ U[S[A[ "Received 1 May 0886 ^ in revised form 2 September 0886#
ABSTRACT This paper deals with the quantitative prediction of the volume fraction of strain!induced martensite produced in a steel that undergoes a thermomechanical loading[ This issue is relevant for several steels with a low stacking fault energy\ where a signi_cant amount of transformed martensite drives many mechanical properties[ Practical situations range from the optimization in the rolling process of a sheet to the improvement of the toughness of the _nal product[ The model relies on the assumption that the martensite "a?# is nucleated within a grain at the intersections of {{shear bands|| formed by the movement of partial dislocations on certain of the twelve "000# gð1 Þ00Łg systems "subscript g refers to the austenitic\ or mother\ phase#[ A modi_ed Taylor!based numerical calculation is performed on a polycrystalline aggregate in order to obtain the intensity of the shear\ and hence the volume fraction of martensite in each grain[ Results are found to model and predict various experimental results obtained mainly on 293L stainless steel under di}erent strain states[ Þ 0887 Elsevier Science Ltd[ All rights reserved[ Keywords ] phase transformation\ twinning\ viscoplastic material\ crystal plasticity\ polycrystalline materials[
0[ INTRODUCTION When dealing with transformation!induced plasticity "TRIP#\ one may be willing to address the _rst question raised by macroscopic observations ] {{How should one model the additional transformation plastic strain appearing during a ther! momechanical loading and incorporate this model into an existing classical plasticity theory<|| This question has been often investigated in the past "Greenwood and Johnson\ 0854 ^ Magee\ 0855#\ and\ recently\ with re_nements "Leblond et al[\ 0878 ^ Stringfellow et al[\ 0881 ^ Diani et al[\ 0884# enabled by numerical implementation of the necessarily complicated models resulting from the description of such a non!trivial phenomenon[ Nonetheless one should go a step further to solve the coupled problem referred to as the {{second problem|| in Diani et al[ "0884#[ This latter question centers basically in modeling the in~uence of the thermomechanical loading parameters $ To whom all correspondence should be addressed[ 0502
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J[ M[ DIANI and D[ M[ PARKS
"stress\ strain and temperature# on the evolution of internal variables characterizing the transformation\ and particularly the volume fraction of the transformed phase "martensite#[ Any answer to such a question should depend on the transformation regime\ be it stress!assisted or strain!induced[ In the former case\ where the applied stress is less than the yield limit of the parent phase\ it appears that the nucleation of the martensite takes place beside pre!existing defects that are not easy to predict[ Despite the early work of Koistinen and Marburger "0848# who proposed a for! mulation for the cooling of carbon steels that has been widely used\ recent authors have mostly focused on the second regime\ where the nucleation is driven by inelastic deformation mechanisms "plasticity#[ Olson and Cohen "0864# made an attempt in this direction using a sigmoidal law to _t experimentally results obtained by Angel "0843# on a 293!type "07:7# stainless steel[ These results\ which were obtained using tension experiments\ are given in Fig[ 0[ Since the OlsonÐCohen model is not sensitive to the state of stress\ Stringfellow et al[ "0881# updated it to account for stress triaxiality\ which is a critical parameter if one is willing to enhance the toughness of metallic or ceramic alloys using the TRIP e}ect[ On the other hand\ recent experiments conducted by Miller and McDowell "0885# and DeMania "0884#\ both on "low carbon# 293L type stainless steel\ tend to highlight the e}ect of the state of strain[ DeMania "0884# found that at −39>C\ more a?!martensite is produced during simple tension than during plane!strain tension\ for a given equivalent strain\ leading correspondingly to more transformation hardening for the simple tension[ This result is contrary to stress!dependent models predicting that\ the triaxiality being greater in plane!strain than in tension\ the former test should produce more martensite[ The aim of the present paper is to deal with such an apparent paradox by describing the in~uence of the strain state on the quantitative production of strain!induced
Fig[ 0[ Formation of martensite by plastic tensile strain at various deformation temperatures in 07:7 CrÐ Ni stainless steel " from Angel\ 0843#[
Kinetics of strain!induced martensite
0504
martensite in TRIP steels during a multiaxial thermomechanical loading[ Our approach is su.ciently general to apply to a wide range of steels[ Nonetheless\ since we focus on the deformation mechanism within the grain\ attention is restricted to steels with a low stacking fault energy "SFE# in austenite where\ to the best of our knowledge\ experimental evidence is fairly well established[ In Section 1 the motivation for de_ning a mesoscopic scale to deal with the kinematics of the deformation by using a {{shear band|| concept is discussed[ This concept is used in Section 2 to build a micromechanical model derived from a modi_ed version of the Taylor model in order to perform numerical thermomechanical loadings on a polycrystalline aggre! gate[ Results of these simulations are then compared\ in Section 3\ to experimental data obtained mainly on 293 stainless steels\ for di}erent strain states and tempera! tures[ Advantages\ limitations and possible improvements of our model are reviewed in Section 4[
1[ DESCRIPTION OF THE KINETICS AND KINEMATICS OF a?! MARTENSITE FORMATION IN STEELS WITH LOW STACKING FAULT ENERGY 1[0[ Physical considerations The mechanisms involved in the production of martensite in steels under thermo! mechanical loadings have focused the attention of several authors\ especially in e}orts directed at explaining the process of quenching[ Lecroisey and Pineau "0861# dealt with the FeÐNiÐCrÐC system[ In the case of high contents of chromium and nickel "about 29)wt[ pct[#\ they argued that a low value of stacking fault energy "SFE# of the austenite "below 29 erg:cm1# promotes the nucleation of a?!martensite at the intersection of {{shear bands|| that could be localized dislocation bands\ twins or o! platelets\ depending on the temperature at which the quasi!static loading is performed[ In the case of a 05"Cr#Ð02"Ni# alloy\ Fig[ 1 gives the nature of the active {{shear system|| as a function of the temperature[ Here we shall use the term {{shear bands|| to denote any localized dislocation bands\ twins or o!platelets\ because none of these deformation mechanisms involves a signi_cant volume change[ Murr et al[ "0871# also reported evidence of such shear!band formation\ leading to a?!martensite at their intersections in a 293 stainless steel[ Finally\ Nishiyama "0867# presents an extended survey of the mechanisms leading to the _nal formation of a?! martensite in steels using the o!martensite relay[ 1[1[ A mesoscopic kinematical description For austenitic steels with low SFE\ the deformation process leading to {{shear band|| formation is achieved\ at the atomic level\ by partial dislocation movements on successive "000# g lattice planes for twins\ and on alternate such planes for o! martensite\ with subscript g referring to the fcc austenite mother!phase "Yang and Wayman\ 0881#[ We do not attempt\ however\ to deal at this very _ne level of description[ Rather\ we focus at a mesoscopic level\ where the {{shear band|| formation
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J[ M[ DIANI and D[ M[ PARKS
Fig[ 1[ 05"Cr#Ð02"Ni# alloy[ Deformation structures as a function of strain o and temperature T ">C#[ The symbols T\ o and a? indicate the presence of\ respectively\ twinning in the austenite\ o!martensite\ and a?! martensite[ The curve corresponds to the formation of about 0 pct a?!martensite "from Lecroisey and Pineau\ 0861#[
mechanism is seen to be a _rst!order transformation\ characterized by its shear ` and its Schmid tensor R de_ned as ] R 01 "s & n¦n & s#\
"0#
where n is the normal to the slip plane and s the direction of the slip "see Fig[ 2#[ The previous description holds for both the "austenite# :"o!martensite#\ "Fig[ 2a#\ and the "austenite# :"twin#\ "Fig[ 2c#\ transformations\ on the same twelve "000# gð100Łg systems " for a comprehensive list\ see Table 1 of the Appendix#[ In the following\ these systems are denoted by superscript a\ with a ranging from 0Ð01[ If the total shear\ occurring on system a is ga\ in a given grain undergoing a ther! momechanical loading\ one can write ga f Ta `T ¦f oa `o \
"1#
where f Ta "resp[ f Ta # is the volume fraction "lattice plane fraction# of twins a "resp[ o! platelets a# and `T 9[6 "resp[ `o 9[24# is the characteristic shear for twins "resp[ o! platelets#[ For simpli_cation\ and because it is di.cult in an experiment to distinguish quantitatively between twins and o!martensite\ we homogenize the twinning defor! mation mechanism with the o!martensitic one by re!writing "1# as ga f a `\
"2#
where the homogenized parameter ` "uniform within a grain# is now indented to account for both previous mechanisms\ and is therefore viewed as a temperature! dependent model parameter[ This is consistent with Fig[ 1\ which shows that the
Kinetics of strain!induced martensite
0506
Fig[ 2[ Schematic 1!D!representation of a grain|s zone "Fig[ a# undergoing an o!martensitic transformation "Fig[ b#\ a twinning "Fig[ d# and a virtual transformation of magnitude ` "Fig[ c#[
predominance of one mechanism over another is a function of temperature in the following way ] at high temperatures\ dislocation!based shear bands are present\ replaced by twins as the temperature lowers\ as well as by o!martensite for the lowest temperatures[ Conceptually\ we are dealing with a two!phase material ] a mother phase and a {{phase|| consisting of {{virtual shear bands|| having one of twelve possible orientations and a parametric shear ` varying between 9[24 and 9[6 "see Fig[ 2c#[ This level of description allows us to link with a polycrystalline aggregate boundary problem through the homogenized e}ective shear ga[ Remark 0 ] The formation of a {{shear band|| is taken to be irreversible\ since it is presumed to be locked by the a?!martensite phase created at its intersections with shear bands of other families[ Remark 1 ] The applicability of the previous kinematic description in modeling the case of a dislocation shear band\ with a large characteristic value for `\ is not clear\ but in the following we will deal only with su.ciently low temperatures that\ according to Fig[ 1\ only twins and o!martensite plates are activated[ Also we do not attempt to describe what is happening when two shear bands "o!platelets or twins# are crossing ]
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J[ M[ DIANI and D[ M[ PARKS
that is\ how the a?!martensite forms and interacts with the {{shear bands||[ So\ in view of these modeling simpli_cations\ in order to retain a reasonable kinematic description\ the volume fraction of a?!martensite has to be kept low\ the material being e}ectively taken as quasi!dilute for a?!martensite embryos[ The previous physical description "1[0# accounts therefore for the nucleation of a?!martensite\ but not necessarily its growth "which is probably a much more complicated problem\ especially when the di}erent variants of a?!martensite are interacting with each other#[ In order to remain consistent with the quasi!dilute hypothesis of the model\ cal! culations "see Section 3# are stopped before reaching signi_cant amount of a?!marten! site[ We should note that while martensite continues to be nucleated with large on! going deformation "see Fig[ 0#\ the full kinematic details of a realistic model would be prohibitively complex[ Remark 2 ] At this point\ it can be noted that\ for a given grain\ the value of f a is constrained by relations "see Table 1 in Appendix for a list of systems a# 2
s f a ¾ 0\ a 0
5
s f a ¾ 0\ a 3
8
s f a ¾ 0 and a 6
01
s fa ¾0
"3#
a 09
since the lattice plane fraction of {{shear bands|| cannot exceed unity on any "000# g plane in the grain[ This type of constraint is very strong in the case of shape memory alloys "see Patoor et al[\ 0882#\ but it will not be the case here since we restrict attention to the {{nucleation|| process of the a? phase[ In the polycrystalline model calculations described below\ we set the limit on evolved a? volume fraction to be 14)\ which results in a negligible fraction of grains having a slip plane reaching the constraint "3# of being {{fully sheared||[ 1[2[ On the choice of active systems When the deformation mechanisms "such as twinning or martensitic trans! formation# involve coordinated atom movements\ it is clear that twelve distinct systems cannot be simultaneously activated within a given grain[ This is mostly due to the strong interaction existing between a couple of unfavorable oriented systems[ An important issue to address\ therefore\ is how to determine the systems a that are activated under a given state of stress\ strain and temperature[ Kelly "0854# considered o!martensite formation in steels with low SFE "in that case\ a 01)MnÐ09)CrÐ3)Ni and a 06)CrÐ8)Ni steel#[ By evaluating the strain energy that accompanies the formation of a?!martensite variants using stacking faults\ he Þ00Łg orientations not to be equivalent\ and concluded that found the twelve "000# gð1 Þ0Ł g "and its crystallographic only one direction per "000# g plane\ namely "000# g ð01 equivalents#\ has to be considered as potentially active[ It should be noted that this result was experimentally corroborated under cooling without macroscopic stress\ and may apply mainly for stress!assisted type martensite[ Nemirovskiy and Nemirovskiy "0875#\ trying to minimise some overall deformation strain energy\ seem to _nd more than four active systems per grain but their results are not clear and will not be discussed further here[ Since we are dealing with strain!induced martensite\ the results found by Kelly "0854# do not seem to apply to our case[ Instead\ we found it convenient to look at
Kinetics of strain!induced martensite
0508
Fig[ 3[ Formation of o!martensite bands in a 4)!compressed FeÐMnÐCrÐC steel "from Diani\ 0881#[
micrographs and to make some simple assumptions[ In Fig[ 3\ one can see a TRIP steel that has undergone a deformation of 4) "here in compression# ^ in almost all grains\ less than three systems have been activated[ The latter conclusion can be made in many other situations "see Lecroisey and Pineau\ 0861# and in any event\ it seems that a maximum of two systems are active within a given grain at a given increment of the loading[ At this point\ we can summarize the hypotheses "denoted H in the following# of the kinematical model within a grain as follows ] Þ00Łg systems H0 ] under imposed macroscopic deformation\ only the 01 "000# gð1 can be activated\ i[e[ dislocations movements along the "000# gð00 Þ9Łg systems are neglected[ The model will therefore be most realistic at low temperatures[ H1 ] all of the "000# gð1 Þ00Łg systems are taken to be equivalent and have same resistance and hardening law[ This is not a kinematical assumption but it is made here to _x the ideas[ H2 ] only two or less of those 01 potentially active systems are active at a time[ They will be selected according to a criterion given in the next section[ H3 ] for each shear system\ the strain is supported by a set of platelets of _xed width w "twins or o!martensite# equally spaced within a grain ^ we take the intersections of those platelets "see Appendix# as equally potent in producing a?!martensite[ Clearly\ further speci_c assumptions based on details of bands\ intersections\ crys! tallography\ etc[ could be incorporated within the overall model to match speci_c
0519
J[ M[ DIANI and D[ M[ PARKS
features for di}erent classes of steels[ For the present\ however\ we want to keep the description as simple as possible\ in order to capture the strain!state e}ect "see Section 3# with only one physical parameter "`#[ In Section 2\ we will impose the kinematic restrictions on the deformation of a given grain in a polycrystalline aggregate[
2[ MICROMECHANICAL MODEL 2[0[ General ~ame Since the crystallographic change accompanying the "austenite# :"a?!martensite# transformation is neglected "Remark 1# and since the {{shear band|| mechanisms "o! martensite transformation and twinning # occur without signi_cant volume change\ our material can be modeled using the classical tools of polycrystalline plasticity or viscoplasticity[ The use of the term {{classical plasticity or viscoplasticity|| is contrasted here with {{transformation plasticity||\ as de_ned clearly by Leblond et al[ "0875# for small strains ^ we do not need\ according to Remark 1\ to take explicitly into account an additional strain "called transformation strain# due to the accommodation of a Bain\ or eigen!strain "Diani et al[\ 0884#[ The relevance of the tools given by the plasticity theory will be constrained however by the limited number of systems "two or less\ see Section 1[2# active at a given time within a grain belonging to the aggregate undergoing a thermomechanical loading[ In particular\ the use of a Taylor assumption of uniform deformation is inappropriate here since it requires _ve independent crystallographic slip systems "see Hutchinson\ 0866#[ We use a modi_ed version of the Taylor model in order to accommodate the imposed deformation at the boundary of the macroscopic sample while respecting local kinematic constraints[ The model is based on the work of Parks and Ahzi "0889#\ who developed a constrained!hybrid approach to deal will polycrystalline materials whose crystals lack _ve independent slip systems to accommodate arbitrary imposed strain[ Lee et al[ "0884# reconstructed the model in a di}erent way\ which we will apply in the following to the case of two or less active systems per grain[ Bold fonts will stand for tensors or vectors[ 2[1[ The modi_ed Taylor model for two or less available systems per `rain Consider a grain with 01 potentially active systems a\ each system being char! acterized by its Schmid tensor Ra de_ned as Ra 01 "sa & na ¦na & sa #\
"4#
na being the unit normal to the slip plane and sa the unit vector in the direction of the slip[ Considering large plastic deformation compared to elastic strain\ elasticity is neglected and crystallographic slip "as operationally de_ned in Section 1[1 for our present purposes# is the only mechanism of plastic deformation assumed to be opera! ting in the grains[ Letting g¾ a − 9 "only easy!twinning is considered here# be the homogenized shear rate on the ath slip system within a given grain\ the traceless
Kinetics of strain!induced martensite
0510
deformation rate generated by two available slip systems\ a and b\ within the grain is therefore given by D g¾ a Ra ¦g¾ b Rb [
"5#
In the case of only one system a\ "5# is replaced by D g¾ a Ra [
"6#
Gab Ra [ [ Rb \
"7#
ab
Let us de_ne now the scalars G by
where [ [ denotes the double contracted product between the two second!order tensors Ra and Rb\ de_ned as Ra [ [ Rb R ija R bij [ In the case of two independent systems\ the 1×1 matrix G de_ned by the scalars Gab is invertible\ having matrix inverse G−0\ with components "G−0# ab\ and the shear rate on the ath slip system can be simply expressed "no sum on repeated superscript b# g¾ a "G−0 # aa "Ra [ [ D#¦"G−0 # ab "Rb [ [ D#[
"8#
For one active system\ one _nds that g¾ a 1"Ra [ [ D#[
"09#
The way of choosing the e}ective slip pair "or single slip# within a grain\ at a given time\ will be treated in Part 2[2[ At this point it can be noted that 55 pairs have to be considered in the case of double slip\ and 01 single slips[ Our material is described as a polycrystalline aggregate subjected to a macroscopic strain[ The traceless "incompressible# macroscopic velocity gradient L Þ is given by Þ D Þ ¦W Þ\ L
"00#
where D Þ and W Þ are the macroscopic deformation rate and spin\ respectively[ A central concept of the modi_ed Taylor model is\ for each grain in the aggregate\ to {{project|| the macroscopic quantity D Þ into a suitable subspace of the 4!dimensional traceless second!order tensor space[ This subspace is generated by the N independent Schmid tensors "each de_ned by "4## related to the N slip systems we considered previously "one or two in our case#[ Using a relay parameter "see Lee et al[ "0884# for a complete construction#\ one _nds Þ D "p [ [ ðpŁ−0 # [ [ D
"01#
to be the relation between the macroscopic deformation rate D Þ and the local one "assumed uniform within the grain\ and denoted D#[ In eqn "01#\ ðAŁ stands for the mean value "over the di}erent grains# of the tensor A de_ned as ðAŁ
0 Ngrains
Ngrains
s A\
"02#
0
Ngrains being the total number of grains in the aggregate[ For each grain\ p is a fourth! order tensor de_ned as
0511
p "G
J[ M[ DIANI and D[ M[ PARKS −0 ab
a
b
b
# "R & R ¦R & Ra #\ for two systems a and b "no sum on a and b# p 1Ra & Ra \ for one system a[
"03#
Also\ in eqn "01#\ the double!contracted product of two fourth!order tensors is given by "using Einstein notation# ] "p [ [ R# ijmn pijkl Rklmn [
"04#
To complete the interaction law in the modi_ed Taylor model\ the local spin W is equated to the macroscopic one as "05#
Þ[ W W
The rate of change of lattice orientation is determined by the spin tensor W ^ given by W W−Wp \
"06#
where the plastic spin Wp\ within a grain\ is given by N
Wp s g¾ a a 0
0
1
0 a "s & na −na & sa # [ 1
"07#
The current orientation for system a orientation is given by integrating set "08# ] s¾a W sa n¾a W na [
"08#
Remark 3 ] Note that the transformation rotation of the lattice within the shear bands is not taken into account in W ðsee "06#Ł\ although it can reach 69>29? for a twin ^ this simpli_cation is mostly justi_ed by the macroscopic scale at which the mechanical model operates\ which does not take into account the substructure formed by the bands[ Also\ once a band is formed\ it is assumed not to act any more in the simulation\ and new bands are occurring only in the mother phase[ In this sense\ from a mechanical point of view\ only the mean plastic rotation of the yet!to!be deformed portion of the austenite lattice\ given by "06#\ is relevant[ On the other hand\ we do not attempt to predict a _nal texture of our material ^ the latter should depend precisely on an accurate treatment of the lattice rotation within the bands and within their inter! sections[ Texture evolution due to twin bands operating in conjunction with dis! location slip has recently been treated by Staroselski and Anand "0885#[ In order to get the stresses we now specify viscoplastic constitutive equations for the shear rates\ g¾ a[ Following Asaro and Needleman "0874#\ we assume that g¾ a depends on the corresponding resolved shear stress\ denoted ta\ through a power law relation of the form ta sa
g¾ a m sign"g¾ a #\ g¾ 9
bb
"19#
Kinetics of strain!induced martensite
0512
where m is a rate exponent\ g¾ 9 is a pre!multiplying strain rate and sa is the slip system hardness[ The evolution of sa depends classically on the active shears ðsingle "b disappears# or doubleŁ rate through hardening moduli\ hab\ as s¾a haa =g¾ a =¦hab =g¾ b =
"no sum on a and b#[
"10#
Hardening moduli are given\ following Brown et al[ "0878#\ by sa a sb \ hab qh9 0− s½ s½
0 1
haa h9 0−
a
0 1
for a b
"11#
where h9\ a and s½ are slip hardening parameters which are taken to be identical for all slip systems[ q is the ratio of the latent hardening to the self!hardening rate "see Kalindidi et al[\ 0881# and is set to more than one "see Section 3# for non!coplanar systems ^ for coplanar systems " family a or b or c or d\ see Appendix 2# q is unity[ Finally "Lee et al[\ 0884# the macroscopic deviatoric stress\ noted Þ S\ is given by S ðpŁ−0 [ [ ðS Ł Þ
"12#
where S is a projected grain stress de_ned by S "G−0 # aa ta Ra ¦"G−0 # bb tb Rb ¦"G−0 # ab "tb Ra ¦ta Rb #\
"13#
where only _rst term "a# remains in the sum in "13# for a single activated system[ Remark 4 ] It is important to note that the rigid!plastic version of the modi_ed Taylor model is kinematically!based[ This means that\ assuming the active systems are known\ all the local kinematic variables can be determined from the global Þ#\ and can be incremented during calculation over a history of macroscopic ones "L deformation without any feedback of the stress[ 2[2[ Numerical resolution The set of equations described in Section 2[1 can be resolved using an explicit numerical scheme[ Let us assumed that for each increment ðt\ tŁ\ all variables are updated at time t\ the strain rate within each grain being noted D"t#[ The calculation starts with selection of potentially two active systems for each grain in the following manner ^ since "see 1[2# we assume that no more than two systems are active at a time in a given system\ one has to choose among 61 possibilities ] 55 pairs and 01 possible single slips[ Among the 55 pairs\ the optimum choice is de_ned to be the one selecting the pair having one of its two shears maximum over all possible pairs[ This can be expressed analytically\ using "8#\ by the following criterion ] "a\ b# $ "0\ 01#×"0\ 01# so that s "G−0 # ad "Rd [ [ D#"t# d a\b
or
s "G−0 # bd "Rd [ [ D#"t# d a\b
maximum and positive[
"14#
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J[ M[ DIANI and D[ M[ PARKS
On the other hand\ among the 01 single slips\ the optimum one has the maximum positive resolved shear strain and is selected by the criterion a $"0\ 01# so that Ra "t# [ [ D"t# maximum and positive[
"15#
The _nal candidate "pair or single slip# is the one with the largest maximum shear rate[ The _rst increment "t 9# is solved using the imposed macroscopic strain rate D Þ instead of D in criteria "14# and "15#[ The previous criterion expresses simply our choice that the active system is the one having maximum {{resolved shear strain rate|| calculated according to "8#[ From grain to grain\ the criterion will select di}erent active systems\ depending on the grain orientation with respect to the macroscopic strain rate[ The active systems having been chosen within each grain\ the projection tensor p"t# is calculated using "03#\ as well as its average by "02#[ The update of local strain rate D"t# is then obtained using "01#[ The shears g¾ a "t# being known\ rotations in each grain can be performed using "05#\ "06# and "07#\ providing the new Schmid tensor Ra "t# for each system[ The selection of the new pairs of systems "or single slip# within each grain can then start for the next increment[ At the end of each increment all variables of the model have to be updated[ First\ resolved shear stresses are calculated using "19# as ta "t# sa "t#
g¾ a "t# m sign"g¾ a "t##\ g¾ 9
b b
"16#
S "t# and Þ S "t# being obtained using "13# and "12#[ Equivalent macroscopic stress is then usually de_ned by Seq z21 Þ S[[Þ S[ Note that simple tension simulations "in direction 2# require imposed macroscopic strain rate D Þ to be of the form D Þ 00
Þ 22 −D \ D Þ 00 \ Þ 11 RD 0¦R
"17#
with scalar R being an adjustable value in order to keep ratio "S Þ11−S Þ00#:S Þeq\ lower than 9[90 at each increment[ f a "t# is also updated at the end of increment ðt\ tŁ using the rate form of eqn "2# ] g¾ a "t# ¾fa "t#`[
"18#
The shear ` is not derived in "18# "through it is temperature dependent# because we will deal in the following with loadings at _xed temperatures[ The values f a\ 0 ¾ a ¾ 01\ being known for each grain at each step of the simulation\ one can _nd in the Appendix a geometrical construction used to obtain the volume of each {{shear band||[ A characteristic length w "shear band thickness# is used in order to transform a volume fraction into a volume ^ the results have shown\ however\ not to depend on this characteristic length when it remains small in comparison to grain size\ h[ The volume of a {{tube|| of a?!martensite formed at the intersection of two shear bands is then deduced "Appendix#\ leading _nally to the value of the volume fraction of a?! martensite in a grain that is derived by assuming simply that all bands of system a are equidistant and periodically spaced[
Kinetics of strain!induced martensite
0514
3[ RESULTS In all the results shown in this section the equivalent plastic strain is taken to be Þ [[D Þ dt\ D Þ being the macroscopic strain rate tensor[ o¹ p Ðz12 D 3[0[ In~uence of initial texture As seen in Section 2[2\ the active systems\ whose slips are triggering the overall kinematics\ are selected during the calculation according to a {{maximum resolved shear strain rate||\ that depends naturally on the orientation of the considered grain[ The behavior of a complete polycrystalline aggregate might therefore depend on its initial texture[ To check the latter\ we performed the following numerical experiments ] we applied a pre!rolling on a {{virtual|| 293L stainless steel up to 29) in order to obtain an initial texture on which to perform a further uniaxial tension[ This initial pre!rolling texture was obtained using data from a _nite element calculation on a brass model developed by Staroselsky and Anand "0885#[ These authors deal with a visco!plastic f[c[c[ Þ9Łg slip systems polycrystalline aggregate\ in each grain of which the twelve "000# gð00 Þ00Łg systems[ This description is are active\ as well as some twinning on the "000# gð1 consistent with the warm!rolling of 293L stainless steels\ where the temperature is higher than the maximum temperature below which twins are prominently produced\ and\ as a result\ mainly dislocation motion is involved in the evolution of the crys! tallographic texture[ One can compare in Fig[ 4 the experimental texture of a 293L stainless steel rolled sheet " from DeMania\ 0884# with those obtained using the brass
Fig[ 4[ Experimental "left\ from DeMania\ 0884# and simulated "middle ] 29) brass!rolling\ 228 grains ^ right\ 29) copper!rolling\ 228 grains# initial textures "equal area projections#[
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J[ M[ DIANI and D[ M[ PARKS
Fig[ 5[ Prediction of the volume fraction of a?!martensite for simple tension in the rolling direction starting with di}erent initial rolling textures "isotropic\ 29) copper!rolling\ 29) brass!rolling# composed of 228 grains[
model[ Another pre!rolling "based on the work of Kalindidi and Anand\ 0881# was done using a Taylor assumption in a visco!plastic f[c[c[ polycrystalline aggregate\ where the deformation was carried out by only dislocations "twelve "000# gð00 Þ9Łg systems#\ and where most of the results presented by Kalindidi and Anand "0881# are derived for copper[ The brass!rolling results shown in Fig[ 4 were determined to be su.ciently discriminating for our purposes[ The model warm!rolling textures presented in Fig[ 4 were then used as initial textures to perform subsequent lower!temperature uniaxial tension in the rolling direction\ using the model presented in Section 2[ Results are reported in Fig[ 5[ The model shows the in~uence of the texture vs isotropic sample[ 3[1[ In~uence of the strain!state ] plane!strain tension vs simple tension DeMania "0884# performed experiments which indicate a strong in~uence of the strain state on the evolution of strain!induced martensite[ For that purpose\ she made two types of tensile specimen from sheet stock of 293L stainless steel\ oriented with the loading axis at di}erent angles with respect to the rolling direction[ One specimen type was for simple tension "ST#\ and the other was for plane!strain tension "PST# tests[ One can see in Fig[ 6 the measured amount of a?!martensite produced vs the equivalent strain at a temperature of −39>C ^ note that experimental values were downshifted by 9[8) since the original specimen already exhibited such a mag! netization ^ therefore\ the experimental amount of martensite shown in Fig[ 6 is
Kinetics of strain!induced martensite
0516
Fig[ 6[ Volume fraction of a?!martensite vs equivalent tensile strain of 293L stainless steel at −39>C "experimental data from DeMania\ 0884#[ The numbers in degrees denote the angle between the tensile axis and the rolling direction "RD#[ Results are shown for simple tension "ST# and plane strain tension "PST#[
representative of martensite produced during the testing "not the rolling#[ A simulation was done using our model\ since the predominant deformation mechanism is known to be o!martensite formation at −39>C\ with the nucleation of a?!martensite as a consequence "see Fig[ 1#[ The results are also reported in Fig[ 6 by solid lines[ The mean value of the shear in the bands\ which was found to be ` 9[34\ was adjusted to _t the experimental uniaxial tension curve of volume fraction of a?!martensite vs strain[ The corresponding curve for plane!strain tension is then obtained without further modi_cation of the model[ In this simulation we used an initial texture corresponding to 29) rolling brass texture "see Section 3[0#[ Although the dilute a? limitation of the model prevented simulation to values of o¹ − 9[1\ where the experimental evidence of reduced martensite under plane tension is clearest\ the model exhibits a consistent trend for smaller deformation[ This trend is con_rmed by stress:strain curves plotted in Fig[ 7[ The experimental data obtained by DeMania are reproduced by the simulation "see Section 2# using the following values for parameters "identical for simple tension and plane strain tension# ] ` is kept equal to 9[34 ^ g¾ 9 9[990:s and m 9[901 are typical values for metals and low strain! rate range experiments "Kalidindi et al[\ 0881# ^ s9 009 MPa drives the initial stress value\ s½ 799 MPa and a 15 driving the shape of the curves[ Parameters to be discussed are h9 29\999 Mpa and q 09 ] self!hardening h9 is usually large\ shear being here achieved through a coordinated movement between atoms\ involving therefore more energy than for dislocation motion ^ also\ hardening
0517
J[ M[ DIANI and D[ M[ PARKS
Fig[ 7[ Equivalent stress vs equivalent tensile strain of 293L stainless steel at −39>C "experimental data from DeMania\ 0884#[ The numbers in degrees denote the angle between the tensile axis and the rolling direction "RD#[ Results are shown for simple tension "ST# and plane strain tension "PST#[
on the _rst system due to transformation strain on the second one "latent hardening ratio q# is inferred here to be ten times larger than self!hardening in order for the simple tension stressÐstrain curve to be above plane strain one[ On the other hand\ a?!transformation is favored in simple tension "Fig[ 6# due to a greater amount of shear intersections ^ as a consequence the role of latent hardening is enhanced in simple tension in regard as plane strain one that is coherent with the result of Fig[ 7[ One could argue that the di}erence is not important and is increasing with the deformation\ the last point being in contradiction with experiment[ Since q is a constant and since the di}erence depends on the di}erence in f that is also increasing with deformation\ there is no way at this point to relieve such a contradiction[ One should remember\ however that a?!transformation is enhanced by stresses so that latent hardening should decrease with stresses or\ in our case\ with deformation ^ using an adjustable formula for q this should help things to _t more quantitatively with experiment[ 3[2[ Temperature!dependence of the model We have not yet discussed the in~uence of the shear amplitude in the band\ `\ which is a key mesoscopic parameter in our description[ The correlation with the microscopic deformation mechanisms is in fact very strong "see Section 1# although slightly ambiguous ] two di}erent sets of micro!mechanisms could lead to the same value of `[ To check further the in~uence of this parameter\ our model reproduced essential
Kinetics of strain!induced martensite
0518
Fig[ 8[ Experimental " from Angel\ 0843# and calculated formations of a?!martensite by plastic strain at various deformation temperatures in 07:7 CrÐNi stainless type steel[
features of the experimental results obtained by Angel "0843# and shown in Fig[ 0 by changing the value of ` to _t for di}erent temperatures[ The results are given in Fig[ 8[ The simulation was stopped when a volume fraction of a?!martensite of about 09) was reached\ since our method is only valid for {{small|| amounts of transformed phase[ The results range between ` 9[24 at the lowest temperatures\ which is the theoretical value for o!martensite\ and ` 9[6\ the theoretical value when only twin! ning operates[ From the _gure\ it appears that the latter condition occurred at a temperature below 9>C[ One can notice also that the curve corresponding to ` 9[24 does not _t the experimental data at a temperature of −077>C ^ this disagreement can either be considered from an experimental or a modeling point of view[ One can in fact argue that one deals here with stress!assisted martensite\ where the deformation mechanisms are slightly di}erent from the strain!induced ones "here o!band for! mation# are not taken into account in the model[ Finally\ by phenomenologically extending the range of the `!parameter to even greater values\ in some sense account! ing for dislocation!based shear banding\ the temperature trend shown in Fig[ 0 can be followed with the model to higher temperatures than shown in Fig[ 8[
4[ CONCLUSION The tool developed in this work for predicting the amount of strain!induced a?! martensite produced in a steel with low stacking fault energy relies on very simple physical hypotheses[ Detailed in Section 1 "H0ÐH3#\ these hypotheses assume that
0529
J[ M[ DIANI and D[ M[ PARKS
the inelastic displacement "including the transformation# is governed by coordinated movement of atoms along twelve equivalent "000# gð100Łg directions accounting for twinning and formation of o!bands ^ since the interaction of such bands is often strong\ only two of them are seen to cross within a grain\ producing a?!martensite at their intersections[ Finally\ using a modi_ed Taylor!based polycrystalline numerical model to simulate di}erent strain states\ we restricted the domain to small amounts of produced a?!martensite in order to minimize the e}ects of volume change[ Being so simple\ the previous description will apply {{as is|| only in few situations and for few materials[ We can see for example in Fig[ 6 that the experimental data we compare with are at rather high martensite values where the e}ect of volume change and interactions between a?!martensite tubes is perhaps not negligible ^ however\ we did not _nd smaller values in the literature\ and we believe that trends at the beginning of the a?!martensite nucleation probably in~uence the end[ In any case\ the amount of martensite produced during a loading of such a material appears to be closely a}ected by the strain state\ temperature and the initial texture of the material[ As a prospect the authors would like to suggest that the kinematic and kinetic description of the transformation "austenite to a?!martensite# seems to be a good skeleton\ at a good scale\ in order to build more sophisticated models[ These models could incorporate\ for example\ the e}ects of dislocation movement\ by using phenom! enologically the concept of a band[ In any event\ such a model could account pro_tably for higher temperatures[
ACKNOWLEDGEMENTS The authors would like to acknowledge the French DoD:DRET for support of this work[ Thanks are due to L[ Anand and A[ Staroselsky for helpful suggestions related to the use of Taylor!based numerical simulations[ We also thanks S[ Ahzi for interesting discussions on the use of the constrained!hybrid model as well as A[ D[ DeMania for information relating to her master thesis on 293L[
REFERENCES Angel\ T[ "0843# Formation of martensite in austenitic stainless steels[ J[ of Iron Steel Inst[ 027\ 054Ð063[ Asaro\ R[ J[ "0874# Texture development and strain hardening in rate dependent polycrystals[ Acta Metallica 22\ 812[ Brown\ S[ B[\ Kim\ K[ H[ and Anand\ L[ "0878# An internal variable constitutive model for hot working of metals[ International Journal of Plasticity 4\ 84Ð029[ DeMania\ A[ D[ "0884# The in~uence of martensite transformation on the formability of 293 stainless steel sheet\ M[Sc[ thesis\ Department of Mechanical Engineering\ Massachusetts Institute of Technology[ Diani\ J[ M[ "0881# Contribution to the study of the transformation induced plasticity at high strain rates\ Ph[D[ thesis\ Ecole Nationale Superieure des Mines de Paris\ France\ pp[ 43[ Diani\ J[ M[\ Sabar\ H[ and Berveiller\ M[ "0884# Micromechanical modelling of the trans! formation induced plasticity "TRIP# phenomenon in steels[ Int[ J[ En`n` Sci[ 22\ 0810Ð0823[ Greenwood\ G[ W[ and Johnson\ R[ H[ "0854# The deformation of metals under small stresses during phase transformations[ Proceedin`s of the Royal Society A172\ 392Ð311[
Kinetics of strain!induced martensite
0520
Hecker\ S[ S[\ Stout\ M[ G[\ Staudhammer\ K [P[ and Smith\ J[ L[ "0871# E}ects of strain state and strain rate on deformation!induced transformation in 293 stainless steel ] Part I[ Magnetic measurements and mechanical behavior[ Met[ Trans[ 02A\ 508Ð515[ Hutchinson\ J[ W[ "0866# Creep and plasticity of hexagonal polycrystals as related to single crystal slip[ Met[ Trans[ 7A\ 0354Ð0358[ Kalindidi\ S[ R[\ Bronkhorst\ C[ A[ and Anand\ L[ "0881# Crystallographic texture evolution in bulk deformation processing of FCC metals[ Journal of the Mechanics and Physics of Solids 39\ 426Ð458[ Kelly\ P[ M[ "0854# The martensite transformation in steels with low stacking fault energy[ Acta Metallica 02\ 524Ð535[ Koistinen\ D[ and Marburger\ R[ "0848# A general equation prescribing the extent of the austeniteÐmartensite transformation in pure ironÐcarbon alloys and plain carbon steels[ Acta Metallica 6\ 48Ð59[ Leblond\ J[ B[\ Devaux\ J[ and Devaux\ J[ C[ "0878# Mathematical modelling of transformation plasticity in steels*I ] Case of ideal!plastic phases[ International Journal of Plasticity 4\ 462Ð 477[ Leblond\ J[ B[\ Mottet\ G[ and Devaux\ J[ C[ "0875# A theoretical and numerical approach to the plastic behavour of steels during phase transformations*I ] Derivation of general relations[ Journal of the Mechanics and Physics of Solids 23\ 284Ð309[ Lecroisey\ F[ and Pineau\ A[ "0861# Martensitic transformations induced by plastic deformation in the FeÐNiÐCrÐC system[ Met[ Trans[ 2\ 276Ð285[ Lee\ B[ J[\ Ahzi\ S[ and Asaro\ R[ J[ "0884# On the plasticity of low symmetry crystals lacking _ve independent slip systems[ Mech[ Mat[ 19\ 0Ð7[ Magee\ C[ L[ "0855# Transformation kinetics\ microplasticity and aging of martensite in Fe!20 Ni[ Ph[D[ thesis\ Carnegie Institute of Technology[ Miller\ M[ P[ and McDowell\ D[ L[ "0885# The e}ect of stress!state on the large strain inelastic deformation behavior of 293L stainless steel[ J[ En`r[ Matls and Techn[ 007\ 17Ð25[ Murr\ L[ E[\ Staudhammer\ K[ P[ and Hecker\ S[ S[ "0871# E}ects of strain state and strain rate on deformation!induced transformation in 293 stainless steel ] Part II[ Microstructural study[ Met[ Trans[ 02A\ 517Ð524[ Nemirovskiy\ Yu[ R[ and Nemirovskiy\ M[ R[ "0875# Analysis of the interrelation of shear processes in intersecting planes "000# g>"990# o and the formation of a?!martensite during deformation of steels with low stacking fault energy of the austenite[ Fiz[ metal[ metalloved[ 51\ 642Ð647[ Olson\ G[ B[ and Cohen\ M[ "0864# Kinetics of strain!induced martensitic nucleation[ Met[ Trans[ 5A\ 680Ð684[ Parks\ D[ M[ and Azhi\ S[ "0889# Polycrystalline plastic deformation and texture evolution for crystals lacking _ve independent slip systems[ Journal of the Mechanics and Physics of Solids 27\ 690Ð613[ Patoor\ E[ "0875# Contribution a l|etude del la plasticite de transformation dans le mono et les polycristaux metalliques[ Ph[D[ thesis\ LPMM\ France[ Patoor\ E[\ Bensalah\ M[ O[\ Eberhardt\ A[ and Berveiller\ M[ "0882# Thermomechanical behaviour determining of shape memory alloys using a thermodynamical potential opti! mizing\ ed[ La Revue de Metallurgie:Science et Genie des Materiaux\ Decembre 0882\ pp[ 0477Ð0481[ Powell\ G[ W[\ Marshall\ E[ R[ and Backofen\ W[ A[ "0847# Strain hardening of austenitic stainless steel[ Transactions of the ASM 49\ 367Ð386[ Staroselsky\ A[ and Anand\ L[ "0885# Large deformation plasticity of FCC Metals due to slip and twinning[ Displacive Phase Transformations and Their Applications in Materials En`ineerin`\ ed[ K[ Inoue[ Stringfellow\ R[ G[\ Parks\ D[ M[ and Olson\ G[ B[ "0881# A constitutive model for trans! formation plasticity accompanying strain!induced martensitic transformations in metastable austenitic steels[ Acta Metall[ Mater[ 39\ 0692Ð0605[ Yang\ J[ H[ and Wayman\ C[ M[ "0881# Self!accommodation and shape memory mechanism of o!martensite*II[ Theoretical considerations[ Mat[ Char[ 17\ 26Ð36[
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J[ M[ DIANI and D[ M[ PARKS
APPENDIX A[0[ Geometrical calculation of {{shear bands|| volume fraction in a `rain In the following\ the grain is described as a cube of volume h2 containing {{shear bands|| of width w parallel to the four "000# planes "see Fig[ 09#[ The equations of those planes in the crystal frames are family a "systems 0\ 1\ 2# ] x¦y−z−a 9\ −h ¾ a ¾ 1h family b"systems 3\ 4\ 5# ] x−y¦z−b 9\ −h ¾ b ¾ 1h family c "systems 6\ 7\ 8# ] x−y−z−c 9\ −1h ¾ c ¾ h family d "systems 09\ 00\ 01# ] x¦y¦z−d 9\ 9 ¾ d ¾ 2h[
"29#
The intersection pattern of any one of these planes with the cubic crystal "see Fig[ 09# is an equilateral triangle or a hexagon\ the area of which\ S\ can be calculated as a function of the plane|s location\ according to the value of parameters a\ b\ c and d[ To _x the ideas\ the calculation has been done for the family a "see schematic in Fig[ 09#\ and the result is given by system "20# in the range −h ¾ a ¾"h:1# ^ results for range "h:1# ¾ a ¾ 1h are found to be symmetric\ with S being an even function of ða−"h:1#Ł[
Fig[ 09[ Cubic crystal where three planes of family a are drawn[
Kinetics of strain!induced martensite
S"a#
z2 "a¦h# 1 \ −h ¾ a ¾ 9 1
S"a#
z2 h ""a¦h# 1 −2a1 #\ 9 ¾ a ¾ 1 1
0522
"triangles#\ "hexagons#[
"20#
For the three other "000# family planes "respectively\ b\ c and d#\ similar results can be found by replacing parameters a\ respectively\ with b\ c¦h and d−h in "20#[ The volume fraction of Þ# platelet " family a# of width w at a location a is therefore given by "21# an o!martensite "000 and is derived from "21# for the other family planes[ "a# f 0\1\2 o
wS"a# 2
h
h \ −h ¾ a ¾ \ f 0\1\2 "h−a# f 0\1\2 "a#[ o o 1
"21#
After each increment of a simulation "see Section 2[2# the number of o!platelets is calculated using "2# and "21# by assuming the platelets to be equally spaced within the grain for a given system a[ A[1[ Geometrical calculation of a?!martensite volume fraction at the intersection of shear bands platelets The next step is to calculate the volume fraction of a {{tube|| of a?!martensite nucleating at the intersection of two {{shear bands||\ each of width w within a given cubic grain[ To _x the ideas\ we consider the families a and b[ According to "29#\ the equation of the line L forming the intersection of the planes containing families a and b can be written as line L \ x
a−b a¦b and z y− [ 1 1
"22#
In Fig[ 00\ one can _nd a cut of a cubic crystal according to a plane x constant\ where the
Fig[ 00[ Two di}erent traces in a plane x constant of intersection of two {{shear bands|| of family a and b[
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J[ M[ DIANI and D[ M[ PARKS
intersection of di}erent lines L with the crystal is drawn[ The length l of the intersection of those lines L with the cubic crystal can now be easily determined to be
0 b b1
l z1 h−
a−b 1
"23#
where =x= stands for the absolute value of scalar x[ Since the width w of both {{shear bands|| is assumed to be small compared to the crystal dimension h\ the length l is used to construct a tube of a?!martensite {{around|| line L[ According to Fig[ 01\ the area of the cross!sectional of such a tube is easily found to be "2:1z1#w1 [
Fig[ 01[ Cross!sectional of the intersection of two shear bands "tube of a?!martensite#[ Using "23#\ one _nds _nally the volume fraction of a tube of a?!martensite to be given by Table 0[
Table 0[ Volume fraction of a?!martensite as a function of the position of the selected intersectin` planes –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– — f a?a\b a b c d ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐ * 2 w1 a−b 2 w1 a¦c 2 w1 a¦d h− h− a 9 1 h2 1 1 h2 1 1 h2 1
0 b b1
b
c
9
0 b b1
2 w1 b¦c h− 1 h2 1
0 b b1 9
b b
2 w1 b¦d 1 h2 1
b b
2 w1 c−d 1 h2 1
b b
–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– —
Kinetics of strain!induced martensite
0524
A[2[ Shear systems in crystals coordinate "cubic lattice#
Table 1[ Characteristics values of the twelve "000# gð1 Þ00Łg stackin` faults directions in FCC materials –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– — Family System a na sa Ð*ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐ 0 0 1 0 0 a 0 0 − − − z2 z2 z2 z5 z5 z5 0 0 0 0 1 1 0 − z2 z2 z2 z5 z5 z5 0 0 0 1 0 2 0 − − − z2 z2 z2 z5 z5 z5 b
3
0
4
z2 0
5
z2 0 z2
c
6
0
7
z2 0
8
z2 0
09
− −
− − −
z2 d
−
0
0
0
0
−
0
1
z2 z2 0 0
z5
z2 z2 0 0
z5 z5 z5 1 0 0 − − z5 z5 z5
z2 z2 0
z2 0 z2 0 z2 0
− − −
0
0
0
z2 0
z5 1
z2 0 z2
0
−
z5
0
0
1
z5 z5 0 0 − − − z5 z5 z5 0 1 0 − z5 z5 z5 0
0
−
1
z5 z5 z5 1 0 0 − z2 z2 z2 z5 z5 z5 0 0 0 1 0 01 0 − z2 z2 z2 z5 z5 z5 –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– — 00
z2 z2 z2 0 0 0
z5 1
−
Pergamon
J[ Mech[ Phys[ Solids\ Vol[ 35\ No[ 8\ pp[ 0502Ð0524\ 0887 Þ 0887 Elsevier Science Ltd[ All rights reserved Printed in Great Britain 9911Ð4985:87 ,*see front matter
PII ] S9911Ð4985"87#99990Ð4
EFFECTS OF STRAIN STATE ON THE KINETICS OF STRAIN!INDUCED MARTENSITE IN STEELS J[ M[ DIANI Centre de Recherche et d|Etude d|Arcueil\ 05 bis av[ Prieur de la co¼te d|or\ 83003 Arcueil\ France
D[ M[ PARKS Department of Mechanical Engineering\ Massachusetts Institute of Technology\ 66 Massachusetts Avenue\ Cambridge MA 91028\ U[S[A[ "Received 1 May 0886 ^ in revised form 2 September 0886#
ABSTRACT This paper deals with the quantitative prediction of the volume fraction of strain!induced martensite produced in a steel that undergoes a thermomechanical loading[ This issue is relevant for several steels with a low stacking fault energy\ where a signi_cant amount of transformed martensite drives many mechanical properties[ Practical situations range from the optimization in the rolling process of a sheet to the improvement of the toughness of the _nal product[ The model relies on the assumption that the martensite "a?# is nucleated within a grain at the intersections of {{shear bands|| formed by the movement of partial dislocations on certain of the twelve "000# gð1 Þ00Łg systems "subscript g refers to the austenitic\ or mother\ phase#[ A modi_ed Taylor!based numerical calculation is performed on a polycrystalline aggregate in order to obtain the intensity of the shear\ and hence the volume fraction of martensite in each grain[ Results are found to model and predict various experimental results obtained mainly on 293L stainless steel under di}erent strain states[ Þ 0887 Elsevier Science Ltd[ All rights reserved[ Keywords ] phase transformation\ twinning\ viscoplastic material\ crystal plasticity\ polycrystalline materials[
0[ INTRODUCTION When dealing with transformation!induced plasticity "TRIP#\ one may be willing to address the _rst question raised by macroscopic observations ] {{How should one model the additional transformation plastic strain appearing during a ther! momechanical loading and incorporate this model into an existing classical plasticity theory<|| This question has been often investigated in the past "Greenwood and Johnson\ 0854 ^ Magee\ 0855#\ and\ recently\ with re_nements "Leblond et al[\ 0878 ^ Stringfellow et al[\ 0881 ^ Diani et al[\ 0884# enabled by numerical implementation of the necessarily complicated models resulting from the description of such a non!trivial phenomenon[ Nonetheless one should go a step further to solve the coupled problem referred to as the {{second problem|| in Diani et al[ "0884#[ This latter question centers basically in modeling the in~uence of the thermomechanical loading parameters $ To whom all correspondence should be addressed[ 0502
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J[ M[ DIANI and D[ M[ PARKS
"stress\ strain and temperature# on the evolution of internal variables characterizing the transformation\ and particularly the volume fraction of the transformed phase "martensite#[ Any answer to such a question should depend on the transformation regime\ be it stress!assisted or strain!induced[ In the former case\ where the applied stress is less than the yield limit of the parent phase\ it appears that the nucleation of the martensite takes place beside pre!existing defects that are not easy to predict[ Despite the early work of Koistinen and Marburger "0848# who proposed a for! mulation for the cooling of carbon steels that has been widely used\ recent authors have mostly focused on the second regime\ where the nucleation is driven by inelastic deformation mechanisms "plasticity#[ Olson and Cohen "0864# made an attempt in this direction using a sigmoidal law to _t experimentally results obtained by Angel "0843# on a 293!type "07:7# stainless steel[ These results\ which were obtained using tension experiments\ are given in Fig[ 0[ Since the OlsonÐCohen model is not sensitive to the state of stress\ Stringfellow et al[ "0881# updated it to account for stress triaxiality\ which is a critical parameter if one is willing to enhance the toughness of metallic or ceramic alloys using the TRIP e}ect[ On the other hand\ recent experiments conducted by Miller and McDowell "0885# and DeMania "0884#\ both on "low carbon# 293L type stainless steel\ tend to highlight the e}ect of the state of strain[ DeMania "0884# found that at −39>C\ more a?!martensite is produced during simple tension than during plane!strain tension\ for a given equivalent strain\ leading correspondingly to more transformation hardening for the simple tension[ This result is contrary to stress!dependent models predicting that\ the triaxiality being greater in plane!strain than in tension\ the former test should produce more martensite[ The aim of the present paper is to deal with such an apparent paradox by describing the in~uence of the strain state on the quantitative production of strain!induced
Fig[ 0[ Formation of martensite by plastic tensile strain at various deformation temperatures in 07:7 CrÐ Ni stainless steel " from Angel\ 0843#[
Kinetics of strain!induced martensite
0504
martensite in TRIP steels during a multiaxial thermomechanical loading[ Our approach is su.ciently general to apply to a wide range of steels[ Nonetheless\ since we focus on the deformation mechanism within the grain\ attention is restricted to steels with a low stacking fault energy "SFE# in austenite where\ to the best of our knowledge\ experimental evidence is fairly well established[ In Section 1 the motivation for de_ning a mesoscopic scale to deal with the kinematics of the deformation by using a {{shear band|| concept is discussed[ This concept is used in Section 2 to build a micromechanical model derived from a modi_ed version of the Taylor model in order to perform numerical thermomechanical loadings on a polycrystalline aggre! gate[ Results of these simulations are then compared\ in Section 3\ to experimental data obtained mainly on 293 stainless steels\ for di}erent strain states and tempera! tures[ Advantages\ limitations and possible improvements of our model are reviewed in Section 4[
1[ DESCRIPTION OF THE KINETICS AND KINEMATICS OF a?! MARTENSITE FORMATION IN STEELS WITH LOW STACKING FAULT ENERGY 1[0[ Physical considerations The mechanisms involved in the production of martensite in steels under thermo! mechanical loadings have focused the attention of several authors\ especially in e}orts directed at explaining the process of quenching[ Lecroisey and Pineau "0861# dealt with the FeÐNiÐCrÐC system[ In the case of high contents of chromium and nickel "about 29)wt[ pct[#\ they argued that a low value of stacking fault energy "SFE# of the austenite "below 29 erg:cm1# promotes the nucleation of a?!martensite at the intersection of {{shear bands|| that could be localized dislocation bands\ twins or o! platelets\ depending on the temperature at which the quasi!static loading is performed[ In the case of a 05"Cr#Ð02"Ni# alloy\ Fig[ 1 gives the nature of the active {{shear system|| as a function of the temperature[ Here we shall use the term {{shear bands|| to denote any localized dislocation bands\ twins or o!platelets\ because none of these deformation mechanisms involves a signi_cant volume change[ Murr et al[ "0871# also reported evidence of such shear!band formation\ leading to a?!martensite at their intersections in a 293 stainless steel[ Finally\ Nishiyama "0867# presents an extended survey of the mechanisms leading to the _nal formation of a?! martensite in steels using the o!martensite relay[ 1[1[ A mesoscopic kinematical description For austenitic steels with low SFE\ the deformation process leading to {{shear band|| formation is achieved\ at the atomic level\ by partial dislocation movements on successive "000# g lattice planes for twins\ and on alternate such planes for o! martensite\ with subscript g referring to the fcc austenite mother!phase "Yang and Wayman\ 0881#[ We do not attempt\ however\ to deal at this very _ne level of description[ Rather\ we focus at a mesoscopic level\ where the {{shear band|| formation
0505
J[ M[ DIANI and D[ M[ PARKS
Fig[ 1[ 05"Cr#Ð02"Ni# alloy[ Deformation structures as a function of strain o and temperature T ">C#[ The symbols T\ o and a? indicate the presence of\ respectively\ twinning in the austenite\ o!martensite\ and a?! martensite[ The curve corresponds to the formation of about 0 pct a?!martensite "from Lecroisey and Pineau\ 0861#[
mechanism is seen to be a _rst!order transformation\ characterized by its shear ` and its Schmid tensor R de_ned as ] R 01 "s & n¦n & s#\
"0#
where n is the normal to the slip plane and s the direction of the slip "see Fig[ 2#[ The previous description holds for both the "austenite# :"o!martensite#\ "Fig[ 2a#\ and the "austenite# :"twin#\ "Fig[ 2c#\ transformations\ on the same twelve "000# gð100Łg systems " for a comprehensive list\ see Table 1 of the Appendix#[ In the following\ these systems are denoted by superscript a\ with a ranging from 0Ð01[ If the total shear\ occurring on system a is ga\ in a given grain undergoing a ther! momechanical loading\ one can write ga f Ta `T ¦f oa `o \
"1#
where f Ta "resp[ f Ta # is the volume fraction "lattice plane fraction# of twins a "resp[ o! platelets a# and `T 9[6 "resp[ `o 9[24# is the characteristic shear for twins "resp[ o! platelets#[ For simpli_cation\ and because it is di.cult in an experiment to distinguish quantitatively between twins and o!martensite\ we homogenize the twinning defor! mation mechanism with the o!martensitic one by re!writing "1# as ga f a `\
"2#
where the homogenized parameter ` "uniform within a grain# is now indented to account for both previous mechanisms\ and is therefore viewed as a temperature! dependent model parameter[ This is consistent with Fig[ 1\ which shows that the
Kinetics of strain!induced martensite
0506
Fig[ 2[ Schematic 1!D!representation of a grain|s zone "Fig[ a# undergoing an o!martensitic transformation "Fig[ b#\ a twinning "Fig[ d# and a virtual transformation of magnitude ` "Fig[ c#[
predominance of one mechanism over another is a function of temperature in the following way ] at high temperatures\ dislocation!based shear bands are present\ replaced by twins as the temperature lowers\ as well as by o!martensite for the lowest temperatures[ Conceptually\ we are dealing with a two!phase material ] a mother phase and a {{phase|| consisting of {{virtual shear bands|| having one of twelve possible orientations and a parametric shear ` varying between 9[24 and 9[6 "see Fig[ 2c#[ This level of description allows us to link with a polycrystalline aggregate boundary problem through the homogenized e}ective shear ga[ Remark 0 ] The formation of a {{shear band|| is taken to be irreversible\ since it is presumed to be locked by the a?!martensite phase created at its intersections with shear bands of other families[ Remark 1 ] The applicability of the previous kinematic description in modeling the case of a dislocation shear band\ with a large characteristic value for `\ is not clear\ but in the following we will deal only with su.ciently low temperatures that\ according to Fig[ 1\ only twins and o!martensite plates are activated[ Also we do not attempt to describe what is happening when two shear bands "o!platelets or twins# are crossing ]
0507
J[ M[ DIANI and D[ M[ PARKS
that is\ how the a?!martensite forms and interacts with the {{shear bands||[ So\ in view of these modeling simpli_cations\ in order to retain a reasonable kinematic description\ the volume fraction of a?!martensite has to be kept low\ the material being e}ectively taken as quasi!dilute for a?!martensite embryos[ The previous physical description "1[0# accounts therefore for the nucleation of a?!martensite\ but not necessarily its growth "which is probably a much more complicated problem\ especially when the di}erent variants of a?!martensite are interacting with each other#[ In order to remain consistent with the quasi!dilute hypothesis of the model\ cal! culations "see Section 3# are stopped before reaching signi_cant amount of a?!marten! site[ We should note that while martensite continues to be nucleated with large on! going deformation "see Fig[ 0#\ the full kinematic details of a realistic model would be prohibitively complex[ Remark 2 ] At this point\ it can be noted that\ for a given grain\ the value of f a is constrained by relations "see Table 1 in Appendix for a list of systems a# 2
s f a ¾ 0\ a 0
5
s f a ¾ 0\ a 3
8
s f a ¾ 0 and a 6
01
s fa ¾0
"3#
a 09
since the lattice plane fraction of {{shear bands|| cannot exceed unity on any "000# g plane in the grain[ This type of constraint is very strong in the case of shape memory alloys "see Patoor et al[\ 0882#\ but it will not be the case here since we restrict attention to the {{nucleation|| process of the a? phase[ In the polycrystalline model calculations described below\ we set the limit on evolved a? volume fraction to be 14)\ which results in a negligible fraction of grains having a slip plane reaching the constraint "3# of being {{fully sheared||[ 1[2[ On the choice of active systems When the deformation mechanisms "such as twinning or martensitic trans! formation# involve coordinated atom movements\ it is clear that twelve distinct systems cannot be simultaneously activated within a given grain[ This is mostly due to the strong interaction existing between a couple of unfavorable oriented systems[ An important issue to address\ therefore\ is how to determine the systems a that are activated under a given state of stress\ strain and temperature[ Kelly "0854# considered o!martensite formation in steels with low SFE "in that case\ a 01)MnÐ09)CrÐ3)Ni and a 06)CrÐ8)Ni steel#[ By evaluating the strain energy that accompanies the formation of a?!martensite variants using stacking faults\ he Þ00Łg orientations not to be equivalent\ and concluded that found the twelve "000# gð1 Þ0Ł g "and its crystallographic only one direction per "000# g plane\ namely "000# g ð01 equivalents#\ has to be considered as potentially active[ It should be noted that this result was experimentally corroborated under cooling without macroscopic stress\ and may apply mainly for stress!assisted type martensite[ Nemirovskiy and Nemirovskiy "0875#\ trying to minimise some overall deformation strain energy\ seem to _nd more than four active systems per grain but their results are not clear and will not be discussed further here[ Since we are dealing with strain!induced martensite\ the results found by Kelly "0854# do not seem to apply to our case[ Instead\ we found it convenient to look at
Kinetics of strain!induced martensite
0508
Fig[ 3[ Formation of o!martensite bands in a 4)!compressed FeÐMnÐCrÐC steel "from Diani\ 0881#[
micrographs and to make some simple assumptions[ In Fig[ 3\ one can see a TRIP steel that has undergone a deformation of 4) "here in compression# ^ in almost all grains\ less than three systems have been activated[ The latter conclusion can be made in many other situations "see Lecroisey and Pineau\ 0861# and in any event\ it seems that a maximum of two systems are active within a given grain at a given increment of the loading[ At this point\ we can summarize the hypotheses "denoted H in the following# of the kinematical model within a grain as follows ] Þ00Łg systems H0 ] under imposed macroscopic deformation\ only the 01 "000# gð1 can be activated\ i[e[ dislocations movements along the "000# gð00 Þ9Łg systems are neglected[ The model will therefore be most realistic at low temperatures[ H1 ] all of the "000# gð1 Þ00Łg systems are taken to be equivalent and have same resistance and hardening law[ This is not a kinematical assumption but it is made here to _x the ideas[ H2 ] only two or less of those 01 potentially active systems are active at a time[ They will be selected according to a criterion given in the next section[ H3 ] for each shear system\ the strain is supported by a set of platelets of _xed width w "twins or o!martensite# equally spaced within a grain ^ we take the intersections of those platelets "see Appendix# as equally potent in producing a?!martensite[ Clearly\ further speci_c assumptions based on details of bands\ intersections\ crys! tallography\ etc[ could be incorporated within the overall model to match speci_c
0519
J[ M[ DIANI and D[ M[ PARKS
features for di}erent classes of steels[ For the present\ however\ we want to keep the description as simple as possible\ in order to capture the strain!state e}ect "see Section 3# with only one physical parameter "`#[ In Section 2\ we will impose the kinematic restrictions on the deformation of a given grain in a polycrystalline aggregate[
2[ MICROMECHANICAL MODEL 2[0[ General ~ame Since the crystallographic change accompanying the "austenite# :"a?!martensite# transformation is neglected "Remark 1# and since the {{shear band|| mechanisms "o! martensite transformation and twinning # occur without signi_cant volume change\ our material can be modeled using the classical tools of polycrystalline plasticity or viscoplasticity[ The use of the term {{classical plasticity or viscoplasticity|| is contrasted here with {{transformation plasticity||\ as de_ned clearly by Leblond et al[ "0875# for small strains ^ we do not need\ according to Remark 1\ to take explicitly into account an additional strain "called transformation strain# due to the accommodation of a Bain\ or eigen!strain "Diani et al[\ 0884#[ The relevance of the tools given by the plasticity theory will be constrained however by the limited number of systems "two or less\ see Section 1[2# active at a given time within a grain belonging to the aggregate undergoing a thermomechanical loading[ In particular\ the use of a Taylor assumption of uniform deformation is inappropriate here since it requires _ve independent crystallographic slip systems "see Hutchinson\ 0866#[ We use a modi_ed version of the Taylor model in order to accommodate the imposed deformation at the boundary of the macroscopic sample while respecting local kinematic constraints[ The model is based on the work of Parks and Ahzi "0889#\ who developed a constrained!hybrid approach to deal will polycrystalline materials whose crystals lack _ve independent slip systems to accommodate arbitrary imposed strain[ Lee et al[ "0884# reconstructed the model in a di}erent way\ which we will apply in the following to the case of two or less active systems per grain[ Bold fonts will stand for tensors or vectors[ 2[1[ The modi_ed Taylor model for two or less available systems per `rain Consider a grain with 01 potentially active systems a\ each system being char! acterized by its Schmid tensor Ra de_ned as Ra 01 "sa & na ¦na & sa #\
"4#
na being the unit normal to the slip plane and sa the unit vector in the direction of the slip[ Considering large plastic deformation compared to elastic strain\ elasticity is neglected and crystallographic slip "as operationally de_ned in Section 1[1 for our present purposes# is the only mechanism of plastic deformation assumed to be opera! ting in the grains[ Letting g¾ a − 9 "only easy!twinning is considered here# be the homogenized shear rate on the ath slip system within a given grain\ the traceless
Kinetics of strain!induced martensite
0510
deformation rate generated by two available slip systems\ a and b\ within the grain is therefore given by D g¾ a Ra ¦g¾ b Rb [
"5#
In the case of only one system a\ "5# is replaced by D g¾ a Ra [
"6#
Gab Ra [ [ Rb \
"7#
ab
Let us de_ne now the scalars G by
where [ [ denotes the double contracted product between the two second!order tensors Ra and Rb\ de_ned as Ra [ [ Rb R ija R bij [ In the case of two independent systems\ the 1×1 matrix G de_ned by the scalars Gab is invertible\ having matrix inverse G−0\ with components "G−0# ab\ and the shear rate on the ath slip system can be simply expressed "no sum on repeated superscript b# g¾ a "G−0 # aa "Ra [ [ D#¦"G−0 # ab "Rb [ [ D#[
"8#
For one active system\ one _nds that g¾ a 1"Ra [ [ D#[
"09#
The way of choosing the e}ective slip pair "or single slip# within a grain\ at a given time\ will be treated in Part 2[2[ At this point it can be noted that 55 pairs have to be considered in the case of double slip\ and 01 single slips[ Our material is described as a polycrystalline aggregate subjected to a macroscopic strain[ The traceless "incompressible# macroscopic velocity gradient L Þ is given by Þ D Þ ¦W Þ\ L
"00#
where D Þ and W Þ are the macroscopic deformation rate and spin\ respectively[ A central concept of the modi_ed Taylor model is\ for each grain in the aggregate\ to {{project|| the macroscopic quantity D Þ into a suitable subspace of the 4!dimensional traceless second!order tensor space[ This subspace is generated by the N independent Schmid tensors "each de_ned by "4## related to the N slip systems we considered previously "one or two in our case#[ Using a relay parameter "see Lee et al[ "0884# for a complete construction#\ one _nds Þ D "p [ [ ðpŁ−0 # [ [ D
"01#
to be the relation between the macroscopic deformation rate D Þ and the local one "assumed uniform within the grain\ and denoted D#[ In eqn "01#\ ðAŁ stands for the mean value "over the di}erent grains# of the tensor A de_ned as ðAŁ
0 Ngrains
Ngrains
s A\
"02#
0
Ngrains being the total number of grains in the aggregate[ For each grain\ p is a fourth! order tensor de_ned as
0511
p "G
J[ M[ DIANI and D[ M[ PARKS −0 ab
a
b
b
# "R & R ¦R & Ra #\ for two systems a and b "no sum on a and b# p 1Ra & Ra \ for one system a[
"03#
Also\ in eqn "01#\ the double!contracted product of two fourth!order tensors is given by "using Einstein notation# ] "p [ [ R# ijmn pijkl Rklmn [
"04#
To complete the interaction law in the modi_ed Taylor model\ the local spin W is equated to the macroscopic one as "05#
Þ[ W W
The rate of change of lattice orientation is determined by the spin tensor W ^ given by W W−Wp \
"06#
where the plastic spin Wp\ within a grain\ is given by N
Wp s g¾ a a 0
0
1
0 a "s & na −na & sa # [ 1
"07#
The current orientation for system a orientation is given by integrating set "08# ] s¾a W sa n¾a W na [
"08#
Remark 3 ] Note that the transformation rotation of the lattice within the shear bands is not taken into account in W ðsee "06#Ł\ although it can reach 69>29? for a twin ^ this simpli_cation is mostly justi_ed by the macroscopic scale at which the mechanical model operates\ which does not take into account the substructure formed by the bands[ Also\ once a band is formed\ it is assumed not to act any more in the simulation\ and new bands are occurring only in the mother phase[ In this sense\ from a mechanical point of view\ only the mean plastic rotation of the yet!to!be deformed portion of the austenite lattice\ given by "06#\ is relevant[ On the other hand\ we do not attempt to predict a _nal texture of our material ^ the latter should depend precisely on an accurate treatment of the lattice rotation within the bands and within their inter! sections[ Texture evolution due to twin bands operating in conjunction with dis! location slip has recently been treated by Staroselski and Anand "0885#[ In order to get the stresses we now specify viscoplastic constitutive equations for the shear rates\ g¾ a[ Following Asaro and Needleman "0874#\ we assume that g¾ a depends on the corresponding resolved shear stress\ denoted ta\ through a power law relation of the form ta sa
g¾ a m sign"g¾ a #\ g¾ 9
bb
"19#
Kinetics of strain!induced martensite
0512
where m is a rate exponent\ g¾ 9 is a pre!multiplying strain rate and sa is the slip system hardness[ The evolution of sa depends classically on the active shears ðsingle "b disappears# or doubleŁ rate through hardening moduli\ hab\ as s¾a haa =g¾ a =¦hab =g¾ b =
"no sum on a and b#[
"10#
Hardening moduli are given\ following Brown et al[ "0878#\ by sa a sb \ hab qh9 0− s½ s½
0 1
haa h9 0−
a
0 1
for a b
"11#
where h9\ a and s½ are slip hardening parameters which are taken to be identical for all slip systems[ q is the ratio of the latent hardening to the self!hardening rate "see Kalindidi et al[\ 0881# and is set to more than one "see Section 3# for non!coplanar systems ^ for coplanar systems " family a or b or c or d\ see Appendix 2# q is unity[ Finally "Lee et al[\ 0884# the macroscopic deviatoric stress\ noted Þ S\ is given by S ðpŁ−0 [ [ ðS Ł Þ
"12#
where S is a projected grain stress de_ned by S "G−0 # aa ta Ra ¦"G−0 # bb tb Rb ¦"G−0 # ab "tb Ra ¦ta Rb #\
"13#
where only _rst term "a# remains in the sum in "13# for a single activated system[ Remark 4 ] It is important to note that the rigid!plastic version of the modi_ed Taylor model is kinematically!based[ This means that\ assuming the active systems are known\ all the local kinematic variables can be determined from the global Þ#\ and can be incremented during calculation over a history of macroscopic ones "L deformation without any feedback of the stress[ 2[2[ Numerical resolution The set of equations described in Section 2[1 can be resolved using an explicit numerical scheme[ Let us assumed that for each increment ðt\ tŁ\ all variables are updated at time t\ the strain rate within each grain being noted D"t#[ The calculation starts with selection of potentially two active systems for each grain in the following manner ^ since "see 1[2# we assume that no more than two systems are active at a time in a given system\ one has to choose among 61 possibilities ] 55 pairs and 01 possible single slips[ Among the 55 pairs\ the optimum choice is de_ned to be the one selecting the pair having one of its two shears maximum over all possible pairs[ This can be expressed analytically\ using "8#\ by the following criterion ] "a\ b# $ "0\ 01#×"0\ 01# so that s "G−0 # ad "Rd [ [ D#"t# d a\b
or
s "G−0 # bd "Rd [ [ D#"t# d a\b
maximum and positive[
"14#
0513
J[ M[ DIANI and D[ M[ PARKS
On the other hand\ among the 01 single slips\ the optimum one has the maximum positive resolved shear strain and is selected by the criterion a $"0\ 01# so that Ra "t# [ [ D"t# maximum and positive[
"15#
The _nal candidate "pair or single slip# is the one with the largest maximum shear rate[ The _rst increment "t 9# is solved using the imposed macroscopic strain rate D Þ instead of D in criteria "14# and "15#[ The previous criterion expresses simply our choice that the active system is the one having maximum {{resolved shear strain rate|| calculated according to "8#[ From grain to grain\ the criterion will select di}erent active systems\ depending on the grain orientation with respect to the macroscopic strain rate[ The active systems having been chosen within each grain\ the projection tensor p"t# is calculated using "03#\ as well as its average by "02#[ The update of local strain rate D"t# is then obtained using "01#[ The shears g¾ a "t# being known\ rotations in each grain can be performed using "05#\ "06# and "07#\ providing the new Schmid tensor Ra "t# for each system[ The selection of the new pairs of systems "or single slip# within each grain can then start for the next increment[ At the end of each increment all variables of the model have to be updated[ First\ resolved shear stresses are calculated using "19# as ta "t# sa "t#
g¾ a "t# m sign"g¾ a "t##\ g¾ 9
b b
"16#
S "t# and Þ S "t# being obtained using "13# and "12#[ Equivalent macroscopic stress is then usually de_ned by Seq z21 Þ S[[Þ S[ Note that simple tension simulations "in direction 2# require imposed macroscopic strain rate D Þ to be of the form D Þ 00
Þ 22 −D \ D Þ 00 \ Þ 11 RD 0¦R
"17#
with scalar R being an adjustable value in order to keep ratio "S Þ11−S Þ00#:S Þeq\ lower than 9[90 at each increment[ f a "t# is also updated at the end of increment ðt\ tŁ using the rate form of eqn "2# ] g¾ a "t# ¾fa "t#`[
"18#
The shear ` is not derived in "18# "through it is temperature dependent# because we will deal in the following with loadings at _xed temperatures[ The values f a\ 0 ¾ a ¾ 01\ being known for each grain at each step of the simulation\ one can _nd in the Appendix a geometrical construction used to obtain the volume of each {{shear band||[ A characteristic length w "shear band thickness# is used in order to transform a volume fraction into a volume ^ the results have shown\ however\ not to depend on this characteristic length when it remains small in comparison to grain size\ h[ The volume of a {{tube|| of a?!martensite formed at the intersection of two shear bands is then deduced "Appendix#\ leading _nally to the value of the volume fraction of a?! martensite in a grain that is derived by assuming simply that all bands of system a are equidistant and periodically spaced[
Kinetics of strain!induced martensite
0514
3[ RESULTS In all the results shown in this section the equivalent plastic strain is taken to be Þ [[D Þ dt\ D Þ being the macroscopic strain rate tensor[ o¹ p Ðz12 D 3[0[ In~uence of initial texture As seen in Section 2[2\ the active systems\ whose slips are triggering the overall kinematics\ are selected during the calculation according to a {{maximum resolved shear strain rate||\ that depends naturally on the orientation of the considered grain[ The behavior of a complete polycrystalline aggregate might therefore depend on its initial texture[ To check the latter\ we performed the following numerical experiments ] we applied a pre!rolling on a {{virtual|| 293L stainless steel up to 29) in order to obtain an initial texture on which to perform a further uniaxial tension[ This initial pre!rolling texture was obtained using data from a _nite element calculation on a brass model developed by Staroselsky and Anand "0885#[ These authors deal with a visco!plastic f[c[c[ Þ9Łg slip systems polycrystalline aggregate\ in each grain of which the twelve "000# gð00 Þ00Łg systems[ This description is are active\ as well as some twinning on the "000# gð1 consistent with the warm!rolling of 293L stainless steels\ where the temperature is higher than the maximum temperature below which twins are prominently produced\ and\ as a result\ mainly dislocation motion is involved in the evolution of the crys! tallographic texture[ One can compare in Fig[ 4 the experimental texture of a 293L stainless steel rolled sheet " from DeMania\ 0884# with those obtained using the brass
Fig[ 4[ Experimental "left\ from DeMania\ 0884# and simulated "middle ] 29) brass!rolling\ 228 grains ^ right\ 29) copper!rolling\ 228 grains# initial textures "equal area projections#[
0515
J[ M[ DIANI and D[ M[ PARKS
Fig[ 5[ Prediction of the volume fraction of a?!martensite for simple tension in the rolling direction starting with di}erent initial rolling textures "isotropic\ 29) copper!rolling\ 29) brass!rolling# composed of 228 grains[
model[ Another pre!rolling "based on the work of Kalindidi and Anand\ 0881# was done using a Taylor assumption in a visco!plastic f[c[c[ polycrystalline aggregate\ where the deformation was carried out by only dislocations "twelve "000# gð00 Þ9Łg systems#\ and where most of the results presented by Kalindidi and Anand "0881# are derived for copper[ The brass!rolling results shown in Fig[ 4 were determined to be su.ciently discriminating for our purposes[ The model warm!rolling textures presented in Fig[ 4 were then used as initial textures to perform subsequent lower!temperature uniaxial tension in the rolling direction\ using the model presented in Section 2[ Results are reported in Fig[ 5[ The model shows the in~uence of the texture vs isotropic sample[ 3[1[ In~uence of the strain!state ] plane!strain tension vs simple tension DeMania "0884# performed experiments which indicate a strong in~uence of the strain state on the evolution of strain!induced martensite[ For that purpose\ she made two types of tensile specimen from sheet stock of 293L stainless steel\ oriented with the loading axis at di}erent angles with respect to the rolling direction[ One specimen type was for simple tension "ST#\ and the other was for plane!strain tension "PST# tests[ One can see in Fig[ 6 the measured amount of a?!martensite produced vs the equivalent strain at a temperature of −39>C ^ note that experimental values were downshifted by 9[8) since the original specimen already exhibited such a mag! netization ^ therefore\ the experimental amount of martensite shown in Fig[ 6 is
Kinetics of strain!induced martensite
0516
Fig[ 6[ Volume fraction of a?!martensite vs equivalent tensile strain of 293L stainless steel at −39>C "experimental data from DeMania\ 0884#[ The numbers in degrees denote the angle between the tensile axis and the rolling direction "RD#[ Results are shown for simple tension "ST# and plane strain tension "PST#[
representative of martensite produced during the testing "not the rolling#[ A simulation was done using our model\ since the predominant deformation mechanism is known to be o!martensite formation at −39>C\ with the nucleation of a?!martensite as a consequence "see Fig[ 1#[ The results are also reported in Fig[ 6 by solid lines[ The mean value of the shear in the bands\ which was found to be ` 9[34\ was adjusted to _t the experimental uniaxial tension curve of volume fraction of a?!martensite vs strain[ The corresponding curve for plane!strain tension is then obtained without further modi_cation of the model[ In this simulation we used an initial texture corresponding to 29) rolling brass texture "see Section 3[0#[ Although the dilute a? limitation of the model prevented simulation to values of o¹ − 9[1\ where the experimental evidence of reduced martensite under plane tension is clearest\ the model exhibits a consistent trend for smaller deformation[ This trend is con_rmed by stress:strain curves plotted in Fig[ 7[ The experimental data obtained by DeMania are reproduced by the simulation "see Section 2# using the following values for parameters "identical for simple tension and plane strain tension# ] ` is kept equal to 9[34 ^ g¾ 9 9[990:s and m 9[901 are typical values for metals and low strain! rate range experiments "Kalidindi et al[\ 0881# ^ s9 009 MPa drives the initial stress value\ s½ 799 MPa and a 15 driving the shape of the curves[ Parameters to be discussed are h9 29\999 Mpa and q 09 ] self!hardening h9 is usually large\ shear being here achieved through a coordinated movement between atoms\ involving therefore more energy than for dislocation motion ^ also\ hardening
0517
J[ M[ DIANI and D[ M[ PARKS
Fig[ 7[ Equivalent stress vs equivalent tensile strain of 293L stainless steel at −39>C "experimental data from DeMania\ 0884#[ The numbers in degrees denote the angle between the tensile axis and the rolling direction "RD#[ Results are shown for simple tension "ST# and plane strain tension "PST#[
on the _rst system due to transformation strain on the second one "latent hardening ratio q# is inferred here to be ten times larger than self!hardening in order for the simple tension stressÐstrain curve to be above plane strain one[ On the other hand\ a?!transformation is favored in simple tension "Fig[ 6# due to a greater amount of shear intersections ^ as a consequence the role of latent hardening is enhanced in simple tension in regard as plane strain one that is coherent with the result of Fig[ 7[ One could argue that the di}erence is not important and is increasing with the deformation\ the last point being in contradiction with experiment[ Since q is a constant and since the di}erence depends on the di}erence in f that is also increasing with deformation\ there is no way at this point to relieve such a contradiction[ One should remember\ however that a?!transformation is enhanced by stresses so that latent hardening should decrease with stresses or\ in our case\ with deformation ^ using an adjustable formula for q this should help things to _t more quantitatively with experiment[ 3[2[ Temperature!dependence of the model We have not yet discussed the in~uence of the shear amplitude in the band\ `\ which is a key mesoscopic parameter in our description[ The correlation with the microscopic deformation mechanisms is in fact very strong "see Section 1# although slightly ambiguous ] two di}erent sets of micro!mechanisms could lead to the same value of `[ To check further the in~uence of this parameter\ our model reproduced essential
Kinetics of strain!induced martensite
0518
Fig[ 8[ Experimental " from Angel\ 0843# and calculated formations of a?!martensite by plastic strain at various deformation temperatures in 07:7 CrÐNi stainless type steel[
features of the experimental results obtained by Angel "0843# and shown in Fig[ 0 by changing the value of ` to _t for di}erent temperatures[ The results are given in Fig[ 8[ The simulation was stopped when a volume fraction of a?!martensite of about 09) was reached\ since our method is only valid for {{small|| amounts of transformed phase[ The results range between ` 9[24 at the lowest temperatures\ which is the theoretical value for o!martensite\ and ` 9[6\ the theoretical value when only twin! ning operates[ From the _gure\ it appears that the latter condition occurred at a temperature below 9>C[ One can notice also that the curve corresponding to ` 9[24 does not _t the experimental data at a temperature of −077>C ^ this disagreement can either be considered from an experimental or a modeling point of view[ One can in fact argue that one deals here with stress!assisted martensite\ where the deformation mechanisms are slightly di}erent from the strain!induced ones "here o!band for! mation# are not taken into account in the model[ Finally\ by phenomenologically extending the range of the `!parameter to even greater values\ in some sense account! ing for dislocation!based shear banding\ the temperature trend shown in Fig[ 0 can be followed with the model to higher temperatures than shown in Fig[ 8[
4[ CONCLUSION The tool developed in this work for predicting the amount of strain!induced a?! martensite produced in a steel with low stacking fault energy relies on very simple physical hypotheses[ Detailed in Section 1 "H0ÐH3#\ these hypotheses assume that
0529
J[ M[ DIANI and D[ M[ PARKS
the inelastic displacement "including the transformation# is governed by coordinated movement of atoms along twelve equivalent "000# gð100Łg directions accounting for twinning and formation of o!bands ^ since the interaction of such bands is often strong\ only two of them are seen to cross within a grain\ producing a?!martensite at their intersections[ Finally\ using a modi_ed Taylor!based polycrystalline numerical model to simulate di}erent strain states\ we restricted the domain to small amounts of produced a?!martensite in order to minimize the e}ects of volume change[ Being so simple\ the previous description will apply {{as is|| only in few situations and for few materials[ We can see for example in Fig[ 6 that the experimental data we compare with are at rather high martensite values where the e}ect of volume change and interactions between a?!martensite tubes is perhaps not negligible ^ however\ we did not _nd smaller values in the literature\ and we believe that trends at the beginning of the a?!martensite nucleation probably in~uence the end[ In any case\ the amount of martensite produced during a loading of such a material appears to be closely a}ected by the strain state\ temperature and the initial texture of the material[ As a prospect the authors would like to suggest that the kinematic and kinetic description of the transformation "austenite to a?!martensite# seems to be a good skeleton\ at a good scale\ in order to build more sophisticated models[ These models could incorporate\ for example\ the e}ects of dislocation movement\ by using phenom! enologically the concept of a band[ In any event\ such a model could account pro_tably for higher temperatures[
ACKNOWLEDGEMENTS The authors would like to acknowledge the French DoD:DRET for support of this work[ Thanks are due to L[ Anand and A[ Staroselsky for helpful suggestions related to the use of Taylor!based numerical simulations[ We also thanks S[ Ahzi for interesting discussions on the use of the constrained!hybrid model as well as A[ D[ DeMania for information relating to her master thesis on 293L[
REFERENCES Angel\ T[ "0843# Formation of martensite in austenitic stainless steels[ J[ of Iron Steel Inst[ 027\ 054Ð063[ Asaro\ R[ J[ "0874# Texture development and strain hardening in rate dependent polycrystals[ Acta Metallica 22\ 812[ Brown\ S[ B[\ Kim\ K[ H[ and Anand\ L[ "0878# An internal variable constitutive model for hot working of metals[ International Journal of Plasticity 4\ 84Ð029[ DeMania\ A[ D[ "0884# The in~uence of martensite transformation on the formability of 293 stainless steel sheet\ M[Sc[ thesis\ Department of Mechanical Engineering\ Massachusetts Institute of Technology[ Diani\ J[ M[ "0881# Contribution to the study of the transformation induced plasticity at high strain rates\ Ph[D[ thesis\ Ecole Nationale Superieure des Mines de Paris\ France\ pp[ 43[ Diani\ J[ M[\ Sabar\ H[ and Berveiller\ M[ "0884# Micromechanical modelling of the trans! formation induced plasticity "TRIP# phenomenon in steels[ Int[ J[ En`n` Sci[ 22\ 0810Ð0823[ Greenwood\ G[ W[ and Johnson\ R[ H[ "0854# The deformation of metals under small stresses during phase transformations[ Proceedin`s of the Royal Society A172\ 392Ð311[
Kinetics of strain!induced martensite
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Hecker\ S[ S[\ Stout\ M[ G[\ Staudhammer\ K [P[ and Smith\ J[ L[ "0871# E}ects of strain state and strain rate on deformation!induced transformation in 293 stainless steel ] Part I[ Magnetic measurements and mechanical behavior[ Met[ Trans[ 02A\ 508Ð515[ Hutchinson\ J[ W[ "0866# Creep and plasticity of hexagonal polycrystals as related to single crystal slip[ Met[ Trans[ 7A\ 0354Ð0358[ Kalindidi\ S[ R[\ Bronkhorst\ C[ A[ and Anand\ L[ "0881# Crystallographic texture evolution in bulk deformation processing of FCC metals[ Journal of the Mechanics and Physics of Solids 39\ 426Ð458[ Kelly\ P[ M[ "0854# The martensite transformation in steels with low stacking fault energy[ Acta Metallica 02\ 524Ð535[ Koistinen\ D[ and Marburger\ R[ "0848# A general equation prescribing the extent of the austeniteÐmartensite transformation in pure ironÐcarbon alloys and plain carbon steels[ Acta Metallica 6\ 48Ð59[ Leblond\ J[ B[\ Devaux\ J[ and Devaux\ J[ C[ "0878# Mathematical modelling of transformation plasticity in steels*I ] Case of ideal!plastic phases[ International Journal of Plasticity 4\ 462Ð 477[ Leblond\ J[ B[\ Mottet\ G[ and Devaux\ J[ C[ "0875# A theoretical and numerical approach to the plastic behavour of steels during phase transformations*I ] Derivation of general relations[ Journal of the Mechanics and Physics of Solids 23\ 284Ð309[ Lecroisey\ F[ and Pineau\ A[ "0861# Martensitic transformations induced by plastic deformation in the FeÐNiÐCrÐC system[ Met[ Trans[ 2\ 276Ð285[ Lee\ B[ J[\ Ahzi\ S[ and Asaro\ R[ J[ "0884# On the plasticity of low symmetry crystals lacking _ve independent slip systems[ Mech[ Mat[ 19\ 0Ð7[ Magee\ C[ L[ "0855# Transformation kinetics\ microplasticity and aging of martensite in Fe!20 Ni[ Ph[D[ thesis\ Carnegie Institute of Technology[ Miller\ M[ P[ and McDowell\ D[ L[ "0885# The e}ect of stress!state on the large strain inelastic deformation behavior of 293L stainless steel[ J[ En`r[ Matls and Techn[ 007\ 17Ð25[ Murr\ L[ E[\ Staudhammer\ K[ P[ and Hecker\ S[ S[ "0871# E}ects of strain state and strain rate on deformation!induced transformation in 293 stainless steel ] Part II[ Microstructural study[ Met[ Trans[ 02A\ 517Ð524[ Nemirovskiy\ Yu[ R[ and Nemirovskiy\ M[ R[ "0875# Analysis of the interrelation of shear processes in intersecting planes "000# g>"990# o and the formation of a?!martensite during deformation of steels with low stacking fault energy of the austenite[ Fiz[ metal[ metalloved[ 51\ 642Ð647[ Olson\ G[ B[ and Cohen\ M[ "0864# Kinetics of strain!induced martensitic nucleation[ Met[ Trans[ 5A\ 680Ð684[ Parks\ D[ M[ and Azhi\ S[ "0889# Polycrystalline plastic deformation and texture evolution for crystals lacking _ve independent slip systems[ Journal of the Mechanics and Physics of Solids 27\ 690Ð613[ Patoor\ E[ "0875# Contribution a l|etude del la plasticite de transformation dans le mono et les polycristaux metalliques[ Ph[D[ thesis\ LPMM\ France[ Patoor\ E[\ Bensalah\ M[ O[\ Eberhardt\ A[ and Berveiller\ M[ "0882# Thermomechanical behaviour determining of shape memory alloys using a thermodynamical potential opti! mizing\ ed[ La Revue de Metallurgie:Science et Genie des Materiaux\ Decembre 0882\ pp[ 0477Ð0481[ Powell\ G[ W[\ Marshall\ E[ R[ and Backofen\ W[ A[ "0847# Strain hardening of austenitic stainless steel[ Transactions of the ASM 49\ 367Ð386[ Staroselsky\ A[ and Anand\ L[ "0885# Large deformation plasticity of FCC Metals due to slip and twinning[ Displacive Phase Transformations and Their Applications in Materials En`ineerin`\ ed[ K[ Inoue[ Stringfellow\ R[ G[\ Parks\ D[ M[ and Olson\ G[ B[ "0881# A constitutive model for trans! formation plasticity accompanying strain!induced martensitic transformations in metastable austenitic steels[ Acta Metall[ Mater[ 39\ 0692Ð0605[ Yang\ J[ H[ and Wayman\ C[ M[ "0881# Self!accommodation and shape memory mechanism of o!martensite*II[ Theoretical considerations[ Mat[ Char[ 17\ 26Ð36[
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J[ M[ DIANI and D[ M[ PARKS
APPENDIX A[0[ Geometrical calculation of {{shear bands|| volume fraction in a `rain In the following\ the grain is described as a cube of volume h2 containing {{shear bands|| of width w parallel to the four "000# planes "see Fig[ 09#[ The equations of those planes in the crystal frames are family a "systems 0\ 1\ 2# ] x¦y−z−a 9\ −h ¾ a ¾ 1h family b"systems 3\ 4\ 5# ] x−y¦z−b 9\ −h ¾ b ¾ 1h family c "systems 6\ 7\ 8# ] x−y−z−c 9\ −1h ¾ c ¾ h family d "systems 09\ 00\ 01# ] x¦y¦z−d 9\ 9 ¾ d ¾ 2h[
"29#
The intersection pattern of any one of these planes with the cubic crystal "see Fig[ 09# is an equilateral triangle or a hexagon\ the area of which\ S\ can be calculated as a function of the plane|s location\ according to the value of parameters a\ b\ c and d[ To _x the ideas\ the calculation has been done for the family a "see schematic in Fig[ 09#\ and the result is given by system "20# in the range −h ¾ a ¾"h:1# ^ results for range "h:1# ¾ a ¾ 1h are found to be symmetric\ with S being an even function of ða−"h:1#Ł[
Fig[ 09[ Cubic crystal where three planes of family a are drawn[
Kinetics of strain!induced martensite
S"a#
z2 "a¦h# 1 \ −h ¾ a ¾ 9 1
S"a#
z2 h ""a¦h# 1 −2a1 #\ 9 ¾ a ¾ 1 1
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"triangles#\ "hexagons#[
"20#
For the three other "000# family planes "respectively\ b\ c and d#\ similar results can be found by replacing parameters a\ respectively\ with b\ c¦h and d−h in "20#[ The volume fraction of Þ# platelet " family a# of width w at a location a is therefore given by "21# an o!martensite "000 and is derived from "21# for the other family planes[ "a# f 0\1\2 o
wS"a# 2
h
h \ −h ¾ a ¾ \ f 0\1\2 "h−a# f 0\1\2 "a#[ o o 1
"21#
After each increment of a simulation "see Section 2[2# the number of o!platelets is calculated using "2# and "21# by assuming the platelets to be equally spaced within the grain for a given system a[ A[1[ Geometrical calculation of a?!martensite volume fraction at the intersection of shear bands platelets The next step is to calculate the volume fraction of a {{tube|| of a?!martensite nucleating at the intersection of two {{shear bands||\ each of width w within a given cubic grain[ To _x the ideas\ we consider the families a and b[ According to "29#\ the equation of the line L forming the intersection of the planes containing families a and b can be written as line L \ x
a−b a¦b and z y− [ 1 1
"22#
In Fig[ 00\ one can _nd a cut of a cubic crystal according to a plane x constant\ where the
Fig[ 00[ Two di}erent traces in a plane x constant of intersection of two {{shear bands|| of family a and b[
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J[ M[ DIANI and D[ M[ PARKS
intersection of di}erent lines L with the crystal is drawn[ The length l of the intersection of those lines L with the cubic crystal can now be easily determined to be
0 b b1
l z1 h−
a−b 1
"23#
where =x= stands for the absolute value of scalar x[ Since the width w of both {{shear bands|| is assumed to be small compared to the crystal dimension h\ the length l is used to construct a tube of a?!martensite {{around|| line L[ According to Fig[ 01\ the area of the cross!sectional of such a tube is easily found to be "2:1z1#w1 [
Fig[ 01[ Cross!sectional of the intersection of two shear bands "tube of a?!martensite#[ Using "23#\ one _nds _nally the volume fraction of a tube of a?!martensite to be given by Table 0[
Table 0[ Volume fraction of a?!martensite as a function of the position of the selected intersectin` planes –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– — f a?a\b a b c d ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐ * 2 w1 a−b 2 w1 a¦c 2 w1 a¦d h− h− a 9 1 h2 1 1 h2 1 1 h2 1
0 b b1
b
c
9
0 b b1
2 w1 b¦c h− 1 h2 1
0 b b1 9
b b
2 w1 b¦d 1 h2 1
b b
2 w1 c−d 1 h2 1
b b
–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– —
Kinetics of strain!induced martensite
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A[2[ Shear systems in crystals coordinate "cubic lattice#
Table 1[ Characteristics values of the twelve "000# gð1 Þ00Łg stackin` faults directions in FCC materials –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– — Family System a na sa Ð*ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐ 0 0 1 0 0 a 0 0 − − − z2 z2 z2 z5 z5 z5 0 0 0 0 1 1 0 − z2 z2 z2 z5 z5 z5 0 0 0 1 0 2 0 − − − z2 z2 z2 z5 z5 z5 b
3
0
4
z2 0
5
z2 0 z2
c
6
0
7
z2 0
8
z2 0
09
− −
− − −
z2 d
−
0
0
0
0
−
0
1
z2 z2 0 0
z5
z2 z2 0 0
z5 z5 z5 1 0 0 − − z5 z5 z5
z2 z2 0
z2 0 z2 0 z2 0
− − −
0
0
0
z2 0
z5 1
z2 0 z2
0
−
z5
0
0
1
z5 z5 0 0 − − − z5 z5 z5 0 1 0 − z5 z5 z5 0
0
−
1
z5 z5 z5 1 0 0 − z2 z2 z2 z5 z5 z5 0 0 0 1 0 01 0 − z2 z2 z2 z5 z5 z5 –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– — 00
z2 z2 z2 0 0 0
z5 1
−