Strain gradient based analysis of transformation induced plasticity in multiphase steels

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transformation induced plasticity in multiphase steels

THESE

pr´esent´ee en vue de l’obtention du grade de Docteur en Sciences de l’Ing´enieur

de l’Universit´e Libre de Bruxelles

Louise Mazzoni Leduc

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Thesis carried out within at the Universit´e Libre de Bruxelles (Faculty of Applied Sciences) BATir department

Supervisor : T.J. Massart.

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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 . . . 6

2 Transformation Induced Plasticity in multiphase steels 9 2.1 Heat treatment for TRIP steels: stabilization of austenite and microstructure optimization . . . 10

2.2 Characteristics of the martensitic transformation . . . 12

2.2.1 Crystallographic point of view . . . 12

2.2.2 A thermo-mechanical criterion for transformation induced plasticity . . 14

2.2.3 Nucleation and growth of martensitic plates . . . 16

2.3 Experimental investigations of TRIP effect . . . 17

2.3.1 Microstructural investigations . . . 17

2.3.2 Effect of the stress state on the mechanical properties of TRIP-assisted steels . . . 18

2.3.3 Influence of the microstructure on the transformation induced plasticity 19 2.4 Micro-mechanical modelling of nucleation and/or growth of martensite . . . . 19

2.4.1 Constitutive model developments . . . 20

2.4.2 Numerical multi-scale calculation . . . 25

2.5 Adopted strategy . . . 28

3 Strain gradient plasticity theories and related applications 31 3.1 Size effects at the micron scale . . . 31

3.2 Dislocation density and hardening of metals . . . 35

3.3 Size independent plasticity . . . 38

3.4 Overview on the available gradient plasticity formulations . . . 40

3.4.1 Thermodynamics restrictions for strain gradient plasticity theories . . . 41

3.4.2 Lower order models . . . 41

3.4.3 Higher order stress models . . . 44

3.4.4 Non local theories . . . 56

3.4.5 Recent assessments of higher order strain gradient plasticity theories. . 59

3.4.6 Discussion on specific boundary conditions related to higher order the- ories. . . 61

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4 Computational modelling of the phase transformation 65

4.1 Physical model for the transformation of a retained austenite inclusion . . . 65

4.2 Embedded cell model . . . 66

4.3 Selection of material parameters and model parameters . . . 67

5 Phase transformation effect in multiphase steels with a single-parameter strain gradient plasticity theory under small strain assumption 71 5.1 Strain gradient plasticity theory . . . 71

5.1.1 The generalized effective plastic strain rate . . . 71

5.1.2 Governing and constitutive equations . . . 72

5.1.3 Implementation issues . . . 73

5.2 Results . . . 74

5.2.1 Preliminary study: general picture without size effect . . . 74

5.2.2 Influence of the boundary conditions at the elasto-plastic boundary . . . 75

5.2.3 Influence of the microstructural parameters . . . 80

5.3 Discussion . . . 85

5.3.1 Discussion about the boundary conditions . . . 85

5.3.2 Size dependent strengthening from composite effect . . . 90

5.3.3 Size dependent transformation strain effect . . . 92

5.3.4 Microstructure optimization . . . 93

5.4 Conclusions . . . 93

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 95 6.1 Introduction . . . 95

6.2 Strain gradient plasticity model . . . 97

6.2.1 Generalized effective plastic strain rate . . . 97

6.2.2 Governing and constitutive equations . . . 98

6.2.3 Implementation issues . . . 99

6.2.4 Evolving plastic boundary conditions . . . 100

6.3 Results . . . 101

6.3.1 Material and loading parameters . . . 101

6.3.2 Separate effects of the gradient terms on the overall transformation hardening . . . 102

6.3.3 Effect of the gradient terms on each transformation hardening contribu- tions . . . 103

6.3.4 Effect of the austenitic grain size . . . 105

6.4 Discussion . . . 107

6.5 Conclusions . . . 113

7 Phase transformation effects in multiphase steels with a finite strain gradient plas- ticity theory 115 7.1 Strain gradient plasticity theory . . . 115

7.1.1 The generalized effective plastic strain rate . . . 115 vi

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7.2 Results . . . 120

7.2.1 Specific assumptions and fixed parameters . . . 120

7.2.2 Use of the single-parameter theory . . . 123

7.2.3 Multi-parameter parameter theory . . . 124

7.2.4 Effect of the shear component of the transformation strain. . . 126

7.2.5 Effect of the ausenitic grain size. . . 128

7.3 Discussion . . . 132

7.4 Conclusion . . . 142

8 Conclusion 145

Bibliography 152

List of publications 167

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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 have been carried out to study these grades of steels. In particular, from a numerical point of view, earlier studies within the framework 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 dependent 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 resistance of the martensite, in agreement with the typical grain size of the current TRIP-assisted steels (Jacques et al., 2007).

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Introduction

1.1 Industrial context

An increasing interest is nowadays devoted to the design of new lightweight and high resis- 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 have 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 Manganese which allows to stabilize the austenite at room temperature 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 neces- sitates specific 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 correlation between the microstructure (evo- lution during deformation or morphology) on one hand and the mechanical properties on the other hand. Some efforts have been presented in the literature, see for instance the contributions from Bayram et al. (1999); Zaefferer et al. (2004); Uchic et al. (2006); Barbier et al. (2009).

This work lies in the same orientation.

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Figure 1.1: Use of different steels grades for the manufacturing of a car, from 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 temperature, obtained thanks to an appro- priate heat treatment of cold rolled multiphase steels. The strain-induced transformation from austenite into harder 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 result, high strength levels are obtained. The TRIP effect may also lead to a delay in the onset of necking. Indeed, necking occurs when the hardening of a loaded specimen is not enough to compensate for the area reduction. The phase transformation preferentially occurs in a zone where necking is likely to happen, which leads to improved total elongations (Gr¨assel et al., 2000). Deformation further 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 regions of higher deformation, closer 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 specific advantages offered by TRIP-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.

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It was shown that TRIP-type sheets show larger fracture elongation, delay of necking and thus better formability. In Matsumura et al. (1992), a tensile strength of 1000 MPa and a total elongation of30% were reported for low carbon steels alloyed with silicium (with a chemical composition containing less than 0.4 wt% C). In (Gr¨assel et al., 2000), the mechanical proper- ties of austenitic Fe-Mn steels with addition of aluminium and silicon 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 mechanical properties, are likely to occur. For low contents on Manganese, namely less than20%, TRansformation Induced Plas- ticity occurs in a range of temperatures from50Cto200C. With an initial volume fraction of austenite of80%, due to the high amount of manganese, the sample reaches a total elongation of about80%and an ultimate tensile strength of about830MPa during a quasi-static tensile test at room temperature. 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 elongations (Gr¨assel 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 composite type response. In addition to the improvements explained by the relative volume fractions 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. Delinc´e et al., 2006, 2007). In the case of TRIP-assisted steels, several investigations have reported that smaller grain sizes tend to increase the stability of the austenite in fully austenitic steels (Reisner et al., 1996; Jimenez et al., 2007) and also play a role in the overall hardening enhancement (Ulvan and Koursaris, 1988). The main objective of the present manuscript is the computational analysis of this size effect in the context of TRIP- assisted steels. This work presents investigations which are purely numerical, 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 two mechanisms (e.g. Fischer and Reisner, 1998; Fischer et al., 2000; Furn´emont, 2003; Lani et al., 2007):

• the increase of the volume fraction of the harder martensitic phase contributing to an elevation of the global hardening through a composite type effect;

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

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 1.2, reproduced from (Jacques, 2004), shows the accumulation of dislocation, generated by the

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phase transformation, at the phase boundary between martensite and austenite which resists plastic slip.

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

Typical TRIP steels have austenitic grain sizes on the order of1µm(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 present case the mismatch of properties and shape change as well as the appearance of an interface impenetrable to dislocations, will dominate the statistically stored dislocations. As a result, strain gradient effects can significantly affect the response leading to an additional strengthening contribution. Hence the motivation of this work is to assess whether the behaviour of real TRIP steels involving retained austenite in the micrometer diameter range is significantly affected by strain gradients effects without which the remarkable improvement of the strength/ductility balance cannot be quantitatively captured.

1.3 Original aspects of the thesis and content

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

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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 resis- 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 connected to the representative 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 competition of the three mechanisms cited above, namely a composite type effect, the effect of the transfor- mation strain and the appearance of an impenetrable boundary. The composite type effect is supposed to be coupled to the change of plastic confinement as a consequence of the growth of the martensitic region, this association results in a higher order composite effect. An ad- ditional uncovered aspect is the quantification of the specific 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 impenetrable 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 present 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 γtsf are carefully assessed and their influence on the overall hardening are discussed. However, the most important issue addressed here is to determine 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 subsequent 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-effect mechanisms is given. The specific heat treatment required to allow the appearance of strain-induced martensitic transformation is set out. The martensitic transforma- tion is then described by commenting the crystallographic aspects as well as the mechanisms affecting plastic flow. Then, some general insight about the TRIP effect is provided. First, ex- perimental efforts to characterize the TRIP effects are briefly summarized, followed by the state

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of the art in multiscale and micromechanical computational modelling.

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

Among the various possibilities, the choice of the Fleck and Hutchinson (2001) strain gradient plasticity will be motivated for the present study, as a good trade off between the complexity of 3D Discrete Dislocation Dynamics in terms of computational time (Devincre and Robert, 1996) and more simple size dependent theories 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 impenetrable 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 have 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 from the transformation strain it- self. It is also revealed that the boundary conditions postulated at the elastic-plastic boundary have 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 presents 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 appreciable 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 role in the strengthening enhancement when the multi-parameter theory is used, which was already concluded under the small strain assumption. However, some differences in the trends are revealed, concerning 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 previous chapters are not to eliminate, they are just fitted to account for large strain field around the austenite inclusion. As a result, 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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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 have 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 subsequent chapters, a physical understanding of the phase trans- formation is proposed in terms of crystallographic, thermodynamic and plasticity aspects.

The subsequent 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 temperature as well as the applied stresses define the transformation regime:

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

• When the temperature ranges fromMs toMsσ > Ms, 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 consequence, plastic deformation will be induced by martensite transformation rather than by slip; the trans- formation is then denoted a stress-assisted transformation;

• Above theMsσtemperature, a significant plastic flow occurs prior to the martensitic trans- formation and promotes it by creating new nucleation sites; this mechanism is called strain-induced transformation. This regime lasts until a temperatureMdis 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 consequence 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 consequences 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 temperature, 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 temperature standing between Ac1 and Ac3 (which limits the α/γ 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 temperature. Indeed, increasing the carbon content depresses the martensite temperature start below zero (Oliver et al., 2007) and thereby increases the austenite stability.

Temperature

Time Intercritical annealing

Isothermal bainite treatment

To room temperature Ac3

T1 Ac1

T2

Figure 2.1: Schematic representation 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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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 heat 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 of600MPa and elongation of31%

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 heat treatment allowing to stabilize the austenite at room temperature 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 temperature were assessed for the case of a low alloyed steel. A sufficiently 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 temperature 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 heat treatment was used, the remaining austenite after bainitic for- mation was completely stabilized at room temperature and transformed to martensite gradually during a long straining range. For holding temperatures of400C and a holding time of400s, a 739MPa tensile strength and a30%maximum uniform elongation were reached.

Alloying elements also play a role 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 precipitation 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 manganese (Mn) or Si also enables to obtain austenite and ferrite at lower intercritical temperatures. 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 retention 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 harder phase grains (bainite and martensite) in the soft ferrite matrix.

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2.2 Characteristics of the martensitic transformation

At room temperature, the metastable austenitic phase transforms into martensite which can appear under different crystallographic structures. In the case of TRIP-steels, theα 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 ofǫmartensite, with hexagonal-packed cubic structure, is responsible for the perfect shape recovery. Many issues in material science field have 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 have 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 about17%along thea3direction and an identical expansion of about12%along thea2 anda1 direction (see also Van Rompaey , 2004).

a2

a3

a1

(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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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 macroscopic 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 γtsf along the habit plane and a dilatational componentδtsf, along the normal to the habit plane. The dilatation component is assumed to range between 0.03−0.04while 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 for a single cristal γtsf can attain20% (Van Rompaey , 2004). Bhadeshia (2001b) exposed a theoritical development for the transformation strain.

The combination between the Bain strain and an appropriate rigid body rotation result in a strain leaving a line undistorted and unrotated, which is referred to as an invariant line strain and will be denoted asRB in the following. As a result, the experimentally observed macro- scopic 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 strainP1 permits to get the observed macroscopic shape deformation but the wrong lattice structure (from figure 2.3a to figure 2.3b). WhenP1is combined to another invariant plane strainP2, an invariant-line strain (equivalent toRB) is applied to the structure which permits to obtain the correct crystal structure but the wrong macroscopic shape (from fig- ure 2.3b to figure 2.3c). To solve these discrepancies, another deformation should be applied in order to renderP2 invisible as far as the shape change is concerned. 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 LI in figure 2.3): it can be either slip or twinning. As a result, twinned or slipped martensite are likely to appear after phase transformation as shown is figure 2.3d. Note thatP2 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 than10variants can usually be found within an austenite grain (Jacques et al., 2007).

To complete this short overview, it must be mentioned that the inelastic processes related to the phase transformation are governed 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 internal stresses act for the variant selection;

• 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

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RB

(a) Austenite

P1

(b) Right shape;

wrong structure

P2

(c) Wrong shape; right structure

LI LI

Twinned martensite Slipped martensite

(d) Correct shape; correct structure

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

strain rates.

2.2.2 A thermo-mechanical criterion for transformation induced plastic- 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 nucleation 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 theT0 temperature, at which the stress-free austenite and martensite phases have the same

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Helmholtz free energyφchem, when no other work than Pressure-Volume work is involved (Van Rompaey , 2004). If the austenite phase is retained at temperatures significantly belowT0, the concept of driving forces and dragging forces is introduced. The reasoning below satisfies the 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 (∆φchem). For the transformation to occur, the driving forces must exceed a critical activation barrier, noted∆φC (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 (Banerjee and Mukhopadhyay, 2007). As shown in Figure 2.4, the difference of free energy between the parent and the product phases is higher than∆φC when the temperature is lower thanMs, the martensite transformation start temperature, which means that the transformation occurs on un- dercooling without any applied forces. If the temperature exceedsMs, 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 temperatures betweenM, the temperature at which plastic slip appears andMd, the upper bound temperature at which martensitic transfor- mation can occur.

∆φC(Ms)

∆φchem(T)

MDF

Ms T T0

α

γ

T emperature φchem

Figure 2.4: Energetic balance for martensitic transformation

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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- ant selection, while the temperature dictates the chemical driving force. This point will be commented in the next session reviewing the experimental investigations led on this topic.

2.2.3 Nucleation and growth of martensitic plates

The martensitic transformation can occur at very low temperatures at which other processes in alloy cease (Kurdyumov, 1997) : for instance Bhadeshia (2001b) reported a martensite-start temperatureMs less than4K for an alloy with composition by weightF e−34Ni−0.22C%.

Furthermore, the martensitic transformation is characterized by a high rate of nucleation and growth, at speeds approaching that of sound in the metal. 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 incoherent 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 nucleation 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 nucleus 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 ∆φC. 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 nucleation sites. In particular, Olson and Cohen (1975) pinpointed the generation of nucleation 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 nucleation sites (Van Rompaey , 2004). The growth of the martensitic transformed region is governed 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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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 have 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 electron 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.

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In multiphase carbon steels, the austenite can be found under different forms, i.e. as “blocky- type” grains or as “film-type” lamellae intertwined 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 after cold-rolling, whereas granular “blocky-type” austenite is found after intercritical annealing in the austenite/ferrite range, leading to smaller austenitic grain sizes (Matsumura et al., 1992). The austenite present 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 to10variants 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 representative volume element as well as realistic flow properties of the constituent phases measured in situ by a combination of dig- ital image correlation and neutron diffraction. Austenite appeared harder than ferrite due to the carbon enrichment. At the grain level, it was observed that the transformation occurs in a discrete manner, with the appearance of3to10variants, whereas at the macroscopic scale, the heterogeneity of the microstructure caused the transformation rate to be continuous. 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 role in fully 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 consequence 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 resistance 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 nucleation 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 depends on the hydrostatic stress due to the dilatation component of the transformation strain.

Perdahcio˘glu 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 volumetric expansion accompanying the martensite formation and acting on the other variants. This results in an isotropic 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 (Perdahcio˘glu et al., 2008b), the effect of the plastic strain was assessed: austenitic

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metastable stainless steel samples were heated to a temperature at which the martensitic trans- formation is suppressed, and were subsequently plastically strained to different levels. No evi- 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 specifically, 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 TRIP-aided steels.

At the macroscopic 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 present in the envisioned application, the strength/ductility balance could be optimized. Experimentally, the austenite stability can be adjusted by changing the testing temperature.

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 depends 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 turn, 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 resistance 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 precipitates 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 elongation, the total elongation and the strain hardening coefficient, which decreased with decreasing grain size.

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

As mentioned previously, the TRIP effect arises from 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 impenetrable to disloca- tions;

• the orientation effect, or Magee effect, which corresponds to the variant selection by internal as well as externally 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 from the transformation strain .

In the past decades, several attempts have 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- scopic models with empirical observations at the microscopic 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 selection 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 external 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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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 TRIP 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).

Olson and Cohen (1975) presented a model in which the intersection of shear bands in the austenite is considered to be the predominant mechanism for strain-induced martensitic trans- formation. In this model, the volume fraction of martensite is influenced only by plastic strains and temperature. 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 equations 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 equations 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 generation 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 metal forming. Improved constitutive laws based on the Tomita and Iwamoto model have recently been developed, in which temperature 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 selection, 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 precipitates in a copper matrix, experimentally and by numerical simulations of cold rolling tests. A thermomechanical transformation criterion was adopted: the difference of Gibbs free energy between the transformed and the initial states constituted the driving force for the martensitic transformation which has to be negative. The microstructure was described by a unit cell approach: the Representative Volume Element (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 dependent from size. As a consequence, the effects of the austenitic 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:

Vµρ∆φchem(Tγ) + Z

Vµ

σij|tsfǫtsfji dV =Fc+ ∆Γ +Welτ +Wplτ. (2.1) The different terms of this energetic criterion are now described. The driving force is com- posed of

• a chemical part,Vµρ∆φchem(Tγ)which is directly related to the difference of free Gibbs energy between the parent and product phases, and which has been evaluated in the liter- ature to range from30to100MJ/m3(Lani et al., 2007; Fischer et al., 2000) .

• a mechanical part related to the transformation deformation R

Vµσij|tsfǫtsfji dV, which is the work associated to internal stresses inside the transforming zoneσij|tsf and the trans- formation strain. Among the24possible crystallographic variants of the martensite trans- formation, the one maximizing the driving force is selected.

where Vµ is the volume of the transforming microregion, ρ is the density, φ is the specific Helmholtz free energy andTγis the temperature of the transforming austenitic region.

The resistive barrier is the sum of

• the energy required to rebuild the product latticeFc, which is often assumed to be constant over the volume of the transforming zone with a value ranging from100to200MJ/m3 (Fischer et al., 2000);

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• the energy to create the interface between parent and product phases,∆Γwhich is often neglected when compared to the other terms of the resistive 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 internal stress Welτ and the plastic dissipation term due to the same internal stressWplτ. 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 than2%, the values of(Welτ +Wplτ)/Vµare reported to range from40to150MJ/m3.

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 216finite elements, each element being understood as an austenite grain with a distinct lattice orientation. The unit cell was subjected to a cooling underMs temperature combined with an external 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 reduction of the irreversible TRIP strain, which cannot be explained by the available constitu- tive equations. This arises from a backstress effect, related to the orientation effect: the internal stress state is responsible for the selection of new variants formed by transformation. As a re- sult, the authors proposed to replace the plastic strain increment and the TRIP strain increment by an extended plastic strain increment accounting for both the Greenwood-Johnson effect and the Magee effect. This modified constitutive equation 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 various 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 externally applied load only affected the driving force.

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

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the dissipation increment per unit volume of transforming region, related to the transformation process, is the driving force which must exceed a threshold value for the phase transformation to occur. A new thermo-mechanical postulate, the postulate of realizability, was introduced, stat- ing that ”if only some dissipative process (plastic flow, phase transformation) can occur, it will occur, i,e. the first fulfilment of the necessary energetic condition is sufficient for the beginning of the dissipative process” (statement reported from Levitas, 1995a). Based on the postulate of realizability, new extremum principles allowed to determine the tensor of transformation strain active in the transforming volume. Different micromechanical situations, namely nucleation, interface propagation or reorientation of martensitic variants were addressed. Later, Levitas (1998) developed a general thermomechanical theory for the phase transformation in inelastic materials for both small and finite strains descriptions. The theory was expressed in a homoge- neously deformed material point, undergoing the phase transformation, and which could belong to a nucleus or a moving interface. The nucleation criterion took into account plastic dissipation, temperature variation during phase transformation and the variation of internal variables. The principle of realizability allowed to determine the unknowns parameters such as the position, shape, orientation of nuclei, etc. This model was used by Levitas et al. (1998a), who presented a numerical solution of the problem of transformation at shear-band intersection. It was shown that some experimental observations, such as the fact that shear-band intersections are major nucleation sites, could be at least qualitatively captured. Levitas et al. (1998b) proposed a new nucleation condition, incorporating the history of local stresses variation in the nucleus during the transformation process. Simple mechanical models accounting for fixed or moving inter- faces between old and new phases, which can be noncoherent (with discontinuous tangential displacements across the interface) or exhibit fracture mechanism (with crack at the interface) were also proposed. Solution algorithms together with numerical results for elasto-plastic prob- lems exhibiting phase transformation were detailed. The authors pointed out that noncoherence and fracture at the interface affect the phase transformation process and promote considerably nucleation. For elastic materials, the growth of a single region of new phase occurred whereas for elasto-plastic materials, a discrete microstructure, with complex multiple connected trans- forming regions was obtained. Note that no crystallographic consideration was accounted for in this study, the transformation strain was rather taken to be volumetric. Idesman et al. (1999) presented algorithmic and computational aspects of the implementation of a simple isotropic deformation model for the description of phase transformation and twinning in elasto-plastic materials at finite strains.

At finer scales, Cherkaoui et al. (1998) derived constitutive equations, in the small strain for- malism and with a uniform temperature assumption, of a transforming single crystal of austenite within a thermo-micromechanical framework, and considering the micromechanics of moving boundaries. Both the orientation effect (Magee effect) and the accommodation effect (Green- wood and Johnson (1965) effect) were accounted for. According to the crystallographic theory of Wechsler et al. (1953), the transformation strain was given by a set of 24 variants, unaffected by the austenitic plastic strain. This micromechanical model provided driving forces governing the nucleation and instantaneous growth of new martensitic ellipso¨ıdal shaped domains, with a given transformation strain and aspect ratio, as well as the progress of plastic strains. These

Figure

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