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Mazzoni, L. (2010). Strain gradient based analysis of transformation induced plasticity in multiphase steels (Unpublished doctoral dissertation). Université libre de Bruxelles, Faculté des sciences appliquées – Construction, Bruxelles.

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BIBLIOTHEQUE DES SCIENCES ET TECHNIQUE.

CPI 74 Av. A, Depage. 30 e-1000 BRUXELLES

“r*! : 650.20.54

Strain gradient based analysis of transformation induced plasticity in

multiphase steels

THESE

présentée en vue de l’obtention du grade de Docteur en Sciences Appliquées de TUniversité

Libre de Bruxelles

Louise Mazzoni Leduc

Université Libre de Bruxelles

003441a 47?

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This research was supported by the Région Wallonne de Belgique in the frame of the ACIETRIP programme, under grant number 0415961 (RW-WINNOMAT).

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transformation induced plasticity in

multiphase steels

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Thesis carried out within at the Université Libre de Bruxelles (Faculty of Applied Sciences).

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Contents

Abstract 1

1 Introduction 3

1.1 Industrial context... 3

1.2 Scope of the thesis ... 5

1.3 Original aspects of the thesis and content... 7

2 Transformation Induced Plasticîty in multiphase steels 11 2.1 Heat treatment for TRIP steels: stabilization of austenite and microstructure optimization... 12

2.2 Characteristics of the martensitic transformation... 14

2.2.1 Crystallographic point of view... 14

2.2.2 A thermo-mechanical criterion for transformation induced plasticity . . 16

2.2.3 Nucléation and growth of martensitic plates... 18

2.3 Experimental investigations of TRIP effect... 19

2.3.1 Microstructural investigations... 19

2.3.2 Effect of the stress State on the mechanical properties of TRIP-assisted steels... 20

2.3.3 Influence of the microstmcture on the transformation induced plasticity 21 2.4 Micro-mechanical modelling of nucléation and/or growth of martensite .... 21

2.4.1 Constitutive model developments... 22

2.4.2 Numerical multi-scale calculation... 27

2.5 Adopted strategy... 30

3 Strain gradient plasticity théories and related applications 33 3.1 Size effects at the micron scale... 33

3.2 Dislocation density and hardening of metals... 38

3.3 Size independent plasticity... 40

3.4 OverView on the available gradient plasticity formulations ... 42

3.4.1 Thermodynamics restrictions for strain gradient plasticity théories ... 42

3.4.2 Lower order models ... 43

3.4.3 Higher order stress models... 46

3.4.4 Non local théories ... 57

3.4.5 Recent assessments of higher order strain gradient plasticity théories. . 60

3.4.6 Discussion on spécifie boundary conditions related to higher order thé­ ories... 62

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4 Computational modelling of the phase transformation 67

4.1 Physical model for the transformation of a retained austenite inclusion... 67

4.2 Embedded cell model... 68

4.3 Sélection of material parameters and model parameters... 69

5 Phase transformation effect in multiphase steels with a single-parameter strain gradient plasticity theory under small strain assumption 73 5.1 Strain gradient plasticity theory... 73

5.1.1 The generalized effective plastic strain rate... 73

5.1.2 Goveming and constitutive équations... 74

5.1.3 Implémentation issues... 75

5.2 Results... 76

5.2.1 Preliminary study: general picture without size effect... 76

5.2.2 Influence of the boundary conditions at the elasto-plastic boundary . . . 77

5.2.3 Influence of the microstructural parameters... 82

5.3 Discussion... 87

5.3.1 Discussion about the boundary conditions... 87

5.3.2 Size dépendent strengthening from composite effect... 92

5.3.3 Size dépendent transformation strain effect... 94

5.3.4 Microstructure optimization... 95

5.4 Conclusions... 95

6 Analysis of size effects associated to the transformation strain in TRIP steels with a multi-parameter strain gradient plasticity theory under small strain assumption 97 6.1 Introduction... 97

6.2 Strain gradient plasticity model... 99

6.2.1 Generalized effective plastic strain rate... 99

6.2.2 Goveming and constitutive équations... 100

6.2.3 Implémentation issues...101

6.2.4 Evolving plastic boundary conditions... 102

6.3 Results... 103

6.3.1 Material and loading parameters... 103

6.3.2 Separate effects of the gradient terms on the overall transformation hardening...104

6.3.3 Effect of the gradient terms on each transformation hardening contribu­ tions ... 105

6.3.4 Effect of the austenitic grain size... 107

6.4 Discussion... 110

6.5 Conclusions... 115

7 Phase transformation effects in multiphase steels with a finite strain gradient plas­ ticity theory 117 7.1 Strain gradient plasticity theory... 117

7.1.1 The generalized effective plastic strain rate...117

VI

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7.1.2 Goveming and constitutive équations... 118

7.1.3 Implémentation issues... 122

7.2 Results... 122

7.2.1 Spécifie assumptions and fixed parameters... 122

7.2.2 Use of the single-parameter theory... 125

7.2.3 Multi-parameter parameter theory... 126

7.2.4 Effect of the shear component of the transformation strain... 128

7.2.5 Effect of the ausenitic grain size... 130

7.3 Discussion... 134

7.4 Conclusion... 144

8 Conclusion 147

Bibliography 154

List of publications 169

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This thesis is devoted to the micromechanical study of the size-dependent strengthening in Transformation Induced Plasticity (TRIP) steels. Such grades of advanced high-strength steels are compelling for the automotive industry, due to their improved mechanical properties. Among others, they combine a good strength versus ductility balance. In this context, many research Works bave been carried out to study these grades of steels. In particular, from a numerical point of view, earlier studies within the ffamework of classical plasticity do not properly reproduce the strengthening levels characterizing TRIP steels and obtained experimentally. In this study, the strain gradient plasticity theory presented by Fleck and Hutchinson (2001) is chosen to ac- count for the strengthening effect resulting from the phase transformation. A two-dimensional embedded cell model of a simplified microstructure composed of small cylindrical metastable austenitic inclusions, partially undergoing the phase transformation, within a ferritic matrix is used. First, the single-parameter version of the strain gradient plasticity theory under small strain assumption is used for the simulations. The impact of the higher order boundary condi­

tions is assessed. It is shown that, when the plastic flow is unconstrained at the elasto-plastic boundaries, the transformation strain has no significant impact on the overall strengthening.

The strengthening is essentially coming from the composite effect with a marked inclusion size effect resulting from the appearance during deformation of new boundaries (at the interface between parent and product phases) constraining the plastic flow. Second, the multi-parameter version of the strain gradient plasticity theory, incorporating separately the rotational and ex- tensional gradients in the formulation, is employed under small strain assumption. The effect of the plastic strain gradients resulting from the transformation strain is better captured. In particu­

lar, the results show a significant influence of the shear component of the transformation strain.

An implicit confinement effect is revealed at the elasto-plastic boundaries which is partly re- sponsible for the transformation strain effect. Size effects on the overall strengthening are also revealed, due to a combined size dépendent effect of the transformation strain and of the evolv- ing composite structure. Third, the extension of the strain gradient plasticity theory to a finite strain description is applied. A significant effect of the transformation strain is obtained with the multi-parameter version of the theory as well as an optimal austenite grain size improving the damage résistance of the martensite, in agreement with the typical grain size of the current TRIP-assisted steels (Jacques et al., 2007).

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Chapter 1 Introduction

1.1 Industrial context

An increasing interest is nowadays devoted to the design of new lightweight and high résis­

tance materials in transportation industry, especially for automotive applications. Decreasing the weight of a vehicle leads to both economical and environmental benefits. On the other hand, the increase of safety standards requires the use of high energy absorbing materials. To fit these industrial requirements, advanced high strength Steel grades hâve been developed during the past decade such as (this is not an exhaustive list of possibilities):

• Dual phase (DP-) steels, the microstructure of which contains hard martensite islands in a soft ferritic matrix,

• Transformation Induced Plasticity (TRIP-) assisted steels, composed of soft ferrite, marten­

site, bainite and retained austenite, likely to transform partially into martensite,

• Twinning Induced Plasticity (TWIP-) steels, containing large amounts of Manganèse which allows to stabilize the austenite at room température and to decrease the stack- ing fault energy, promoting the mechanical twinning as the prominent deformation mode in such grades of Steel.

For the purpose of illustration. Figure 1.1 shows the design of an automotive body, using high strength steels.

The experimental procedure required to obtain such grades of high strength steels néces­

sitâtes spécifie treatments which are complex in nature (Huang et al, 2006; Srivastava et al., 2006; Kumar et al., 2008). Moreover a key to better understand the outstanding strengthening enhancement related to high strength steels is the corrélation between the microstructure (évo­

lution during deformation or morphology) on one hand and the mechanical properties on the other hand. Some efforts hâve been presented in the literature, see for instance the contributions from Bayram et al. (1999); Zaefiferer et al. (2004); Uchic et al. (2006); Barbier et al. (2009).

This Work lies in the same orientation.

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Example of parts using High-Strength or Advanced High-Strength Steels

Bumper Beam Boron DP 780-1000 CP 800-1000

Front side members Bainitic (HR)

DP 600-780 TRIP 800-1000

Floor side reinf

DP-TRIP DP500/600 Rear side member

HSLA HSLA TWB

Bainitic 460/600 DP450/S00

reinf.

DP 4S0-S00 Boron Steel

Roof bow 450/500

B-pillar reinfbrcement 550/600 DP 500/600 TRIP 600-800

Boron Steel

Figure 1.1; Use of different steels grades for the manufacturing of a car, ffom e.g. Lacroix (2007).

The interest devoted to TRIP steels is highly motivated by the remarkable mechanical prop- erties offered, combining both strength and ductility. The TRIP effect results from the presence of retained austenite, which is metastable at room température, obtained thanks to an appro- priate beat treatment of cold rolled multiphase steels. The strain-induced transformation from austenite into barder martensite, accompanied with a local transformation strain, induces plas­

tic deformations in the surrounding phases, namely ferrite and bainite. Consequently, the work hardening rate of the surrounding phases increases and so does the global strain hardening. As a resuit, high strength levels are obtained. The onset of necking is delayed: the phase trans­

formation preferentially occurs in a zone where necking is likely to happen, which leads to improved total élongations (Grâssel et al., 2000). Deformation fiirther takes place in local areas possessing lower flow stress. Similar microstructural investigations under dynamic and quasi- static tensile testing revealed that more austenite transformation occurred in régions of higher deformation, doser to a fracture tip, and under dynamic conditions (Oliver et al., 2007). The fracture mechanism is thus delayed, and an improved formability is obtained (Srivastava et al., 2006). To illustrate the spécifie advantages offered by TRlP-assisted steels, one could refer to Oliver et al. (2007) or Huh et al. (2008), where the mechanical properties of low alloyed DP- and of TRIP-type sheets under dynamic tensile tests were compared to get qualitative results for the crashworthiness of an auto-body. It was shown that TRIP-type sheets show larger fracture élongation, delay of necking and thus better formability. In Matsumura et al. (1992), a tensile

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Introduction 5

strength of 1000 MPa and a total élongation of 30% were reported for low carbon steels alloyed with silicium (with a Chemical composition containing less than 0.4 wt% C). In (Grâssel et al., 2000), the mechanical properties of austenitic Fe-Mn steels with addition of aluminium and Sili­

con were investigated. In this grade of Steel, both strain-induced martensitic transformation and strain-induced mechanical twinning of austenite, which is also responsible for increased me­

chanical properties, are likely to occur. For low contents on Manganèse, namely less than 20%, TRansformation Induced Plasticity occurs in a range of températures from 50°C to 200°C7. With an initial volume fraction of austenite of 80%, due to the high amount of manganèse, the sam- ple reaches a total élongation of about 80% and an ultimate tensile strength of about 830 MPa during a quasi-static tensile test at room température. Note that lower alloyed austenitic steels generally contain a maximum of 20% volume fraction of residual austenite, and consequently cannot reach such values of total élongations (Grâssel et al., 2000).

1.2 Scope of the thesis

In multiphase steels the interplay of hard and soft phases results in properties improvements related to a eomposite type response. In addition to the improvements explained by the relative volume fraetions of the phases and their morphologies, an effect related to the particle size has been investigated experimentally by many authors (e.g. Ulvan and Koursaris, 1988; Varma et al., 1994; Reisner et al., 1996). The principle of smaller being stronger also manifests for example in dual-phase steels (e.g. Delincé et al., 2006, 2007). In the case of TRIP-assisted steels, several investigations hâve reported that smaller grain sizes tend to increase the stability of the austenite in ftilly austenitic steels (Reisner et al., 1996; Jimenez et al., 2007) and also play a rôle in the overall hardening enhaneement (Ulvan and Koursaris, 1988). The main objective of the présent manuseript is the computational analysis of this size effeet in the context of TRIP- assisted steels. This work présents investigations which are purely numerieal, no experimental efforts are provided.

For TRIP-assisted steels, the mechanical properties can also be enhanced owing to the TRIP effect which induces an extra strengthening contribution through three mechanisms (e.g. Fischer and Reisner, 1998; Fischer et al., 2000; Fumémont, 2003; Lani et al., 2007):

• the inerease of the volume fraetion of the harder martensitic phase contributing to an élévation of the global hardening through a composite type effeet;

• the génération of extra dislocations around the transformed régions required to accom- modate the relatively large transformation strain occurring in the transforming zone; •

• the appearance of a new boundary impénétrable to dislocations, leading during further deformation to an extra hardening of a région surrounding the transforming zone of the material through a higher-order type effeet. This assumption allows to model the harder characteristic of the martensite product phase with respect to the austenite parent phase.

An in-depth analysis of these effects and of their relative contributions to the TRIP enhance- ment is further motivated from previous experimental investigations. As an illustration. Figure

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1.2, reproduced from (Jacques, 2004), shows the accumulation of dislocation, generated by the phase transformation and which are stopped by the impénétrable phase boundary between martensite and austenite.

Figure 1.2; Dislocations generated in the ferrite at the tip of the martensite variants, from Jacques(2004)

Typical TRIP steels hâve austenitic grain sizes on the order of 1/xm (Jacques, 2004) with an extremely good strength versus ductility balance resulting from an excellent strain hardening capacity (e.g. Van Rompaey et al., 2006; Jacques et al., 2007). It is well admitted nowadays that for such small sizes (e.g. Fleck and Hutchinson, 1997; Fleck et al., 2003; Ma et al., 2006), geometrically necessary dislocations required by the presence of the plastic strain gradients, accommodating in the présent case the mismatch of properties and shape change as well as the appearance of an interface impénétrable to dislocations, will dominate the statistically stored dislocations. As a resuit, strain gradient effects can significantly affect the response leading to an additional strengthening contribution. It is important to note that the experimentally measured TRIP strengthening effects can hardly be modelled with classical théories without adjusting some fitting parameters to artificially raise the strength (see e.g. Delannay et al., 2008). Hence the motivation of this work is to assess whether the hehaviour of real TRIP steels involving retained austenite in the micrometer diameter range is significantly affected by strain gra­

dients effects without which the remarkable improvement of the strength/ductility balance cannot be quantitatively captured.

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Introduction 7

1.3 Original aspects of the thesis and content

Size efFects in the mechanical response of TRIP steels hâve not received much attention in numerical studies except for the work of Reisner et al. (1996); Iwamoto and Tsuta (2000);

Turteltaub and Suiker (2006a), while strong evidence can be found in experimental studies that the austenite grain size affects both tensile properties of TRIP-assisted steels, and the résis­

tance to transformation (see e.g. Ulvan and Koursaris, 1988; Varma et al., 1994; Reisner et al., 1996). This thesis relates to the following original aspects. First, size effects related to TRIP steels are investigated by the use of a continuum theory. Fleck and Hutchinson (2001) strain gradient plasticity theory is used, involving up to three length parameters, setting the scales at which gradients affect the microstructure, and eonnected to the représentative sizes of the mi­

crostructure, such as the austenite grain size. This thesis is thus a non standard application of the Fleck and Hutchinson (2001) theory. The TRIP effect results from the compétition of the three mechanisms cited above, namely a composite type effect, the effect of the transfor­

mation strain and the appearance of an impénétrable boundary. The composite type effect is supposed to be coupled to the change of plastic confinement as a conséquence of the growth of the martensitic région, this association results in a higher order composite effect. An ad- ditional uncovered aspect is the quantification of the spécifie contribution of the two key mechanisms (the effect of the transformation strain and the higher order composite ef­

fect) to the overall strengthening. In particular, the appearance of an impénétrable boundary between the parent and the newly formed product phase is modelled via the introduction of evolving higher order boundary conditions on the plastic strain rate. The high values of the transformation strain requires the use of an extension of the strain gradient theory accounting for large deformations, which is still rare in the literature except for the work of Niordson and coworkers (Niordson and Redanz, 2004; Niordson and Tvergaard, 2005). Consequently the analysis of the TRIP effect using an extension of the Fleck and Hutchinson (2001) strain gradient plasticity theory to the finite strain framework is also a new point brought by the présent thesis.

A two dimensional embedded cell model is employed (Van Rompaey et al., 2006), with an in depth study of the partial transformation of a single austenitic inclusion surrounded by a ferritic matrix. A simplified transformation criterion is assumed. The work consists partly in a systematic study, in order to provide guidelines for the optimization of TRIP-assisted mul- tiphase steels with respect to size effects. The microstructural parameters such as the volume fraction of the retained austenite and of the transforming austenite for instance, as well as the transformation parameters, such as the shearing component of the transformation strain 7*®-^

are carefully assessed and their influence on the overall hardening are discussed. However, the most important issue addressed here is to détermine whether the size effect, introduced by the use of the strain gradient plasticity, contributes to enhance the strengthening resulting from the different sources of the TRIP effect mentioned above. A subséquent question, but also of great interest, is to analyze the other effect of the austenite grain size.

The following work is articulated around eight chapters. In the second chapter, an overview of the TRIP-effeet mechanisms is given. The speeific beat treatment required to allow the

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appearance of strain-induced martensitic transformation is set ont. The martensitic transforma­

tion is then described by commenting the crystallographic aspects as well as the mechanisms afîecting plastic flow. Then, some general insight about the TRIP effect is provided. First, ex­

perimental efforts to characterize the TRIP efifects are briefly summarized, followed by the State of the art in multiscale and micromechanical computational modelling.

In the third chapter, the existing classes of strain gradient plasticity théories are presented.

Among the various possibilities, the choice of the Fleck and Hutchinson (2001) strain gradient plasticity will be motivated for the présent study, as a good trade off between the complexity of 3D Discrète Dislocation Dynamics in terms of computational time (Devincre and Robert, 1996) and more simple size dépendent théories which do not incorporate higher order variables and give thus no option for imposing higher order boundary conditions (Acharya and Bassani, 2000), a feature which is required in this case at the austenite/martensite interface impénétrable to dislocations.

In chapter four, the simplified microstructure, the corresponding embedded cell model and the transformation criterion as well as the varying and fixed material and microstructural pa- rameters used for the study are described.

Chapter five reports the results of a study of the simplified microstructure with varying microstructural and transformation parameters, and where the single-parameter version of the Fleck and Hutchinson strain gradient plasticity theory under small strain assumption is used. It is shown that the size of the retained austenitic inclusion strongly influences the overall strength- ening of the microstructure, and may hâve an impact on the damage process in the martensitic phase. However, the strengthening gain related to the phase transformation seems to originate mainly from a “higher order composite-type” effect and not ffom the transformation strain it- self It is aiso revealed that the boundary conditions postulated at the elastic-plastic boundary hâve a significant impact on the strengthening.

Chapter six extends the results of the study obtained with the multi-parameter version of the Fleck-Hutchinson strain gradient plasticity theory under small strain assumption. It is shown that introducing three length parameters in the model leads to higher strengthening effects re­

lated to the phase transformation, and in particular, the transformation strain impact becomes critical. An implicit confinement effect at the elastic-plastic boundaries appears when the multi- parameter framework is used, and contributes to the strengthening enhancement brought by the transformation strain.

Chapter seven présents the results obtained with an extension of the Fleck and Hutchinson (2001) theory to the finite strains (Niordson and Redanz, 2004; Niordson and Tvergaard, 2005);

allowing to generalize the trends observed in the previous chapters. It is confirmed that when using the multi-parameter theory, the transformation strain has an appréciable impact on the overall strengthening related to the martensitic transformation, which is not the case when using the single-parameter theory. The confinement effect at the elastic-plastic boundary is found to play a major rôle in the strengthening enhancement when the multi-parameter theory is used, which was already concluded under the small strain assumption. However, some différences in the trends are revealed, conceming the size effects on the overall strengthening and on the damage process of the martensitic phase. Generally, it confirms that the trends given in the

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Introduction 9

previous chapters are not to eliminate, they are just fitted to account for large strain field around the austenite inclusion. As a resuit, Chapter five, six and seven give a valuable insight on the use of the different versions of the Fleck-Hutchinson strain gradient plasticity theory for forthcoming applications.

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Chapter 2

Transformation Induced Plasticity in multiphase steels

The goal of this chapter is to give an overview of the physical and mechanical aspects of the Transformation Induced Plasticity, namely the TRIP effect. The attractive mechanical properties characterizing this grade of steels hâve been investigated in the literature, in re­

lation with the Processing aspects aiming at optimizing the transformation effect. In order to set the scene for the subséquent chapters, a physical understanding of the phase trans­

formation is proposed in tenus of crystallographic, thermodynamic and plasticity aspects.

The subséquent strengthening of microstructures is analysed and results into questions to address. Secondly, a review of the modelling approaches available in the literature for this type of steels is given. The relation between phase transformation, dislocations and plas­

ticity naturally suggests the use of strain gradient plasticity to study the mechanical aspects of the phase transformation.

The martensitic transformation can occur under varying thermo-mechanical conditions (see e.g. Cherkaoui et al., 2000; Van Rompaey , 2004). The température as well as the applied stresses define the transformation régime:

• The martensitic transformation can occur upon cooling without applying extemal stresses, at a given start température called Mg. This type of transformation is referred to as a thermally-induced transformation;

• When the température ranges from Mg to Mg > Mg, which marks a change of mode in the transformation process, the transformation occurs due to the applied stresses which contribute to the driving force for the phase transformation. As a conséquence, plastic deformation will be induced by martensite transformation rather than by slip; the trans­

formation is then denoted a stress-assisted transformation; •

• Above the M/ température, a significant plastic flow occurs prior to the martensitic trans­

formation and promotes it by creating new nucléation sites; this mechanism is called strain-induced transformation. This régime lasts until a température Md is reached, at which failure occurs through fracture in the parent phase before the start of the martensitic transformation.

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The focus in this study is set on the conséquence on plastic flow of strain-induced martensitic transformation, referred to in the text as TRansformation Induced Plasticity effect, also noted TRIP effect in the following. The conséquences of the TRIP effect on the balance “strength versus ductility” are the subject of intensive research. The size effects related features of this transformation are investigated here.

2.1 Heat treatment for TRIP steels: stabilization of austenite and microstructure optimization

The Processing route to obtain retained austenite, which is metastable at room température, is described in Sakuma et al. (1991); Jacques et al. (1998); Srivastava et al. (2006) and consists in two steps. The first step is an intercritical annealing: the multiphase Steel sheet is heated in the austenite ferrite range, at a température standing between Aci and Acs (which limits the o;/7 domain in the phase diagram). The obtained microstructure consists of a fine dispersion of austenite grains located both inside ferrite grains and at ferrite grain boundaries (Jacques et al., 1998). After a fast cooling avoiding any major ferrite formation, the second stage of the process is an isothermal bainite treatment. During the bainite formation, the carbon diffuses in the austenite islands. This increases the stability of the austenite, which permits to retain it at room température. Indeed, increasing the carbon content depresses the martensite température start below zéro (Oliver et al., 2007) and thereby increases the austenite stability.

Figure 2.1 : Schematic représentation of the heat treatment used to obtain TRIP microstructures.

Resketched from Srivastava et al. (2006)

In (Srivastava et al., 2006), the effect of the duration of the intercritical annealing and of the

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Transformation Induced Plasticity in multiphase steels 13

isothermal bainite treatment on a C-Mn-Si TRIP cold rolled Steel was investigated. The isother­

mal bainite treatment resulted in an increase of the carbon concentration within a decreasing amount of austenite. Consequently, the austenite stability was improved. Moreover, the vol­

ume fraction and stability of retained austenite were shown to influence the formability of such grades of steel: a more stable austenite usually resulted in a poorer formability. Srivastava et al.

(2006) showed that for a particular chemistry, the beat treatment required to obtain TRIP-steel can be optimized to get a higher volume fraction of austenite combined to an increased stability and still with an increased formability. High tensile strength of 600 MPa and élongation of 31%

were obtained when optimizing the processing route as proposed by these authors.

The attractive mechanical properties of TRIP-assisted steels are obtained from an optimal amount of retained austenite in the microstructure. The design of the beat treatment allowing to stabilize the austenite at room température on the mechanical properties of the TRIP-assisted steels is then crucial, as shown in Zaefferer et al. (2004), in which the isothermal bainite treat­

ment time and température were assessed for the case of a low alloyed Steel. A suflîciently high holding time allowed to reach an optimum Chemical stabilization in terms of high content and homogeneous distribution of the carbon in the austenite. Conversely, decreasing the holding température led to a smaller amount of bainite and consequently a higher amount of retained austenite which was less stabilized, as well as a higher defect density in ferrite, austenite and bainite. When an optimal beat treatment was used, the remaining austenite after bainitic for­

mation was completely stabilized at room température and transformed to martensite gradually during a long straining range. For holding températures of 400°C and a holding time of 400s, a 739 MPa tensile strength and a 30% maximum uniform élongation were reached.

Alloying éléments also play a rôle in the stabilization of austenite (Matsumura et al., 1992;

Jacques et al., 1998; Girault et al., 2001; Zaefferer et al., 2004; Oliver et al., 2007). The addition of Silicon (Si) permits to prevent the précipitation of cementite, which normally occurs during bainite formation and acts also as a solid solution strengthener for the ferrite matrix. Aluminium (Al) is a less potent Carbide retarder but may partially replace Silicon in order to avoid surface quality problems. Note that a full substitution of Si with Al would be detrimental for the strength versus ductility balance since the resulting ferrite matrix would exhibit a weaker behaviour.

The addition of manganèse (Mn) or Si also enables to obtain austenite and ferrite at lower intercritical températures. Phosphorus (P), which also inhibits cementite formation, and Mn are also solid solution strengtheners. However, it was shown in Jacques et al. (1998) or in Jacques et al. (2001a), that controlling the parameter of bainitic tempering stage, led to the rétention of a noticeable amount of retained austenite (nearly 10% are reported in Jacques et al. (1998)) in cold rolled low carbon and low Silicon steels. This was not anticipated in the case of low Silicon contents. The authors reported improved mechanical properties due to both the occurrence of TRIP effect and a composite strengthening effect related to the dispersion of barder phase grains (bainite and martensite) in the soft ferrite matrix.

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2.2 Characteristics of the martensitic transformation

At room température, the metastable austenitic phase transforms into martensite which can appear under different crystallographic structures. In the case of TRIP-steels, the ol martensite, with body-centered cubic structure, appears during plastic deformation and is responsible for high strength combined with an outstanding ductility. In the case of Fe- based shape memory alloys, the formation of e martensite, with hexagonal-packed cubic structure, is responsible for the perfect shape recovery. Many issues in material science field hâve explored the martensitic transformation (see Bhadeshia, 2001a, for instance). A short overview is provided here.

2.2.1 Crystallographic point of view

This section mostly refers to the work of Bhadeshia (see e.g. Bhadeshia, 2001a). The trans­

formation from the austenitic parent phase to the martensitic product is not related to any diffu­

sion process: the carbon in solution in the austenitic phase does not hâve time to diffuse out of the crystal structure. Thus, the transformation results from a mechanical process. A change of crystal lattice accompanies the martensitic transformation: the face-centered cubic (fcc) struc­

ture of the austenite, also represented by a body-centered tetragonal structure (cf. Figure 2.2a) is deformed into the body-centered cubic martensite (cf Figure 2.2b). As proposed by Bain in 1924, the change of crystal lattice occurring during the displacive martensitic transformation can be achieved by a simple homogeneous deformation called the Bain strain which consists of a contraction of the parent lattice of about 17% along the direction and an identical expansion of about 12% along the and ai direction (see also Van Rompaey , 2004).

«3

(a) (b)

Figure 2.2: The change of crystal lattice from (a) fcc/bct austenite to (b) bcc martensite can be achieved by the Bain strain. Resketched from Bhadeshia (2001a)

However, the orientation relationships between the parents and product phases obtained after the application of the Bain strain, do not fit with experimental observations. Indeed, the

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Transformation Induced Plasticity in multiphase steels 15

Bain strain leaves no plane undistorted and unrotated. Experimentally, the martensite forms on particular crystallographic planes known as the habit planes, which remain undistorted and unrotated after the transformation and are then denoted as invariant planes. On a macroscopie scale, the strain accompanying the transformation is then an invariant plane strain, because of the presence of an invariant plane. The transformation strain minimizes the strain energy and is composed of a shearing strain 7*®-^ along the habit plane and a dilatational component along the normal to the habit plane. The dilatation component is assumed to range between 0.03 — 0.04 while the shear component is difficult to estimate, due to the fact that it might be influenced by the twinning process inside the newly formed variant (Ganghoffer and Simonsson, 1998; Van Rompaey , 2004; Van Rompaey et al., 2006; Lani et al., 2007). Experimental studies show that 7*®-^ can attain 20% (Van Rompaey , 2004).

The combination between the Bain strain and an appropriate rigid body rotation resuit in a strain leaving a line undistorted and unrotated, which is referred to as an invariant line strain and will be denoted as RB in the following. As a resuit, the experimentally observed macro­

scopie shape deformation (an invariant plane strain) is inconsistent with the lattice transforma­

tion strain (an invariant line strain). As shown in Figure 2.3, the phenomenological theory of martensite crystallography solves this remaining problem (Wechsler et al., 1953). The applica­

tion of an invariant plane strain Pi permits to get the observed macroscopie shape deformation but the wrong lattice structure (from figure 2.3a to figure 2.3b). When Pi is combined to another invariant plane strain P2, an invariant-line strain (équivalent to RB) is applied to the structure which permits to obtain the correct crystal structure but the wrong macroscopie shape (from fig­

ure 2.3b to figure 2.3c). To solve these discrepancies, another deformation should be applied in order to render P2 invisible as far as the shape change is concemed. This way, the correct shape of figure 2.3b and the correct structure of 2.3c can be obtained simultaneously. The latter de- formation must be lattice-invariant (noted L1 in figure 2.3); it can be either slip or twinning. As a resuit, twinned or slipped martensite are likely to appear after phase transformation as shown is figure 2.3d. Note that P2 must be an homogeneous shear, since it is rendered transparent by lattice invariant strain.

The Work by Wechsler et al. (1953) States that multiple martensite orientations are allowed:

in the case of low-alloyed TRIP-assisted steels, martensite develops along 24 possible orien­

tations, named variants, in the austenite parent phase, due to the symmetry of crystal lattice.

However, less than 10 variants can usually be found within an austenite grain (Jacques et al., 2007).

To complété this short overview, it must be mentioned that the inelastic processes related to the phase transformation are govemed by two effects, namely:

• the Magee effect, which is related to orientations for the newly formed martensitic vari­

ant. This effect translates that both the applied and internai stresses act for the variant sélection;

• the Greenwood and Johnson (1965) effect, related to the accommodation of the transfor­

mation strain, which induces elastic and plastic straining, in order to produce compatible strain rates.

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RB

(d) Correct shape; correct structure

Figure 2.3: Phenomenological theory of martensite crystallography. Resketched from (Bhadeshia, 2001a)

2.2.2 A thermo-mechanical criterîon for transformation induced plastîc-

ity

In this subsection, some standard thermodynamical aspects are briefly recalled, in order to introduce the concepts of driving and dragging forces, which are the basis for the processes involved during nucléation and growth of martensite plates inside the parent phase, and for the development of micromechanical or phenomenological models for phase transformation.

As shown in Fig. 2.4, the martensitic transformation occurs under equilibrium condition at the Tq température, at which the stress-free austenite and martensite phases hâve the same Helmholtz firee energy (j)chem, when no other work than Pressure-Volume work is involved (Van Rompaey , 2004). If the austenite phase is retained at températures significantly below Tq, the concept of driving forces and dragging forces is introduced. The reasoning below satisfies the

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Transformation Induced Plasticity in multiphase steels 17

second law of thermodynamics through the Clausius-Duhem inequality. In this context, the rate of dissipation is related to a mechanical term, which is defined as the mechanical driving force for the phase transformation (MDF), and a term related to variation of the Helmholtz free en- ergy, which stands for the Chemical driving force (Aipchem)- For the transformation to occur, the driving forces must exceed a critical activation barrier, noted A^c (see e.g. Bhadeshia, 2001a;

Van Rompaey , 2004). The activation barrier is related to the interfacial energy between parent and product phase, the energy required for the rapid propagation of the interface, the elastic energy stored to accommodate shape change and volume change during the phase transition and the energy dissipated during plastic deformation of parent and product phases (Baneijee and Mukhopadhyay, 2007). As shown in Figure 2.4, the différence of free energy between the parent and the product phases is higher than A(f)c when the température is lower than Mg, the martensite transfonuation start température, which means that the transformation occurs on un- dercooling without any applied forces. If the température exceeds Mg, the mechanical driving force must bring the extra energy to reach the activation barrier. In our study, the transforma­

tion is preceded by plastic flow, i.e. in a range of températures between the température at which plastic slip appears and M^, the upper bound température at which martensitic transfor­

mation can occur.

Figure 2.4: Energetic balance for martensitic transformation

Patel and Cohen (1953), pointed the fact that among the 24 possible variants, the selected orientation(s) must maximize the mechanical driving force.

In summary, the stress State acts on the mechanical driving force and therefore on the vari­

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ant sélection, while the température dictâtes the Chemical driving force. This point will be commented in the next session reviewing the experimental investigations led on this topic.

2.2.3 Nucléation and growth of martensitic plates

The martensitic transformation can occur at very low températures at which other processes in alloy cease (Kurdyumov, 1997) : for instance Bhadeshia (2001b) reported a martensite-start température Ms less than 4K for an alloy with composition by weight Fe — 34Ni — 0.22C%.

Furthermore, the martensitic transformation is characterized by a high rate of nucléation and growth, at speeds approaching that of sound in the métal. Consequently, no diffusion process can be related to the martensitic transformation. The interface between parent and product phases shows high mobility. Without the help of any thermal activation process, it cannot there- fore be an incohérent interface, i.e. an interface presenting an incompatibility in the atomic configuration of the two adjoining phases (see Porter and Easterling, 1992). In the case of the transformation of fcc austenite into bcc martensite, the interface between parent and prod­

uct phases is semi-coherent, i.e the structural misfit in the interface plane between parent and product phases is periodically accommodated by screw dislocations, in order to minimize the elastic energy associated to the interface (Porter and Easterling, 1992; Bhadeshia, 2001b). The transformation interfaces are then qualified to be glissile, i.e. their motion does not require any diffusion process. The nucléation of martensite is probably related to the dissociation of three dimensional arrays of dislocations. The faulted structure between the partial dislocations is con- sidered to be the martensitic embryo, with a glissile interface (Van Rompaey, 2004; Bhadeshia, 2001a, see). The embryo becomes a nucléus if accurate growth conditions are fulfilled: for the martensite to nucleate, the driving force, which, as explained in the previous subsection, can be supplied by thermal or mechanical loading, must exceed the activation barrier Acpc- This condition permits the rapid movement of dislocations, with a rate limited only by the usual barriers for the dislocation motion (Bhadeshia, 2001a). The plastic strain drives the marten­

sitic transformation by the introduction of deformation defects such as deformation twins or glide planes, which act as nucléation sites. In particular. Oison and Cohen (1975) pinpointed the génération of nucléation sites through shear band intersections. The transformation is also enhanced by an additional effect: the plastic flow accommodating the shape change associated to a martensitic variant induces new nucléation sites (Van Rompaey , 2004). The growth of the martensitic transformed région is govemed by the dislocation motion and proceeds as long as the transformation interface remains glissile. Furthermore, the growth of martensite plates is observed to stop when it encounters a strong physical barrier, typically a grain boundary or a previously formed martensite plate (Van Rompaey , 2004). Plastic strains induced in the parent phase by the transforming zone or the already formed variants introduce defects perturbing the interface semi-coherence. Therefore, plastic accommodation acts as a dragging process for the growth of martensite.

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Transformation Induced Plasticity in multiphase steels 19

2.3 Experimental investigations of TRIP effect

In this section, some experimental investigations performed previously are recalled. These experimental results are important for the numerical simulations of the mechanical properties of TRIP-assisted steels, since they motivate micromechanical or phenomenological models de- scribed in the following section.

2.3.1 Microstructural investigations

Many experimental issues on multiphase TRIP steels were addressed in the literature. In par- ticular, Jacques and coworkers hâve widely studied the microstructure of TRIP-assisted steels (see e.g. Jacques et al., 2001b; Jacques, 2004; Jacques et al., 2007). Figure 3.5 shows a typical micrograph, obtained by scanning électron microscopy (SEM), of a high Silicon TRIP-assisted multiphase Steel: it consists of a ferritic matrix and a dispersion of austenite and bainite grains located at the ferritic grains boundaries (Jacques et al., 2007).

Figure 2.5: Typical micrograph of a TRIP microstructure - Reproduced from Jacques et al.

(2007)

In multiphase carbon steels, the austenite can be found under different forms, i.e. as “blocky- type” grains or as “film-type” lamellae inteitwined with bainitic plates, (Matsumura et al., 1992;

Jacques et al., 2001b). The film shaped austenite usually forms when heating in the austenitic range is performed aflter cold-rolling, whereas granular “blocky-type” austenite is found after intercritical aimealing in the austenite/ferrite range, leading to smaller austenitic grain sizes (Matsumura et al., 1992). The austenite présent in the bainitic phase generally does not trans-

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form into martensite whereas the “blocky-type” austenite is likely to be subjected to strain- induced martensitic transformation. The transformation is supposed to develop by successive bursts, with up to 10 variants developing consecutively (see Jacques et al., 2007). However, in (Fischer et al., 2000), it was shown that proportional loadings lead to the formation of a unique variant, whereas non-proportional loadings favors the formation of several variants per grain.

Jacques et al. (2007) provided a required information for the development of any microme- chanical model (see Lani et al., 2007), i.e. the size of a représentative volume element as well as realistic flow properties of the constituent phases measured in situ by a combination of dig­

ital image corrélation and neutron diffraction. Austenite appeared barder than ferrite due to the carbon enrichment. At the grain level, it was observed that the transformation occurs in a discrète manner, with the appearance of 3 to 10 variants, whereas at the macroscopie scale, the heterogeneity of the microstructure caused the transformation rate to be continuons. It was also emphasized that the largest austenite grains would transform first during straining. Jacques et al. (2001b) pinpointed the fact that the composite type of strengthening was highly effective in low-alloy TRIP-aided multiphase steels, whereas it played a minor rôle in flilly austenitic steels.

Some investigations highlighted the relation between microstructure of TRIP-assisted steels and their fracture behaviour. In Huo and Gao (2005), the TEM analysis of fatigue fracture surfaces in a high-strength Steel showed that retained austenite transformed into martensite at the fatigue crack tip zone. As a conséquence of the absorbed energy during the transformation and of the crack closure process related to the compressive stresses resulting from the phase change, the propagation rate of the fatigue crack could be reduced. Lacroix et al. (2008) addressed the influence of the retained austenite volume fraction and stability on the fracture résistance of TRIP-assisted steels. The stability of the retained austenite had two major effects on the fracture behaviour of the Steel: it led to an extension of the necking zone and affected the void nucléation mechanism when damage started accumulating only after the austenite had transformed into martensite.

2.3.2 Effect of the stress State on the mechanical properties of TRIP- assisted steels

Patel and Cohen (1953) stated that the mechanical response of TRIP-aided multiphase steels dépends on the hydrostatic stress due to the dilatation component of the transformation strain.

Perdaheioglu et al. (2008a) analysed the effect of the stress state and strain path on the strain induced martensitic transformation in an austenitic stainless Steel. For proportional loadings, it was shown that the amount of tension of the stress State influences linearly the transformation rate which is in agreement with the Patel and Cohen theory: the increase in the transformation speed originates from the increase of the hydrostatic stress, due to the volumétrie expansion accompanying the martensite formation and acting on the other variants. This results in an isotropie increase in the driving force characterizing the transformation of the other variants.

Thus, the applied stress influences the rate of the transformation whereas plastic strains induces the transformation itself

In (Perdaheioglu et al., 2008b), the effect of the plastic strain was assessed: austenitic

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Transformation Induced Plasticity in multiphase steels 21

metastable stainless Steel samples were heated to a température at which the martensitic trans­

formation is suppressed, and were subsequently plastically strained to different levels. No évi­

dence of plastic pre-straining influence was found regarding the kinetics of transformation. The plastic pre-straining rather influenced the hardening of the material, more speciflcally, the driv- ing force necessary to slip became higher than the one necessary to martensitic transformation.

This behaviour was reported as stress-assisted transformation.

In Lebedev and Kosarchuk (2000), different mechanical tests (uniaxial tension, simple shear, Marciniak and equibiaxial testing) showed that the austenite transformation rate and the result- ing hardening behaviour was sensible to the stress triaxiality.

In addition, Jacques et al. (2007) provided insight on the behaviour of the TRJP-aided steels.

At the macroscopie scale, the stress-state dependence of the martensitic transformation rate and consequently on the hardening of TRIP-assisted multiphase steels was shown: a maximum transformation rate was observed at an intermediate level of stress triaxiality. The authors ar- gued that by properly tuning the austenite stability with respect to the stress State présent in the envisioned application, the strength/ductility balance could be optimized. Experimentally, the austenite stability can be adjusted by changing the testing température.

2.3.3 Influence of the microstructure on the transformation induced plas­

ticity

Several factors affect the martensitic transformation rate at the level of the grain of austenite.

First, the transformation rate dépends on the grain orientation with respect to the loading direc­

tion (Oliver et al., 2002), as well as on the austenite stability, which is related to several factors such as the austenite grain size, the carbon content or the stress State (Jacques et al., 2007). In tum, the transformation rate affects the strength versus ductility balance (see e.g. Jacques, 2004;

Jacques et al., 2007; Lacroix, 2007). Size effects are also prominent for the martensitic trans­

formation process: experimental studies show that résistance to the martensitic transformation increases when the austenite size decreases, see for instance the contribution from Reisner et al. (1996), in which the transformation of austenite précipitâtes in a copper matrix is investi- gated. Other experimental investigations on the austenite grain size may be found in Ulvan and Koursaris (1988) or Varma et al. (1994). Ulvan and Koursaris (1988) observed that a variation in grain size did not affect the formability of stainless steels, whereas it impacted the tensile properties of the materials, such as the ultimate tensile stress which slightly increased with decreasing grain size, or the uniform élongation, the total élongation and the strain hardening coefficient, which decreased with decreasing grain size.

2.4 Micro-mechanical modelling of nucléation and/or growth of martensite

As mentioned previously, the TRIP effect arises ffom different mechanisms:

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• the composite-type effect of a hard dispersed phase into a soft matrix, including also the fact that the interface between the old and new phases becomes impénétrable to disloca­

tions;

• the orientation effect, or Magee effect, which corresponds to the variant sélection by internai as well as extemally applied stress fields and the related oriented plastic accom­

modation;

• the accommodation effect, or Greenwood-Johnson effect, related to the additional plastic straining around the newly formed martensitic variant, coming ffom the transformation strain .

In the past décades, several attempts hâve been made to describe the mechanical behaviour of materials exhibiting TRIP effects. However, the models describing the phase transformation do not systematically account for the three mechanisms contributing to the TRIP effect. Three classes of models can be identified (Van Rompaey , 2004; Kouznetsova and Geers, 2008):

• the phenomenological constitutive models, which are based on the enrichment of macro­

scopie models with empirical observations at the microscopie level, using (semi-)analytical homogenization or statistical averaging techniques. These models hardly account for the orientation and the accommodation effects;

• the constitutive models at the microscale (i.e. the scale of the processes involved), which allow to model explicitly the partial transformation of the austenite phase, on the basis of a thermo-mechanical criterion, and for which the orientation and/or the accommodation effect can be analysed;

• the multiscale models, for which the modelling of the relevant microstructural features are upscaled to coarser scales via an appropriate homogenization technique.

A sélection of models describing the martensitic transformation is detailed in the following based on the above described classification. As mentioned in the previous chapter, the TRIP effect is influenced by extemal parameters, such as loading State or plastic pre-straining, as well as microstructural features such as the austenitic grain size. These variables affecting the phase transformation enrich some of the models described above, which will be highlighted in the text.

2.4.1 Constitutive model developments

An indepth review of the different existing constitutive models for the martensitic transfor­

mation, dating from before 2000, is provided in Cherkaoui et al. (1998, 2000). This review is updated in the following.

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Transformation Induced Plasticity in multiphase steels 23

2.4.1.1 Phenomenological modelling

The earliest phenomenological attempt to model the TRIP efFect was lead by Greenwood and Johnson (1965). In their approach, the transformation strain related to the transformation is only responsible for the plastic accommodation and a relation between the TRJP strain, the transformation strain and the applied stress is given in the one dimensional case. Leblond and coworkers proposed a micromechanical model aiming at accounting for the Greenwood Johnson effect, with plastic strains generated in a spherical inclusion of austenite partially undergoing the martensitic transformation (see Leblond et al., 1986a,b).

Oison and Cohen (1975) presented a model in which the intersection of shear bands in the austenite is considered to be the prédominant mechanism for strain-induced martensitic trans­

formation. In this model, the volume fraction of martensite is influenced only by plastic strains and température. This model was later extended by Stringfellow et al. (1992) using a self con­

sistent approach where the martensite has the shape of spherical inclusions and the stress State was included in the mechanical driving force for martensitic transformation. Papatriantafillou et al. (2006) developed constitutive équations for the behaviour of four-phased TRIP-assisted Steel, based on the Stringfellow et al. (1992) model. These laws were used to perform the fi- nite element analysis of necking in uniaxial tension, and to compute forming limit diagrams for sheets containing TRIP-assisted steels. In order to fit with the experimental observation that the number of intersections of shear bands increases with increasing strain rates, Tomita and Iwamoto (1995) developed a phenomenological constitutive thermocoupled model of TRIP steels, generalizing the Stringfellow et al. (1992) constitutive équations to account for the strain rate sensitivity in the shear band formation process. This model is used to study the deformation behaviour of a TRIP-steel cylinder under tension, with a finite element scheme. Experimental investigations on an austenitic stainless Steel under uniaxial tension and compression tests, re- vealed the stress-state dependence of the rate of shear band formation (Iwamoto et al., 1998).

The initially developed Tomita and Iwamoto (1995) transformation kinetics model was later improved by including the stress-state dependence of the génération of shear bands (Iwamoto et al., 1998; Iwamoto and Tsuta, 2000; Tomita and Iwamoto, 2001). Iwamoto and Tsuta (2000) accounted for the austenitic grain size in the deformation behaviour of austenite by means of an Hall-Petch type relation. The deformation behaviour of an austenitic stainless Steel cylinder was simulated with the finite element method, showing that the mechanical properties of TRIP steels, can be controlled by the austenitic grain size. The Tomita and Iwamoto model is the departure point for recent finite element efforts to model phase transformation (see e.g. Serri et al., 2005; Dan et al., 2008, 2007b). In Serri et al. (2005) the different contributions of the martensitic transformation to the overall plastic behaviour were investigated with the aim of as- sessing their influence in sheet métal forming. Improved constitutive laws based on the Tomita and Iwamoto model hâve recently been developed, in which température or plastic pre-strain contribute to the driving force for the martensitic transformation (Dan et al., 2007a; Li et al., 2007). However, these models do not take into account the variant sélection, and are not suitable under pure shear stress State (Iwamoto, 2004). In Iwamoto (2004), this model was incorporated in a multiscale framework, in order to account for the growth of elliptic martensite particles

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with a given orientation (see next subsection).

2.4.1.2 Microscale modelling

Reisner et al. (1996) investigated the strain-induced martensitic transformation of austenite précipitâtes in a copper matrix, experimentally and by numerical simulations of cold rolling tests. A thermomechanical transformation criterion was adopted: the différence of Gibbs free energy between the transformed and the initial States constituted the driving force for the martensitic transformation which has to be négative. The microstructure was described by a unit cell approach: the Représentative Volume Elément (RVE) was composed of a cylindrical copper matrix containing a central spherical austenite inclusion. The flow curve of the alloy was assigned to the copper matrix to address the particle size while the strength of a single crystal Fe particle was supposed not to be dépendent from size. As a conséquence, the effects of the austenitîc grain size were incorporated through the yield strength ratio between copper and austenite, which entered the model as a physical length scale. This simple model successfully captured the effect of the austenite grain size on the stability against martensitic transformation, which was evidenced experimentally.

Fischer and Reisner (1998) proposed a micromechanical criterion driving locally the trans­

formation of small amounts of the retained austenite, by fast movement of the interfaces be­

tween the phases. It has the form of a balance of dragging and driving forces:

V^pA(l>,hem{T'^)+ f aij\tsfe%^dV = Fc + AF+ WJi + W;i. (2.1) The different terms of this energetic criterion are now described. The driving force is com­

posed of

• a Chemical part, V^pA(f)chem{T'^) which is directly related to the différence of free Gibbs energy between the parent and product phases, and which has been evaluated in the liter- ature to range from 30 to lOOMJ/m^ (Lani et al., 2007; Fischer et al., 2000).

• a mechanical part related to the transformation deformation jy which is the Work associated to internai stresses inside the transforming zone (Tij\tsf and the trans­

formation strain. Among the 24 possible crystallographic variants of the martensite trans­

formation, the one maximizing the driving force is selected.

where is the volume of the transforming microregion, p is the density, 0 is the spécifie Helmholtz free energy and T'*' is the température of the transforming austenitic région.

The résistive barrier is the sum of

• the energy required to rebuild the product lattice Fc, which is often assumed to be constant over the volume of the transforming zone with a value ranging from 100 to 200MJ/m^

(Fischer et al., 2000);

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Transformation Induced Plasticity in multiphase steels 25

• the energy to croate the interface between parent and product phases, AT which is often neglected when compared to the other terms of the résistive barrier, partly due to the fact that martensite forms in the shape of needles, in order to minimize the interface energy between the parent and product phases;

• the elastic strain energy due to the fluctuation of the internai stress W^i and the plastic dissipation term due to the same internai stress W^i. Although in (Fischer et al., 2000), the accommodation terms are considered as small enough to be neglected when the vol­

ume fraction of the transforming zone is less than 2%, the values of -f Wpi)/V^^ are reported to range from 40 to 150MJ/m^.

Reisner et al. (1998) used this criterion to describe the onset and/or the kinetics of the strain induced martensitic transformation for two strongly different microstructures. A dilute Cu-Fe alloy containing small austenite particles which partially transform to martensite exhibiting a microstructure containing parallel bands of martensite was investigated. Size effects related to the austenite grain size as well as the load-type sensitivity were reported on the stability against strain-induced martensitic transformation. Simulations on a low-alloyed TRIP-assisted Steel microstructure showed that the kinetics of the transformation is strongly affected by the type of texture given by the orientations of the austenite grains. For instance. Fischer et al.

(2000) performed numerical simulations on a three dimensional unit cell model composed of 216 finite éléments, each element being understood as an austenite grain with a distinct lattice orientation. The unit cell was subjected to a cooling under Mg température combined with an extemal loading. The results showed the prominence of the Magee effect. In particular, during full or partial unloading while cooling continuously, experimental results showed a significant réduction of the irréversible TRIP strain, which cannot be explained by the available constitu­

tive équations. This arises from a backstress effect, related to the orientation effect: the internai stress State is responsible for the sélection of new variants foimed by transformation. As a re­

suit, the authors proposed to replace the plastic strain incrément and the TRIP strain incrément by an extended plastic strain incrément accounting for both the Greenwood-Johnson effect and the Magee effect. This modified constitutive équation is more adapted during unloading since ongoing plastification is then allowed. The energetic criterion of Fischer and Reisner (1998) has also been used in the work of Lani et al. (2007); Van Rompaey et al. (2006). In particular. Van Rompaey et al. (2006) used this phenomenological model in order to describe the strain-induced martensitic transformation in a single austenite grain. The variant orientation maximizing the mechanical driving force was identified for varions loading conditions. A three dimensional embedded cell model of an austenite inclusion surrounded by a ferritic matrix was employed to study the transformation of a single martensite plate with the most favourable variant orien­

tation through finite element simulations. The study showed that the mechanical driving force and the accommodation terms are on the same order of magnitude and are strongly affected by the shearing component of the transformation strain and the plastic straining prior to the trans­

formation. The global stress triaxiality resulting from the extemally applied load only affected the driving force.

Levitas and Stein (1997) described a phase transformation criterion, based on the fact that

Figure

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