A micromechanical study on the coupling effect between microplastic deformation and martensitic transformation

COMPUTATIONAL MATERIALS SCIENCE

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Computational Materials Science 3 (1994) 307-325

A micromechanical study on the coupling effect between microplastic deformation and martensitic transformation F. Marketz, F.D. Fischer Christian Doyplcr Laboratory for Microrncchanics of Materials, University for Mining and Metallurgy , Franz-Josef-Str.18, A-8700 Leoben, Austria

Received 27 March 1994; accepted 19 May 1994

Abstract

Martensitic transformation is simulated in terms of computational micromechanics applying the finite-element method in order to study the effect of microstress and microstrain causing the unusual material behavior associated with stress-assisted maxtensitic transformation (MT). Both the preferred orientation of martensitic crystallographic variants (orientation effect) and microplastic deformation (accommodation effect) may lead to significant macroscopic deformations (transformation-induced plasticity, TRIP). The essential microstructurai features of MT are described by stress-free transformation tensors e__Tn for martensitic variants emerging along habit plane variants j. Transformation-induced microstress and accommodation strain fields are analysed for inclusion-type, plate-shaped martensitic variants transforming within a representative volume element (RVE) under different levels of externally applied stress. The RVE is specified with respect to an average Schmid-factor for the transformation shear 7. The second homogenization step is carried out by a microfield approach coupling the RVEs with antisymmetric boundary conditions. In order to describe the thermodynamic state of the stressed phases we persue a thermodynamic field concept by introducing a thermomechanical potential applicable for martensitic transformation with plasticity effects. Transformation criteria for stress-assisted nucleation and stress-driven growth enable to clarify the role of plastic deformation arising from the progress of transformation. It will be shown that the hydrostatic microstress fields developing at the plate ends of transforming martensitic crystals alter the stress-assisted nucleation characteristics. Microplastic strain stabilizes the austenite thermodynamically. The applied modelling technique also allows to study the coupling between orientation- and accommodation effect. A incremental formulation ~--~Tp E for TRIP will be discussed compared to existing relations.

1. I n t r o d u c t i o n In order to get a better understanding of the mechanical behavior of metallic materials undergoing a solid-solid phase transformation numerical simulations are carried out applying micromechanical methods. The macroscopic material behavior associated with stress-assisted martensitic transElsevier Science B.V. SSDI 0927-0256(94)00054-G

formation (MT) is controlled by the orientation of martensitic crystallographic variants and microplastic deformations caused by transformationinduced microstresses. Both the orientation- and accommodation-effect may lead to significant macroscopic deformations, a phenomenon called transformation-induced plasticity (TRIP). In the case of stress-assisted martensitic transformation

308

F. Marketz, F.D. Fischer~Computational Materials Science 3 (1994) 307-3~5

the macroscopic material behavior depends on rearrangements on the microscale with a typical size scale of micrometers. Since one has to expect inhomogeneous deformation fields on the microscale we apply numerical solution procedures in the context of computational micromechanics in order to achieve an understanding of local stress and strain fields and their effect on the kinetics of MT and TRIP. Micromechanics itself has been presented as a discipline combining microstructural aspects with material behavior on a macroscopic level [1]. Optimization of heat treatment processes or designing materials for desired properties by the aid of computational mechanics methods requires special material constitutive laws for the bulk material which account for the essential microstructural rearrangements occuring under arbitrary thermomechanical loading paths. The mechanical effect of phase transformations is described as a Tl~IPstrain increment which is a part added to elastic, plastic, thermal and creep strain rates for constitutive laws in incremental form. ----I~P has been proposed in forms derived from ge-neralization of uniaxial TRIP-experiments [2] discussed in section 3. In this work the formulation of~p_~ will be discussed in the context of a desiredcoupled micromechanics-thermodynamics-kinetics approach with special emphasis on micromechanics. Thermodynamics aspects are outlined in R.ef. [3], for the description of the evolution of the martensitic volume fraction ~M, termed transformation kinetics, see Ref. [4]. T R I P occurs when the martensitic product phase forms in an initially elastic stress field generated by an applied load. A remarkable elongation is obtained although a tensile specimen is loaded below the yield stress crr, A of the weaker austenitic parent phase. Experimental evidence to T R I P has been given e.g. by Gautier et al. [5,6]. In steels the role of stress in connection with MT results in special mechanisms such as stress-assisted nucleation, that means, selection of favorably aligned martensitic crystallographic variants (orientationeffect or "Magee-effect") and plastic accommodation as a mechanism of internal stress relaxation ("Greenwood-Johnson-effect"). These effects are interacting and can be treated along within a

--~E'-~P

thermomechanical framework including elasticplastic material behavior of the phases in order to investigate mechanical effects on MT which is the object of this paper. In the bulk material martensitic variants are not free to form without any constraints and generate transformation-induced microstress fields :=:/nt a as. a microstructural effect A transformation condition for stress-assisted MT derived from a thermomechanical potential makes the estimation of transformation progress in a microstress-field possible. Transformation-induced microstresses arising due to transformation of certain martensitic crystals can be numerically evaluated by finite element techniques (FE) [7,S], if one considers crystallographic variants as semicoherent particles [9] with transformation strain fields described by "stress-free" transformation tensors [10] which can be derived from kinematic theories [11] in the case of martensitic transformation. The object of this paper is at one hand a combined description of orientation- and accommodation-effects within a thermomechanical framework based on an explicit form of a thermomechanical potential G in the form of the Gibbs free energy formulation for strained solids. The driving force A G y acting on a phase boundary which is the jump [G V] in G across an austenitic-martensitic phase boundary will be calculated in dependence of transformation-induced microstress and -strain. A vector v is written in a bold-italic type style, v ® w (viwj) is the dyadic product and v .w (viwi) is the scalar product of two vectors. A second order tensor T is written with double superscripts and for a fourth order tensor, e.g. /2, capitalized calligraphic letters are used. The scalar product of the tensors S and T is written T : S (7~jSij). The determinant, trace and transpose of T_ are written det T, tr T and T T. An upper index V means that a quantity is given per unit volume.

2. M a r t e n s i t i c transformation m e c h a n i s m s Martensitic transformation (MT) is a polymorphic phase transition in solids [12]. It is diffusionless and sometimes denoted displacive since the constituent atoms do not interchange places.

F. Marketz, F.D. Fischer~Computational Materials Science 3 (1994) 307-325

MT proceeds by nucleation of the martensitic crystals inside the austenitic parent phase. From mechanics point of view a heterogeneous composite develops in the course of transformation with a martensitic product phase of different mechanical properties compared to the parent phase. The typical element of the martensitic microstructure is a plate (two-dimensional heterogeneity) or lath (one-dimensional heterogeneity) with small ratio of thickness to the other dimensions. Microstructure here means the arrangement of martensitic crystals of typical morphology. The first part of a martensitic crystal that forms is probably the midrib [15,14]. The interface between the midrib and the parent phase is a definite plane which is usually taken as the habit plane defined with respect to the axes of the face-centered-cubic (fcc) parent lattice. During MT in steels the lattice changes from the parent to the body-centeredcubic (bcc) or body-centered-tetragonal (bct) product lattice depending on carbon content. The Bain-distortion/3 transforms some cell in the parent lattice into the corresponding product lattice with the smallest uniform distortion defined by the lattice constants [11]. For a Fe-30.9% Ni alloy with lattice constants afcc = 3.591 A, abcc= 2.875 A

=

0

0

0

abcc/afcc

1.132 0 0 0 1.132 0 0

0

,

(1)

F__~=

I+A

.

(3)

0 With the kinematic theory Wechsler et al. [11] bridged the gap between crystallography of MT and mechanics aspects. The Bain distortion itself cannot result in the observed undistorted habit plane. There is, additionally, a fine-scale heterogeneous deformation P which leaves the product lattice invariant and produces a regular internal structure of the martensitic crystal (polysynteticallytwinned domain structure [12]). This lattice invariant deformation mechanism is in many cases a shear (lattice invariant shear, LIS) which occurs on the elements of the twinning system in Fe-Ni alloys with high Ni-contents where the habit plane is of {3 14 10}fee-type. Habit plane indices of the 24 martensitic crystallographic variants predicted by the WLR-theory are listed in table 1. From the knowledge of the Bain-distortion, the plane and direction of the LIS, crystal lattice correspondencies and the assumption that the habit plane which seperates parent and product phase is a plane of zero distortion and rotation, the amount g of the LIS, the amount and direction of shape change d and the habit plane normMs n j of 24 crystallographic variants or habit plane indices {h ]cl}fcc can be determined. In order to ensure both coherent twin boundaries and a coherent interphase boundary the Hadamard jump condition must be fullfilled which allows ~ to be written as [13]

0.801

(4)

The volume change for this alloy is calculated by dV : po = det(B) = 1.026. dVo p --

309

(2)

MT is accompanied by a change in the shape of a region transformed becoming apparent as a surface relief of a specimen described by a deformation gradient ~ written with respect to a local system < z~ >. ~ which can be seen as the average deformation gradient for the transforming unit [13] is a combination of a simple shear 7 along and a dilatation A perpendicular to the habit plane which is left unrotated and undistorted,

d describes the magnitude and direction of the shape change, n is the unit normal vector to the habit plane. With respect to a crystal reference system < x~ > defined by the crystal axes of the parent lattice the habit plane unit normal n l and dl of a variant j = 1 have been determined by the kinematic theory, see [15], n~ = (0.1848 0.7823 0.5948) T ~ (3 14 10) T,

(5)

d~ -- (-0.0472 0.1601 - 0 . 1 5 2 1 ) T,

(6)

The magnitude of shape change is ]dll = 0.2258, the angle between n [ and d~ is fl = 83.4 ° . From

310

F. Marketz, F.D. Fischer~Computational Materials Science 3 (199,~) 307-3~5

Table 1 Habit plane indices of crystallographic variants for a Fe30Ni-alloy and corresponding transformation strains in y-direction of externally applied stress No.

Miller Indices

{:T,33

1 2 3

(3 14 10) (3 1"4 10) ('i'4 3 10)

0.1121 -0.0561 0.0089

4

(14 3 10)

0.0172

5 6 7 8 9 10 11 12 13 14

(5 ~ 10) (5 14 10) (14 3 10) (14 3 10) (14 10 3) ( ' ~ 10 3) (]"0 14 3) (T'O"i4 3) (I4 ~ 3) (14 ~ 3)

15

(10 ~ 3) (10 14 3) (3 10 14)

O.1093 -0.0620 0.0092 0.0258 -0.0799 0.0742 0.1085 -0.0572 -0.0796 0.0830 0.1141 -0.0652

16

17 18 19 20 21 22

23 24

-0.0506

(5 ~'0 14) ('I"03 14) (10 3 14) (10 3 14)

0.0792 -0.0180 0.0279 0.0057 -0.0082

(T'03 14) (3 10 14) (3 T'O14)

-0.0785 0.1110

geometrical relations shown in Fig. 1 7 and A are computed as 7 = 0.22 and A = d l . h i = 0.026, A evidently corresponds to the transformation volume change. The stress-free transformation tensor in Lagrange formulation written with respect to the local reference system < z~ > defined with respect to the habit plane is

ogo 1 FIT Fi - z= = & = ~=_=.~=_=r

6

,

(7)

0 with 6 = A + 1( A2 + (1 + A 2) sin 2 7) = 0.052. represents an invariant plane strain (IPS). T h e condition for an IPS is d e t ( ~ ) = 0. Note that

y,

1

/ ~'

I+A

l

_ X~ i--

1 Fig. 1. Kinematics of invariant plane strain.

the stress-free transformation tensor e_~,j for each of the 24 variants is identical with respect to the local reference system < x~ >j defined with respect to each variant j. Transformed to a common global reference system < x, > by e=TJ = Qj e2ir Q~T the transformation strains become orientation dependent. The components of the orthogonal transformation tensors Q. are the direction cosines be=3 tween < x~ >j and < xi >. The stress-free transformation tensor can be applied in the case that the domain-structure is frozen, that means, the assumption is taken that the domain-boundaries seperating regions with different Bain-variants have a much lower mobility than the interphase boundary between parent and product phase. However, this is not the case for shape memory alloys where the material behavior under loading in the fully martensitic state is governed by a reorientation of martensitic crystallographic variants due to high mobility of twin boundaries. One has to distiguish between the LIS which is a part of the transformation mechanism and plastic deformation in the phases that only occurs if the yield stress in one phase is exceeded. For the continuum mechanical treatment as follows invariant plane strain deformation modes for MT are taking into account.

F. Marketz, F.D. Fischer/Computational Materials Science 3 (1994) 307-3~5

=

=

+

2)

311

(9)

MACRO P

~

MICRO

p(r) .J.

u---M-- ' / 'xL. ',\ . ~ A _I

:

<

t

/\ ~ l

O'ip ~ij

E'j'T

"'~

MESO Fig. 2. Size scales considered: Macroscale (bulk material), rnesoscale (mesodornain) and microscale. 3. M a r t e n s i t i c t r a n s f o r m a t i o n and micromechanics Macroscopic thermomechanical constitutive relations from a continuum point of view for inelastic material behavior are based on essential microscale mechanisms of deformation [16]. Structural rearrangements of material elements on the microscale by a phase transformation can be related to corresponding increments of macroscopic strain ~E--TP accounting for transformation-induced plasticity ( T R I P ) effects. Taking additive decomposition of strain increments for a material point of the bulk material the total strain increment reads as a sum of an elastic, thermal and inelastic part /~ the latter of which comprises terms related to c ~ s i c a l plasticity .=:=p E , creep ~-cr E and an extra term related to T R I P __E___Tp,

The mesodomain state variables, the stress state _~ and the temperature T are volume averages over local quantities. ~_, .4 and B_B_are tensors having to be determined adjusted to microscale mechanisms. Taking into account the crystallographic relations of MT defined with respect to the lattice directions one may switch to the microscale on which a material point is in the martensitic or austenitic state. In order to account for essential microstructural features of MT and for applicability of continuumbased constitutive relations for microscale computations, a micro-meso approach has to be introduced as proposed in Refs. [18,8]. The crystals of a polycrystalline mesodomain are described with respect to their crystal orientation g -- g(51, (I), ~b2) in a global system described by a set of angles ¢1, (I) and ¢2 in Euler space. The mesodomain is regarded as a region with an average crystal orientation determined by calculation of an average maximum Schmid-factor --T m = m* (y) in Euler space for the transformation shear 7 [19]. For finite-element calculations a representative volume element (RVE) takes the role of the mesodomain. If MT interacts with an applied stress ~ the thermodynamic state of the mesodomain can described by a thermomechanical potential G in the form of Gibbs free energy which can be derived by extending the form of Rice [16] for strained solids by internal variables describing microstructural rearrangements by MT which arc obviously the stress-free transformation tensors. G results from volume averaging over thc RVE [18] as

G=G(E,T,(M, , < e_p >)

(8) E__Tpis proportional to the transformation progress described by the increment of transforming volume fraction dM depends on the thermodynamics and kineti~ssof MT affected by stresses and strains on the microstructural scale. Transformation progress described on the size scale of a mesodomain (Fig. 2) is usually denoted as transformation kinetics [4]. A general incremental formulation for _E__Tpis given by Tanaka et al. [17],

[2].--E-----TP

= F a n + FIB + ~ c + Go.

(10)

FG9 and I'tB are interfacial energies related to grain boundaries and interphase boundaries, ~ c is the temperature dependent part of G, which is the free energy of the stress-free phases [3] expressed in terms of temperature dependent material functions [20]. Go is the stress-dependent part being of special interest for computational micromechanits investigations [8] since it depends on the complex microstress and -strain states emerging from the interaction of the orientation- and accommo-

F. Marketz, F.D. Fischer/Computational Materials Science 3 (1994) 307-3~5

312

dation effect. G,, is composed of strain energies U0 from load stress _~ and Uint from transformationinduced microstress ~ n t ' some part of plastic work 6Wp, & ~, 0.05, and II,

a a = Uo + Ui,~t + &Wp + II,

(11)

II is the potential of the loading mechanism for a process under dead loading, II = - V (E_: E) = - V t r ( ~ E ) .

(12)

E is the total strain tensor. Disregarding the interfacial energies G is written finally as

G = ¢ c + Uo + Uint + 5Wp - V (_E: E).

(13)

Assuming an arbitrary stage of transformation the RVE contains a certain volume fraction ~M of martensitic variants grown along habit planes with normals nj with transformation strains e_,rj , ~M = E ~M,j -- VM V -- 91 VM,j, j=l j=l

(14)

and parent phase regions with the volume fraction ~A = 1 -- ~M. VM is the volume of the martensitic phase, VMj the volume of martensitic variants with the habit plane normal n j , ~Mj is the corresponding volume fraction. The average stress-free transformation tensor is calculated as <

1 >= V

24

24

= j----1

(15) j=l

The total strain increment in a material point with position r within the RVE is 24

= L(_r) + ¢='p(_r)+ ~o(r_) + E

~_~rjOj(r), (16)

j=l

where Oj (r__)is a set of indicator functions and r is the position vector.

Oj(_r)=

1 if 0 if

r_ E VM,j r_ ¢ VM,j.

(17)

If martensitic transformation (MT) in steels is assisted by an externally applied stress E transformation kinetics are dominated by stress-assisted

nucleation (SAN) yielding preferentially oriented martensitic variants. Nucleation sites for heterogeneous nucleation exist primarily as interfacial defects at grain boundaries or interphase boundaries. Stress can assist nucleation at the same nucleation sites where MT on cooling starts from. SAN in this point of view can be described by the thermodynamic effect of stress with the mechanical driving force AG~. If the yield stress of the austenitic parent phase o'y,A is exceeded new nucleation sites due to microplastic accommodation deformation are created (strain-induced MT) [21]. The formation of a martensitic microregion can stimulate the nucleation of others. Such a mechanism both involves stress-assisted and strain-induced nucleation governed by stress and plastic accommodation strains. If no accommodation occurs during SAN the orientation effect can be explained in terms of the local mechanical driving force AG y [22]. The driving force for SAN of a variant j is

xay = + AaL = ¢ vC,A -- ¢C,M v +

= + ~nt

):

~--'r,j"

(a8)

which is the sum of the chemical and mechanical driving force AGVa,j dependent on microstress. If microstress coincides with macrostress (¢ri t = 0) as it was assumed by Patel and Cohen ~3q, SAN can be described by the mechanical driving force depending on applied stress E__ v AGa,pc = =~ : e=T(0, ¢ , ¢ ) .

(19)

Maximizing A G V p c with respect to the Euler angles 0, ¢ and ¢ which corresponds to minimizing a thermomechnical potential disregarding elastic and plastic deformation of the phases but taking into account the orientation dependence of transformation strains the change in martensite start temperature M' = M,(E__) has been successfully described [23]. The behavior of bulk materials is taken to be governed by heterogeneous, classical nucleation with a semicoherent coherence state of strain relaxation. Aspects of martensitic nucleation concerning critical nucleus structure, state of overall strain relaxation and degree of heterogeneity are overviewed in Ref. [9]. With the invariant plane

F. Marketz, F.D. Fischer/Computational Materials Science 3 (199~) 307-3P5 strain assumption for a critical martensitic nucleus [24] and using a type of nucleation criterion specifying a critical nucleation level a local transformation condition [3] writes AG[. = + >_ v (20) Experimental evidence to transformation-induced plasticity has been given in most cases by uniaxial experiments (E~y ~ 0, all other Eli = 0), stressing a parent phase specimen, keeping Eyy constant and .providing the driving force for transformation by 7' < 0. Results of such nonisothermal "creep"tests [5,6] show a linearly increasing TRIP-strain ETp,~y with increasing Ey~ for applied stresses significantly lower than the yield stress of the parent phase tyy, A expressed by the Greenwood-Johnson relation 5 A E,rp,uy = K Euy f(~M) with K - 60"y,A (21) for diffusional transformations. For this type of transformation without any orientation relations between the phases an incremental formulation ~_~p has been derived by generalization to the ~-dimensional case [2]

taking several assumptions, one of them being that thc average stress deviator in the parent p h ~ e is equal to the deviator S of applied stress ~. &p in this case describes the accommodation effect. In the extended Greenwood-Johnson concept [25] Eq. (22) is adjusted to martensitic transformation by considering the effect of the transformation shear 7 on accommodation in the parent phase and taking a mean yield stress or} instead of o'v. A so that

32) 1/2

4e v 5( K - 3 Cr----~y- 6 cr-----~y ~2 + 37

313

E__~rp= < e__ r >dVM ~M v~

eT V < S A > ~M,

(24)

-- 2 CrV,A

which is somewhatcontradictoryto concepts working with mechanicaldrivingforces. Inoueet al. [27], [28] have proposed a formulation '~

&. =

+

3

1

E

(25)

i

I represents the transformation type (eg. I = 1 martensitic, I = 2 pearlitic). This formulation considers the transformation volume change expressed by ~3/ determined from experiments. For the accommodation effect the formulation of eq. (22) is used with constants K, determined by experiments. For martensitic transformations in steels we pers u e the concept of splitting E~_Tp in two parts corresponding to the micromech-gnical effects of MT, one corresponding to the orientation- (&t.,or) and the other one to the accommodation-effect (E__TV,.cc). Within this micromechanical approach the strain-increment __£----£---~vv,odue r to the orientationeffect results from the sum of the contributions of each of the variants formed during the increment of transformation progress

&P,o

> vM

=<

=

(26) J

A variant j can be formed if transformation condition (20) in the averaged form 1

VA (23)

In the case of shape memory alloys without plastic accommodation ( £ p = 0 ) t h e inelastic deformation behavior is governed by the orientation effect only. Sun et al. [26] set & p also proportional to ~M and the average transformation strain < e_T _ >dVM of the transforming region, with volume dVM. However, it is assumed that E__q,p is colinear to the average stress deviator < S A > in the matrix phase,

V ¢~i'" = A c V + < g a > : C--T./ -> WN,

(27)

is fullfilled, < cra > is the average stress tensor in austenite. A v~nant on the other hand does not transform (~M y = 0 ) i f AGV,~ < W v c,it. With the assumption to be justified '~ateron that stressassisted nucleation and not the growth of variants is FIGURE 3 the governing mechanism for the orientation effect which leads to ~M,j proportional to A G v j , conditions

F. Marketz, F.D. Fischer/Computational Materials Science 3 (199~) 307-325

314

-~- - 2 ~ -

4. P r o b l e m f o r m u l a t i o n

-t- -2------~--~- 7 ~ _ . . - +

Fig. 3. Periodic microstructure achieved with antisymmetry boundary conditions, one oriented martensitic variant depicted. (solid lines : grain boundaries, dashed lines : edges of representative volume element

In the fully austenitic state the material is taken to be elastically homogeneous and stress free initially. An externally applied load then causes a homogeneous stress state ~ taken to be uniaxial tension Evu as in the case of non-isothermal "creep"tests. At the martensite start t e m p e r a t u r e M~ = Me (Evv) transformation begins by stress-assisted nucleation of a favored variant with corresponding e_rj., see Eq. (31)• The m a x i m u m mechanical driving force is A G~,~. = ~ : ~ , ~ .

: (28)

=

J and

L -

-

AG},j

(29)

lEuu(~sin20pccosc~+6coQOpc).(32)

8pc is the angle between the direction of applied stress Evv and the habit plane normal n j . of a variant with m a x i m u m mechanical driving force [23], Opc= ~ arctan

,

(33)

J hold. The sum is taken over all A G V j enabling a nonzero transformation increment ~M,j. Finally, the strain increment for the orientation effect is written (30)

~M is determined by a macroscopic transformation kinetics relation, the determination of the average stress tensor < a___A > for the mechanical driving force AGaVj is done by computational micromechanics. Also the interaction between E__TP,or and

E____TP,acccan be understood if the local stress state within and around a martensitic variant is ~i " me t hods. c ~ ctu l a t e d by use of numerical

3' sin 20pc cos a + 6 sin 2 OPt ~,i"

=

o~ is the angle enforcing the average Schmid factor m""r = 0.426 [18] - -

1

.

m* = m* (g) = ~ sm 20pc cos a.

(34)

We restrict ourselves to a simple heterogeneous composite problem in the early stage of transformation near M,~. One single plate shaped crystallographic variant j* transforms. The shape of the variant and consequently its volume fraction ~M are prescribed by meshing (Fig. 4). Transformation-induced microstresses build up due to the incompatible strain field emerging due to transformation• Parameter studies with respect to a varying volume fraction ~M and applied stress level IEvy are carried out in order to compare the local fields and estimate their effect on the transformation progress•

-- ~ cos 20pc cos a + ~ sin 20pc --

6 7 cos 20pc cos a + ~ sin 20pc ~ sin 20pc cos a + 6 coQ Opt

0

sin a

0

0

(31)

F. Marketz, F.D. Fischer~Computational Materials Science 3 (199g) 307-325

315

tA : ~=e,A+ ~=p,A'

(a)

tM =

+ #,M +

,

(35)

The elastic strain increment within a transformation strep links with the stress increment of transformation-induccd microstresses by ~e,A = t : - ' : ~nt,A' ~e,M ----f:-I : ~n,,M for E = 0 since t =£-1 :a

= c - ' : (N + a,.,) = £ + '_-0,n,

(b)

Fig. 4. (a) Finite element mesh with depicted plate shaped inclusion-type variant to transform, volume fraction ~M = 0.82% (midrib stage). (b) Finite element mesh with depicted plate shaped inclusion-type variant to transform, volume fraction ~M = 2.61% (plate stage).

(36)

£ is the fourth rank elasticity tensor taken to be identical for both phases. The transformation step is taken to occur isothermally so that thermal strains are disregarded here. Once a certain region has been defined to transform the transformation strains which are the components of e__T,j, are imposed incrementally during a normalized time interval 0 < t' < 1. During this transformation step material properties change from the parent to the martensitic phase so that the current yield stress is try(t') = ttO'y,M + (1 - tt)o'y, a. During an increment of equivalent plastic strain dp,v the increment of current flow stress is ]2 = h~p,v, h is the the plastic hardening function h-1 = Epa _ E - l , t/,'p is the plastic tangent modulus, E the Young modulus. Material input data are listed in Table 1. The yield condition is of the form F=3J2-y2=0,

(37)

J2 = Sij Sij/2 is the second invariant of the Kirchhoff stress deviator _S. The plastic part of the strain increment in each phase satisfies normality t = AS, ~ > 0 being determined by the consistency condition F = 0.

4.1. Plastic constitutive relations Due to the introduced micro-meso linkage deformations of the parent and product phase materials are taken to be described by J2-flow theory with linear isotropic hardening. Infinitesimal elastic strains and the rate of deformation as strain rate measure leads to an additive strain rate decomposition written for parent phase regions and transforming regions in any stage of transformation under Eyy = 0 as

4.2. Micromechanical model In order to study transformation-induced microstress and -strain distributions within and around plate shaped martensitic variants we define a representative volume element (RVE) defined with respect to the polycrystal average of the Schmid factor m* for the transformation shear 7. The material in the early transformation stage

316

F. Marketz, F.D. Fischer~Computational Materials Science 3 (1994) 307-325

Table 2 Material input d a t a for numerical analyses

Young Modulus E [MPa] Poisson Ratio v Yield Stress O'y [MPa] Plastic Tangent Modulus E p [MPa]

is modelled by an antiperiodic arrangement of RVEs enforced by antisymmetry boundary conditions (ABC). An ABC has been introduced by Tvergaard [29] for a micromechanical model with axisymmetric configuration. For the generalplane-strain micromechanical model used here on each edge of the RVE compatibility with neighboring cells is achieved with an ABC. The edge displacement fields of opposite cell edges are related to each other by a 180°-rotation. A detailed formulation of these boundary conditions is given by the authors in Ref. [18]. The resulting periodic microstructure is shown in Fig. 3. The average straining is determined from the average displacement gradient with components calculated from the displacements of two antisymmetry points only. The average stresses are the components of E. The transformation-induced microstress state is therefore self-equilibrating < ~nt >=/~,u

dV = 0

(38)

v

The micromechanical model applied here approximates the microstructural geometries of the actual system, an initially single phase material with an emerging new phase. Although resulting microstructures are usually random we take use of a mutual periodic distribution of crystallographic variants in an early transformation stage. The orientation effect due to interaction of overall stress with transformation strains is assumed to occur equally favored in any grain of a polycrystalline material. Initiation and progress of transformation are controlled by by the chemical and non-chemical free energies of the system. A dominant part of the latter part is the mechanical part considered here. Grain boundaries and

Austenite

Maxtermite

210000 0.30 250 2000

210000 0.30 800 4000

other lattice imperfections can be considered as a preferential sites for martensite nucleation. Grain boundaries on the other hand hinder the growth of a martensitic variant. Material behavior under MT will therefore depend on the distribution of imperfections which is not taken into account here. In order to make local fields visible within a RVE an arbitrary deterministic grain boundary distribution is assumed depicted in Fig. 3 so that the length size of a crystallographic variant is limited. Since we do not want to predict microstructure development and equilibrium martensitic variants but only to determine local fields around a plateshaped variant the model is chosen this way.

5. R e s u l t s 5.1. Transformation-induced microstress distribution

In order to study the development of transformation-induced microstresses (TIMS) we have reduced the problem to a plate shaped inclusiontype variant with transformation strains to be accommodated. In Fig. 5(a) the yy-component of rr is shown. TIMS develop prevalently at plate f:=in t ends but not along the habit plane itself. The stress distribution ayu in direction of externally applied stress Euy = 125 [MPa] is depicted in Fig. 5(b). TIMS states are rather complex if variants transform in the bulk within a surrounding matrix material and are, of course, dependent on the shape of the martensitic phase. There are differnt mechanisms of relaxing TIMS: the change in the domain structure of martensitic variants and self-accommodation (transformation of "compos-

317

F. Marketz, F.D. Fischer//Computational Materials Science 3 (1994) 307-3~5

$22

[ M"~a]

(a) $22

MPa

~yy

(b) Fig. 5. (a) Transformation-induced microstress distribution aint,yy, no externally applied stress ()2yy = 0), ~M (b) Stress distribution cryy, externally applied stress Eyy = 125[MPa], ~M 0.8'2%.

=

0.82%.

=

ite variants" with small average transformation strains) are not considered here. Plastic deformation in the parent and product phase are the only accommodation mode permitted in the current study. The equivalent plastic strain ~P,V arising from T I M S in the case E~y = 0 is shown in Fig. 6(a). The components of the plastic strain tensor :v e are almost colinear to the transformation tensor e=T,j.. This means if the transformation strains average out, < e_.r > = 0, the same can be expected for the plastic accommodation strains. The amount of plastic accommodation strains is

limited by the transformation strains. M a x i m u m plastic strains arising from T I M S are computed to be about one third of the m a x i m u m transformation strain in direction of erternally applied stress. Plastic straining decays from the plate tips to the RVE edges. The situation changes if MT occurs in an applied stress field with the applied stress level taken Ey~ = 125 [MPa] which is half the parent phase yield stress o'y,A. Due to the applied stress level it is easier to exceed cry, A in the parent phasc. This is why plastic strains are higher and thc re-

F. Marketz, F.D. Fischer~Computational Materials Science 3 (199~) 307-3~5

318

PEEQ

%%

(a) PEEQ

~yy

(b) Fig. 6. (a) Plastic equivalent strain distribution ep, v, no externally applied stress (Eyy = 0), ( M = 0.82%. (b) Plastic equivalent s t r a i n d i s t r i b u t i o n ep,v, externally applied stress ~y~ = 125 [MPa], (M = 0.82%.

gion of plastic accommodation is larger as shown in Fig. 6(b). The difference of the two accommodation strain fields of Figs. 6(b) and 6(a) are then to be attributed to the Greenwood-Johnson effect accompanying MT. The local driving force for MT is the difference G~ - G ~ in the Gibbs free energies between the parent phase and the martensitic phase. Deformation of the parent phase changes the mechanical part of GAv. GA y can then be computed as the average over the local area around the centroidM position a finite element,

a ~ -- -~ Cv, A + s w , ~ + v ~

-(~_ + z): (,__0+ ,__,)•

(39)

GVA decreases due to plastic deformation, that means the driving force is reduced, the accommodated parent phase stabilized. In order to gain an estimation on how the TIMS field affects MT an extended Patel-Cohen concept [23] based on the mechanical driving force AGaVj is persued. Since for each one of the 24 stress-free transformation tensors e__T,j the trace is tr(e__Td ) = 6 and, therefore,

F. Marketz, F.D. Fischer~Computational Materials Science 3 (199~) 307-325

SES

319

(a)

{M2a]

(b)

[MPa] -4.82E+02 -4.17E+02 -3.51E+02 -2.86E÷02 -2.20E+02 -1.55E+02 -8.96E÷01

",.

-2.40E+01 ÷4.14E+01 .I.06E+02 *1.72E*02 .2.38E÷02 ÷3.03E*02 +3.69E*02

Fig. 7. (a) Mises equivalent stress distribution av, externally applied stress Eu~ = 125[MPa], ~M ---- 0.82%. (b) Transformation-induced hydrostatic stress distribution p, no externally applied stress Eyy = 0, ~M : 0.82%. under a hydrostatic stress state with hydrostatic pressure p the mechanical driving force is AGaV,j = p 6 which is the same for each variant. The orientation effect vanishes under a hydrostatic stress state. In Figs. 7(a) and 7(b) Mises equivalent stress g V ( ~ n t ) and hydrostatic stress (p > 0 hydrostatic pressure) are depicted. Hydrostatic stress states develop at the plate ends but only hand in hand with a non-zero Mises stress. That means that TIMS do not cancel the orientation effcct.

5.2. Thermodynamic force along phase boundaries Accommodation of the transformation strains and interfacial mobility are considered as two major elements fundamental to the discussion of martensitic growth [30]. tIere we determine the exact size of the relaxed plastic zones dependent on martensite plate shape and applicd load level. It is reported in the literature that first the midrib of a crystallographic variant is formed which is a straight thin region dividing the final plate in

320

F. Marketz, F.D. Fischer~Computational Materials Science 3 (1994) 307-3,$.5

half [14]. Afterwards growth continues sideways on both sides until a restrictive force may stop it locally. The change in the thermomechanical potential G associated with the local growth of a new martensitic phase at the expense of an old one can be treated as a thermodynamic force AG/v acting on a phase boundary which is the jump [G v] across the interface. ~Gv] can be derived by a variation of C applying the time derivative of a material integral which includes the jump in the integrand along the phase boundary [3]. Taking identical elastic constants for austenite and martensite AG/v writes as a sum of a chemical AG~, c : ¢ Cv , A - ¢C,M v and mechanical part AG~a

ac•

= aCY,c + a c L .

10 5 0

-15

-20 -25 -30 -35

(40)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Our point of interest is the trend of the mechanical part along the phase boundary of the plates depicted in Fig. 4 written again for £A = ~-,M [3]

Interface Coordinate d I A --[] O ..... 0

A C I,a v = ~(w~a -- W ; v , M ) --

1 ~(KA + G M ) : (-~,a --~,M -- -~-Tj')'

(41)

and its dependence on the volume fraction ~M of a variant. In order to vary ~M and make the numerical results comparable the finite element mesh is changed by variation of element sizes. Two meshes with plate shaped variants of volume fractions ~M = 0.82% and ~M = 2.61% are depicted in Figs. 4(a) and 4(b). Two further cases with ~M = 1.45% and ~M = 2.13% are taken into account. The upper phase boundary of the plates is taken into account. Local stresses, strains and specific plastic work are taken from the center of corresponding two finite elements on either side of the boundary. Fig. 8(a) shows AG}(,a along the phase boundary described by an interface coordinate dl (dl = 0 : lower right end, d l = 1 : upper left end of plate). In regions of significant TIMS (near plate ends) A G ~ decreases significantly. On the other hand AG/V¢ also decreases with increasing (M but the curves take a similar course. The same holds for for AGy,~ dependent on applied stress level E~v depicted in Fig. 8(b) for ~M = 0.82%. That means during certain growth stages of single variants local fields remain similar. This justifies the TRIPstrain increment for the orientation-effect to be

~M=0.82% (Midrib) ~M=1.45% ~M=2.13% ~M=2.61%(Plate)

(a)

30 20 10

>

0

0

-I0

I 1

-20

~

at- ~ - ~ - b - - g t . .

b

&

-30

':,A"

-4C

, 0.0

i , i 0.1 0.2

\ ,

i , , 0.3 0.4

.

. . 0.5

.

. . . 0.6 0.7

, 0.g

•

, 0.9

• 1.0

Interface Coordinate d I /x - - [] .... C) - -

E=0MPa E= 125 MPa ~3= 225 MPa

(b)

Fig. 8. (a) Mechanical part A G ~ a of thermodynamic force on phase boundary AG v dependent on volume fraction fM. (b) Mechanical part AG/Va of thermodynamic force on phase boundary AG V dependent on externally applied stress Gy u.

F. Marketz, F.D. Fischer~Computational Materials Science 3 (1994) 307-3~5 derived from a criterion based on the mechanical driving force rather than of a transformation condition for martensitic growth.

5.3. Stress-assisted nucleation tendency The coupling mechanism between plastic deformation and martensitic transformation can be considered under various aspects. The effect of plastic deformation on transformation kinetics has been discussed in terms of strain-induced nucleation kinetics [21]. However, in the continuum mechanics treatment aspects to be considered are different. Working with transformation conditions on the mesolevel [31], [3] for the RVE it is possible with the introduced thermomechanical potential of eq.(10) to predict the arrangement of variants in an elastic-plastic parent phase material concerning shape. It has been shown that plate-shaped variants with a larger length-width ration produce least accommodation plasticity and are thermodynamically favoured [18]. To estimate the effect of TIMS on MT the concept of a distribution of the wj . depenmaximum mechanical driving force AGa, dent on the local stress state is persued here as a first step referring the local transformation condition of Eq. (20). AGvb.-distributions in the parent phase are depicted in Fig. 9(a) for Euy = 0, in 9(b) for Eyy = 125 [MPa] and in 9(c) for Eyy = 125 [MPa] and ~M = 2, 61%. The highest values of mechanical driving forces are computed to be about 100 [MPa] in regions of plastic accommodation which is at least 7 times as high as the mechanical driving force from overall load for the first transforming variant ( A G v = Euy eT,j.,yy ,,~ 14 [MPa]). The corresponding maximum decrease in G v is about 70 [MPa]. Stress-assisted nucleation is, therefore, favorable in plastically accommodated regions although the austenite is stabilized mechanically. The distributions of Figs. 9(a) and 9(b) are similar. Stress-assisted nucleation tendency is governed by the TIMS. This tendency is maintained for higher ~M but the region of a vary large AGa,j. is larger as shown in Fig. 9(c). The amounts of maximum AGo,j. are similar for all three considered cases govcrned by thc transformation strains. In Figs.

321

10(a) and 10(b) the variants j" giving AGo,j. are depicted which allows the prediction of the orientation effect in a TIMS field. It is maintained in this stage of transformation since variants defined in table are favored with transformation strains similar to the already transformed one. Locally different variants are favored if the applied stress level Euu is varied which is seen due to the difference of Figs. 10(a) and 10(b). But regardless of Euy and the volume fraction of a variant the same variants are favored but not in the same positions within the RVE.

6. D i s c u s s i o n

Concerning the mechanisms responsible for the material behavior under MT mechanical aspects of this transformation type have been treated within a thermodynamic framework by Sun et al. [26]. On the other hand transformation kinetics relations are developed with respect to mechanical effects of MT [4]. Both approaches require the knowledge of transformation-induced microstress (TIMS) and -strain distributions computed on a size scale where the typical rearrangements of the material take place. This is done here by the help of the finite element method and a suitable micromechanical model. A thermomechanical potential is introduced in order to quantify the state of the strained material. Formulations of local driving forces for transformation on the microscale are derived from thermodynamics considerations here, too. The mechanical driving force AGaV,j and the mechanical part AG/Vo of the thermodynamic force acting on a phase boundary depending on local field quantities are computed and enable to quantify the role of internal stresses with respect to growth and stress-assisted nucleation tendency. It has been shown that the transformation mechanism itself governs the autocatalytic character of the transformation. Therefore, further investigations in order to predict the domain-structure of the martensitic product phase under externally applied stress 5] arc required which could enable to define a stress-dependent transformation strain tensor. Attempts concerning that point of interest

F. Marketz, F.D. Fischer~Computational Materials Science 8 (1994) 307-8~5

322

MPa

(a) I,IDFliAX

MPa

Y]yy

i?e

7

(b) MDFI~kX

MPa

~yy

(c) Fig. 9. (a) Distribution of maximum mechanical driving force AGo,j*, no externally applied stress Evv = 0, ~M = 0.82%. (b) Distribution of maximum mechanical driving force AGo,j*, externally applied stress Eyv = 125[MPa], ~M = 0.82°'/o. (c) Distribution of maximum mechanical driving force A G m j . , externally applied stress Evv = 125[MPa], ~M = 2.61~.

F. Marketz, F.D. Fischer/Computational Materials Science 3 (199,~) 307-3~5

323

(a)

(b)

~yy

Fig. 10. (a) Most favorable crystallographic variant j* in transformation-induced microstress field, no externally applied stress ~'-y~ = O, ~M = 0.82%. (b) Most favorable crystallographic variant j*, externally applied stress ~yy = 125 [MPa], ( M ---- 0 . 8 2 % .

have been carried out by Roytburd and Temkin [31] by introducing a thermomechanical potential with the twin fractions of the lattice invariant shear (LIS) as a degree of freedom dependent on E. The objective of micromechanical models is also to predict the morphology of the transformation product. Both nucleation barriers for stress-assisted and strain-induced nucleation and energy barriers for interface movement remain to be quantified. In the case of heterogeneous nucleation which governs the behavior of bulk materials

critical driving forces are qualitatively determined as a function of the driving force for transformation [9]. The treatment of plastic deformation by J2-flow theory is used here as an approximation. A better understanding between transformation modes and plastic deformation is essential and can be achieved by an advanced theory with a costitutive law for transformation twinning derived from the domain structure scale. Accommodation of the transformation strains is a major element in the discussion of martensite growth having been

324

F. Marketz, F.D. Fischer~Computational Materials Science 3 (199j) 307-3¢5

treated here by determination of the exact size of the relaxed plastic zone. A micromechanical treatment of martensite growth as a dynamic process to investigate local dynamic plastic zone interactions during growth taking into account interface mobility remains to be carried out. If plastic accommodation occurs the components of the plastic strain tensor =p e are almost propor-

Acknowledgement

s

The ABAQUS finite element code was made availiable under academic licence from Hibbitt, Karlsson and Sorensen Inc., Pawtucket, RI, USA. Computations reported on here were carried out on an IBM 580 workstation of the computing center at the University of Leoben, Austria.

tional to those of the transformation tensor _ewj to be a c c o m m o d a t e d

for ~ = 0 References

,#,

(42)

/C(_r) is a factor depending on position since plastic strains b e c o m e smaller with increasing distance from the martensitic plate ends. T h e interaction of E__ with transformation increases the components of plastic strains in direction of applied stress. T h e strain increment E___.TP, acc related to the G r e e n w o o d - J o h n s o n effect will for that reason depend on the m e a n transformation strain and on __E. T o derive an explicit formulation further micromechnical investigations under more complex applied stress states have to be carried out. T h e

average stress state in the parent phase < a a > depends on the transformed volume fraction ~M. It can be concluded from the numerical results that with increasing ~M the hydrostatic part of < a_A > becomes larger which will favor unexpected variants to transform in later transformation stages. The orientation effect is smaller if there are other sources of microstress than that from the transformation itself, for example elastic inhomogeneity. So the knowledge of microstresses on the size scale of critical martensitic nuclei remain to be determined. With the continuum mechanical treatment of martensitic transformation one can arrive at a size scale of typical microstructural features of MT described by the kinematic theories [11]. Since material behavior is also governed by mechanisms in the nucleation and growth stages a better understanding of stress and strain effects will help to develop better constitutive laws for phase transforming materials in the framework of continuum mechanics.

[I] G.K. Haritos, J.W. Hager, A.K. Amos and M.J. SMkind, Int. J. Solids Struct. 24 (1988) 1081. [2] J.B. Lehlond, J. Devaux and J.C. Devaux, Int. J. Plasticity5 (1989) 551. [3] F.D. Fischer, M. Berveiller, K. Tanaka and E.R. Oheraigner, Arch. Appl. Mech. 64 (1994), 54. [4] K. Tanaka and S. Nagaki, hag. Arch. 51 (1982) 287. [5] E. Gautier, A. Simon and G. Beck, Acta Metall. 35 (1987) 1367. [6] E. Gautier and A. Simon, in: Phase Transformations '87, ed. G.W. Lorimer (The Inst. of Metals, London,

1988) 285. [7] J.F. Ganghoffer, S. Denis, E. Gautier, A. Simon, K. Simonsson and S. SjSstr/~m, J. Phys. IV, Coll. C4, suppl. J. Phys. III 1 (1991) C4 77. [8] F. Marketz and F.D. Fischer, Mdcamat 93 Int. Sere. on Micromechanics of Materials , Collection de la Direction des I~tudes et Rechereches d'l~lectricitdde France (Editions Eyrolles,Paris, 1993) 258. [9] G.B. Olson and A.L. Roytburd, in: Martensite - A Tribute to Morris Cohen, eds. G.B. Olson and W.S. Owen (ASM International,1992) 149. [I0] J.D. Eshelby, Proc. R. Soc. London 241 (1957) 376. [11] M.S. Wechsler, D.S. Lieberman and T.A. Read, J. Met. Trans. A I M E 197 (1953) 1503. [12] A.L. Roytburd, in: Solid State Phys. 33, eds. H. Ehrenreich, F. Seitz and D. Turnbull (1978) 317. [13] K. Bhattacharya, Continuum Mech. Thermodyn. 5

(1993) 2431. [14] H.J. Neuhauser and W. Pitsch,Acta Metall. 19 (1971) 337. [I5] Z. Nishiyama, MartensiticTransformations (Academic Press, New York, 1971). [16] J.R. Rice in: Constitutive Equations in Plasticity, ed. A.S. Argon, (The M I T Press, Cambridge, Massachusetts, 1975) 23. [17] K. Tanaka, E.R. Oberaigner and F.D. Fischer, Cont. Mech. Thermodyn. (1994), submitted. [18] F. Marketz and F.D. Fischer,Modelling Simul. Mater. Sci. Eng. 5 (1994), to appear. [19] H. Horikawa , S. Ichinose,K. Morii, S. Miyazaki and K. Otsuka 1988 Metall. Trans. A 19 (1988) 915.

F. Marketz, F.D. Fischer~Computational Materials Science 3 (1994) 307-325 [20] L. Kaufman L and M. Cohen in: Progress in Metal Physics 7, eds. B. Chalmers and R. King (Pergamon Press, London, 1958) 165. [21] G.B. Olson and M. Cohen, Met. Trans. A 6 (1975) 791. [22] F. Marketz and F.D. Fischer, Met. Trans. A (1994), to appear. [23] J.R. Patel and M. Cohen, Acta Metall. 1 (1953) 531. [24] G.B. Olson, in: Deformation, Processing and Structure, ed. G. Krauss (Am. Soc. for Metals, 1984) 391. [25] F.D. Fischer, Eur. J. Mech. A /Solids 11 (1993) 233. [26] Q.P. Sun and K.C. Hwang, J. Mech. Phys. Solids 41 (1993) 1.

325

[27] T. Inoue , Z.-G. Wang and K. Miyao, in: Proc. 2 na Int. Conf. Residual Stresses ICRS2, eds. G. Beck, S. Denis and A. Simon (Elsevier, London, 1989) 606. [28] M. Kohsuke, Z.-G. Wang and T. Inoue, J. Soc. Mater. Sci. Jpn. 35 (1986) 1352. [29] V. Tvergaard, Acta Metall. 38 (1990) 185. [30] M. Grujucic, H.C. Ling, D.M. Haezebrouk, W.S. Owen, in: Martensite - A Tribute to Morris Cohen, eds. G.B. Olson and W.S. Owen (ASM Int., 1992) 175. [31] A.L. Roytburd, Izv. AN SSSR, ser. fiz. 47 (1983) 435.