Youssri El Majaty

Jean-Baptiste Leblond

Djimedo Kondo

A novel treatment of Greenwood-Johnson's mechanism of transformation plasticity -Case of spherical growth of nuclei of daughter-phase

Keywords: Transformation plasticity, Greenwood-Johnson's mechanism, spherical phase growth, limit-analysis, finite element micromechanical simulations

, not only by eliminating the need for ad hoc hypotheses of limited validity, but more importantly by extending its results to more general situations involving large external stresses, comparable in magnitude to the yield stress of the weaker, motherphase. The theoretical results are compared to other theories, experiments, and finite element micromechanical simulations considering a representative volume of shape identical to that in the theory. In addition, the methodology presented paves the way to incorporation, in a future work, of the effect of anisotropies of morphologic type (tied for instance to growth of nuclei of daughter-phase elongated in a certain privileged direction) upon transformation plasticity; this will be done through consideration of representative unit cells of more complex shape.

Introduction

The wording transformation plasticity was coined to denote the anomalous plastic behavior of metals and alloys during solid-solid phase transformations. Such transformations occur notably during the cooling phase of thermomechanical treatments such as welding and quenching. The major impact of transformation plasticity upon residual stresses and distortions resulting from such treatments has been underlined by various authors, see e.g. [START_REF] Miyao | Analysis of temperature, stress and metallic structure in carburized-quenched gear considering transformation plasticity[END_REF]; [START_REF] Fukumoto | Three-dimensional FEM analysis of helical gear subjected to the carburized quenching process[END_REF]; [START_REF] Taleb | Prediction of residual stresses in the heat-affected zone[END_REF], among many others. This impact constitutes the major motivation for developing good models of transformation plasticity, to be implemented in finite element programmes and used for accurate predictions of residual stresses and distortions produced by thermomechanical treatments.

It is widely accepted nowadays that transformation plasticity arises from two distinct physical mechanisms, which were proposed by [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF] and [START_REF] Magee | Transformation kinetics, microplasticity and ageing of martensite in Fe-31 Ni[END_REF] respectively:

• In [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s interpretation, transformation plasticity is due to standard, dislocation-induced plasticity at the microscopic scale in the weaker, motherphase (that prevailing at high temperatures, with a generally much lower yield stress). Microscopic plasticity arises from the difference of specific volume between the phases coexisting during the transformation (volumetric part of the transformation strain), which generates internal stresses of sufficient magnitude to induce plasticity in the weaker one. The effect occurs even in the absence of external stress; but such a stress, when present, takes advantage of the internal "transformation-induced weakness" of the material to deform it plastically. • In contrast, in [START_REF] Magee | Transformation kinetics, microplasticity and ageing of martensite in Fe-31 Ni[END_REF]'s interpretation, transformation plasticity is not due to standard plasticity but to the deviatoric part of the transformation strain of the transforming regions. (In this interpretation the expression "transformation plasticity" is somewhat inappropriate, since the transformation strain responsible for the effect is of essentially pseudo-elastic, reversible nature; but the wording is classical). In Magee's view, when the external stress is zero the deviatoric part of the transformation strain varies randomly in direction in the material, and therefore averages out to zero; but when the external stress is nonzero, it "orients" this deviatoric part and the net result is a nonzero macroscopic strain.

In this paper, we concentrate on [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism. (Some comments on this point will be offered below). The most classical -and still widely used, despite its age -constitutive model of this mechanism is due to Leblond et al. (1986a[START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]. The basis of this model was an additive decomposition of microscopic strains, followed by homogenization and detailed analysis of the decomposition of macroscopic strains (Leblond et al., 1986a); this led to an expression of the "transformation plastic strain rate" as an average value over the domain occupied by the mother-phase, which was finally evaluated explicitly using a simple model of growth of a spherical nucleus of daughter-phase within a concentric spherical shell of mother-phase, considered as a typical representative volume element (RVE) in the transforming material [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF].

The explanation of the popularity of this model probably lies in the fact that the expression of the transformation plastic strain rate it proposes involves only standard mechanical parameters of the two phases, the yield stress of the weaker phase and the volumetric transformation strain, in the absence of any ad hoc parameter to be determined experimentally. Such a feature is essential for applications to the numerical simulation of thermomechanical treatments, since most of the time no experimental data concerning transformation plasticity are available for the specific material considered.1 [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]'s model however suffered from serious drawbacks, which have long been noted. First, it relied on some technical hypotheses of doubtful applicability (recalled below). Also, the disregard of elasticity and the hypothesis of instantaneous total "plastification" of the mother-phase it relied upon, led to an infinite value of the transformation plastic strain rate at the very beginning of the transformation, which has repeatedly been criticized. But the most important drawback was that the treatment was by nature limited to small external stresses only (much smaller in magnitude than the yield stress of the weaker phase); as a consequence it failed to predict the well-documented nonlinear dependence of the transformation plastic strain rate upon the magnitude of the external stress tensor (see e.g. Figs. 3 to 6 in the experimental section of [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s paper).

The model was improved in later years through refinement of the description of the mechanical behaviour of the mother-phase; see the works of [START_REF] Diani | Micromechanical modelling of the transformation induced plasticity (TRIP) phenomenon in steels[END_REF]; [START_REF] Cherkaoui | Couplings between plasticity and martensitic phase transformation : overall behavior of polycrystalline TRIP steels[END_REF]; [START_REF] Taleb | A micromechanical modeling of the Greenwood-Johnson mechanism in transformation induced plasticity[END_REF]; [START_REF] Vincent | Viscoplastic behavior of steels during phase transformations[END_REF]; [START_REF] Fischlschweiger | A mean field model for transformation induced plasticity including backstress effects for non-proportional loadings[END_REF]; Weisz-Paltrault (2017), among others. For instance, the issue of the infinite initial value of the transformation plastic strain rate was settled by [START_REF] Taleb | A micromechanical modeling of the Greenwood-Johnson mechanism in transformation induced plasticity[END_REF] and [START_REF] Weisz-Paltrault | Multiphase model for transformation induced plasticity. Extended Leblond's model[END_REF] by accounting for elasticity and discarding the hypothesis of instantaneous complete plastification of the mother-phase. The very author of the model and coworkers contributed to the effort by introducing viscous effects, which are often important at the temperatures of interest [START_REF] Vincent | Viscoplastic behavior of steels during phase transformations[END_REF]. [START_REF] Weisz-Paltrault | Multiphase model for transformation induced plasticity. Extended Leblond's model[END_REF] extended the model by accounting for the possibility of consecutive or simultaneous formation of several phases. It seems fair to say, however, that in spite of these efforts the model has remained essentially the same over the years and still suffers from serious shortcomings, most notably its lack of generality.

The aim of this paper is to propose an enhanced theoretical framework for the analysis of [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism. This framework will be based on the [START_REF] Mandel | Contribution théorique à l'étude de l'écrouissage et des lois de l'écoulement plastique[END_REF][START_REF] Hill | The essential structure of constitutive laws of metal composites and polycrystals[END_REF] theory of homogenization like before; but [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]'s approximate treatment, based on specific and disputable hypotheses, of the problem of a spherical volume of mother-phase containing a growing spherical core of daughter-phase, will be replaced by a much more rigorous and efficient treatment of the same problem, based on the powerful "kinematic approach" of limit-analysis. (See e.g. [START_REF] Leblond | Classical and sequential limitanalysis revisited[END_REF] for a very recent summary and review of the theory). The idea of using limitanalysis is logical in view of the assumption -in line with [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]'s hypotheses -of negligible effect of elasticity. The main difficulty raised by use of this theory, in the context of transformation plasticity, is the presence within the material of some "internal active mechanism" -the growth, within the matrix of mother-phase, of a daughter-phase with a different specific volume -not accounted for in limit-analysis, which considers only "passive" materials. This difficulty will be settled by including only, in the volume considered, the domain -a hollow sphere -occupied by the plastically deforming motherphase; the internal mechanism just mentioned will then simply become an additional loading condition imposed on the inner boundary of this volume. The presence of this extra loading will constitute the sole difference with elementary cells and loadings considered in theoretical analyses of porous plastic materials (with traction-free voids); and this will permit to use, with some adaptations, the results of some previous works on the ductile necessary to apply the model, are often scarce.

rupture of metals [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I -Yield criteria and flow rules for porous ductile media[END_REF][START_REF] Monchiet | A micromechanics-based modification of the Gurson criterion by using Eshelby-like velocity fields[END_REF].

The benefits to be expected from the new approach are three-fold:

• Use of the variational framework of limit-analysis will permit to dispense with some of [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]'s most criticizable hypotheses. Also, the greater rigour of the treatment will be seen to lead to a finite value of the transformation plastic strain rate at the onset of the transformation without demanding incorporation of the effect of elasticity. • More importantly, the wider generality of the treatment will permit to address not only small external stresses like before, but also stresses comparable in magnitude to the yield stress of the weaker mother-phase. The experimentally well-known nonlinear dependence of the transformation plastic strain rate upon the external stress tensor will thus be given a theoretical interpretation, apparently the first of its kind. • But perhaps the most important benefit will not lie in the improvement of the treatment of the problem of spherical growth of nuclei of daughter-phase, but in the perspectives opened, thanks to the versatility of the method, for future extensions to more complex situations. Such situations could include for instance growth of spheroidal rather than spherical nuclei of daughter-phase; the treatment of such a problem could permit to study and model effects arising from anisotropies of morphological type (tied to development of domains of daughter-phase of anisotropic, for instance elongated shape).

The paper is organized as follows:

• As a necessary prerequisite, Section 2 recalls some elements of Leblond's general theory of transformation plasticity (Leblond et al., 1986a) and approximate treatment of the model problem of growth of a spherical nucleus of daughter-phase within a spherical RVE of mother-phase [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]). • Section 3 then presents the limit-analysis of a spherical cell subjected to conditions of homogeneous boundary stress [START_REF] Mandel | Contribution théorique à l'étude de l'écrouissage et des lois de l'écoulement plastique[END_REF][START_REF] Hill | The essential structure of constitutive laws of metal composites and polycrystals[END_REF] on both its outer and inner boundaries. The treatment makes an extensive use of [START_REF] Monchiet | A micromechanics-based modification of the Gurson criterion by using Eshelby-like velocity fields[END_REF]'s generalization -based on trial velocity fields of [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion, and related problems[END_REF]'s type -of [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I -Yield criteria and flow rules for porous ductile media[END_REF]'s classical analysis of the same problem but in the absence of internal loading. • Section 4 explains how the results derived may be used to predict the value of the transformation plastic strain rate, assumed to arise only from [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism. An analytical, albeit somewhat complex expression of this strain rate is derived in general, and shown to reduce to simple forms in two special cases: (i) small external stresses; (ii) beginning of the transformation, whatever the external stresses. • Finally Section 5 compares the new theoretical predictions to (i) those of [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]'s former theory; (ii) some experimental results of [START_REF] Desalos | Comportement dilatométrique et mécanique de l'austénite métastable d'un acier A533[END_REF]; [START_REF] Coret | Experimental study of the phase transformation plasticity of 16MND5 low carbon steel under multiaxial loading[END_REF][START_REF] Coret | Experimental study of the phase transformation plasticity of 16MND5 low carbon steel induced by proportional and nonproportional biaxial loading paths[END_REF] for the A533 steel; and (iii) the results of some new finite element micromechanical simulations of the mechanism studied, considering the same typical spherical geometry as the theory.

2 Elements of Leblond's classical theory of transformation plasticity

Generalities

We consider, within some metal or alloy, a RVE Ω containing "mother-" and "daughter-" phases occupying respectively the domains Ω M and Ω D , separated by an interface I (Fig. 1). The volume fraction f of the daughter-phase is defined as

f ≡ vol(Ω D ) vol(Ω) (1)
where the symbol "vol" denotes the volume of a domain. The mother-phase may transform into the daughter-phase through motion of the interface I in the direction of Ω M .

Ω D Ω M I Fig. 1. The general RVE considered.
Use is made of the [START_REF] Mandel | Contribution théorique à l'étude de l'écrouissage et des lois de l'écoulement plastique[END_REF][START_REF] Hill | The essential structure of constitutive laws of metal composites and polycrystals[END_REF] homogenization theory assuming conditions of homogeneous boundary stress or strain.

Decomposition of microscopic strains

The behaviour of both phases is thermo-elastic-plastic. The microscopic strain ǫ k in phase k (k = M, D) thus splits into thermal, elastic and plastic parts:

ǫ k = ǫ th k + ǫ e k + ǫ p k (k = M, D).
(2)

• The thermal strain ǫ th k of phase k, depending on the temperature T , is defined as the "free strain" which would exist in a small, unstressed volume of this phase at this temperature, measured in reference to a given phase -in dilatometric diagrams generally the daughter -and a given reference temperature T 0 . It is of the form

ǫ th k (T ) = [α k (T -T 0 ) + β k ] 1 (k = M, D), (3)
where α k is the thermal expansion coefficient of phase k and β k is related to its free strain at the temperature T 0 . The relative change of specific volume ∆V V from the mother-to the daughter-phase (volumetric part of the transformation strain) depends on the temperature T and is given by

∆V V (T ) ≡ tr ǫ th D (T ) -ǫ th M (T ) = 3 [(α D -α M )(T -T 0 ) + β M -β D ] . (4)
Further comments on these strains and their macroscopic counterpart E thm (see below) are provided in Appendix A. • The elastic strain ǫ e k of phase k is related to the local stress σ k in this phase through the relation

ǫ e k = S : σ k (5)
where S is the (isotropic) elastic compliance tensor. No distinction is made here between the compliance tensor of the mother-phase, that of the daughter-phase and the overall compliance tensor; the idea being that the major factors influencing transformation plasticity are the standard plastic strain in the weaker mother-phase [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism), and the transformation strain [START_REF] Magee | Transformation kinetics, microplasticity and ageing of martensite in Fe-31 Ni[END_REF]'s mechanism); not the difference of elastic moduli between the phases nor the anisotropic nature of their crystal elasticity. • The plastic strains ǫ p k (k = M, D) of the phases conventionally include, in Leblond et al. (1986a)'s definition, not only standard, dislocation-induced plastic strains (arising from [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism), but also the deviatoric part of the transformation strain, possibly oriented by the external stress applied [START_REF] Magee | Transformation kinetics, microplasticity and ageing of martensite in Fe-31 Ni[END_REF]'s mechanism). (The inclusion of the latter strain in the plastic strain is admittedly illogical because of its different, pseudo-elastic nature; but it is in line with the equally illogical but widely accepted preservation of the wording "transformation plasticity" to designate Magee (1966)'s pseudo-elastic mechanism). With this definition, the plastic strain is discontinuous across the interface I separating the phases, its discontinuity

[[ǫ p ]] ≡ ǫ p D -ǫ p M (6)
being identical to the deviatoric part of the transformation strain.

Decomposition of macroscopic strains

With the hypothesis of identical elastic compliance tensors of the phases, the [START_REF] Mandel | Contribution théorique à l'étude de l'écrouissage et des lois de l'écoulement plastique[END_REF][START_REF] Hill | The essential structure of constitutive laws of metal composites and polycrystals[END_REF] "localization tensors" appearing in the formulae connecting the microscopic thermal, elastic and plastic strains to their macroscopic counterparts, are identical to the unit tensor. The macroscopic strain tensor E is thus given by

E ≡ ǫ Ω = E thm + E e + E p ,              E thm ≡ ǫ th Ω = (1 -f )ǫ th M (T ) + f ǫ th D (T ) E e ≡ ǫ e Ω = S : Σ E p ≡ ǫ p Ω , (7)
where

Σ ≡ σ Ω (8)
is the macroscopic stress tensor (the symbol . Ω denotes an average value over Ω).

The symbol "thm" in E thm stands for "thermo-metallurgical"; the word "metallurgical" is added to emphasize the fact that this strain does not only depend on the temperature T , but also on the volume fraction f of the daughter-phase, and hence on the "metallurgical structure". Some additional comments on this strain are provided in Appendix A.

Time-differentiation of the expression (7) 4 of the overall plastic strain E p yields, account being taken of the discontinuity of ǫ p across the interface I:

Ėp = ǫp Ω + 1 vol(Ω) I [[ǫ p ]] V n dS = ǫp Ω + ḟ [[ǫ p ]] I/Vn , (9)
where

[[ǫ p ]] I/Vn ≡ I [[ǫ p ]] V n dS I V n dS = I [[ǫ p ]] V n dS ḟ vol(Ω) (10)
denotes the average value of the discontinuity [[ǫ p ]] over the interface I, with a weight identical to the normal velocity V n of this interface.

The next step consists in distinguishing, in the local plastic strain rate ǫp , those parts due to variations of the macroscopic parameters Σ, T and f :

ǫp = δǫ p δΣ : Σ + δǫ p δT Ṫ + δǫ p δf ḟ (11)
where the notations are self-evident. 2 The unusual term proportional to ḟ here arises from the fact that even when the overall stress Σ and the temperature T are fixed, the progress of the transformation induces a local plastic flow through [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism. Inserting the expression (11) of ǫp in equation ( 9), we get

Ėp = Ėcp + Ėtp ,      Ėcp ≡ δǫ p δΣ Ω : Σ + δǫ p δT Ω Ṫ Ėtp ≡ δǫ p δf Ω + [[ǫ p ]] I/Vn ḟ . (12)
The strain rate Ėcp here may be interpreted as corresponding to "classical plasticity", insofar as it arises from the sole variations of the external stress and the temperature. On the other hand the strain rate Ėtp due to the sole variation of the fraction of the daughterphase, that is to the progress of the transformation, may be interpreted as corresponding to "transformation plasticity". Furthermore the latter strain rate naturally appears as a sum of two terms, δǫ p δf Ω

ḟ and [[ǫ p ]] I/Vn ḟ . The first corresponds to [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism since it arises from the standard (dislocation-induced) plastic flow generated by the progress of the transformation, and the second to [START_REF] Magee | Transformation kinetics, microplasticity and ageing of martensite in Fe-31 Ni[END_REF]'s mechanism since it arises from the "plastic strain discontinuity" between the two phases, that is the deviatoric part of the transformation strain. (The dependence of [[ǫ p ]] upon the external stress is implicit in equation ( 12) 3 ).

Additional hypotheses

Although the "classical plastic strain rate" Ėcp has been studied in detail by Leblond et al. (1986b), we shall focus here on the sole "transformation plastic strain rate" Ėtp . The reason is that numerous numerical calculations performed since Leblond et al. (1986a,b)'s work have shown that Ėcp becomes much smaller than Ėtp as soon as the transformation begins: in the former strain rate, the effect of the term δǫ p δΣ Ω : Σ roughly amounts to a slight modification of the value of Young's modulus (Leblond et al., 1986b) Also, for the detailed calculation of Ėtp , [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF] considered only [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism, and we shall follow the same path here. Two points of view are possible with regard to such a simplification.

• One may consider that the paper of [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF] and the present one only provide a partial treatment of the problem, which should be completed by an additional independent modelling of Magee (1966)'s mechanism. (It must be stressed, however, that the theoretical treatment of this mechanism is much more difficult than that of [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism; indeed in contrast to [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism which is of purely "mechanical" nature, that of [START_REF] Magee | Transformation kinetics, microplasticity and ageing of martensite in Fe-31 Ni[END_REF] involves complex couplings between metallurgical and mechanical aspects -so it seems impossible, when modelling it, to avoid introduction of some adjustable ad hoc parameter(s), thus impairing the practical application of the model, as explained above). • One may also consider that [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]'s paper and the present one apply only to materials in which [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism dominates.

The relative importance of the two mechanisms, according to the type of material and situation considered, and their possible experimental separation, are discussed in Appendix B; with the conclusion that although [START_REF] Magee | Transformation kinetics, microplasticity and ageing of martensite in Fe-31 Ni[END_REF]'s mechanism is present in different materials and situations, and may even completely dominate over that of [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF], in many cases the converse is true and Magee (1966)'s mechanism may safely be disregarded.

The spherical growth model and its approximate treatment

To evaluate the contribution of [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism to the expression (12) 3 of the transformation plastic strain rate Ėtp , [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF] consider the simplest possible model of growth of a spherical nucleus of daughter-phase within a concentric spherical matrix of mother-phase, subjected to some overall stress tensor Σ.

Figure 2 shows the RVE; the external radius is denoted R, and that of the central core of daughter-phase r f , where the lower index underlines the connection with the fraction f of this phase:

f = r 3 f R 3 . (13) r =r + u δ f f - r + r f δ f r + r + u f δ f δ Ω M Ω D R Σ + Fig. 2.
The spherical RVE considered by [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF].

Between times t and t + δt, the spherical layer lying between the radii r f and r f + δr f transforms from the mother-to the daughter-phase; the accompanying expansion generates increments of radial displacement δu -, δu + of the inner and outer boundaries of this layer. These displacements induce a plastic strain rate distribution within the RVE, which must be evaluated.

Leaving aside the contribution of Magee (1966)'s mechanism, and introducing the simplifying hypotheses of complete plastification of the weaker mother-phase and absence of plasticity in the stronger daughter-phase (justified by the large difference between their respective yield stresses), we transform the expression (12

) 3 of Ėtp into Ėtp = (1 -f ) δǫ p δf Ω M ḟ , (14)
which reduces the problem to evaluating the average value δǫ p δf Ω M

. The task is made more difficult by presence of the load Σ, which destroys spherical symmetry.

To obviate this difficulty, [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF] σ M where δǫ eq ≡ ( 2 3 δǫ p : δǫ p ) 1/2 denotes the equivalent increment of plastic strain, σ ′ ≡ σ -1 3 (tr σ)1 the local stress deviator and σ M the yield stress of the mother-phase, assumed to be ideal-plastic. They also introduce some approximations:

• Approximation (1): correlations between the quantities δǫeq δf and σ ′ are negligible so that 14); • Approximation (2): the average value of σ ′ over the mother-phase differs little from that over the entire RVE, so that σ ′ Ω M ≃ Σ ′ where Σ ′ ≡ Σ -1 3 (tr Σ)1 is the overall stress deviator. Equation ( 14) then becomes

δǫeq δf σ ′ Ω M ≃ δǫeq δf Ω M σ ′ Ω M in equation (
Ėtp = 3 2 1 -f σ M δǫ eq δf Ω M Σ ′ ḟ . ( 15
)
Whereas the tensorial average value δǫ p

δf Ω M appearing in equation ( 14) vanished with Σ, the scalar average value δǫeq δf Ω M appearing in equation ( 15) does not. Thus a good estimate of this new average value, applicable to small overall stresses, may be derived by taking Σ = 0 -thus restoring spherical symmetry.

Elasticity being neglected, the radial increments of displacement δu -, δu + of the bounding surfaces of the transforming shell are given by

δu -= 0 ; δu + = ∆V V δr f ( 16
)
where the first value results from complete rigidity (no elasticity, no plasticity) of the core of daughter-phase, and the second from the known variation of volume of the transforming shell. Then, with Σ = 0 and spherical symmetry, plastic incompressibility in the motherphase implies (in the absence of elasticity) that the purely radial increment of displacement there is given by

δu r (r) = δu + r 2 f r 2 = ∆V V r 2 f r 2 δr f . ( 17
)
It is then easy to calculate the local value of δǫ eq and the average value δǫeq δf Ω M through integration over r, and the result reads

δǫ eq δf Ω M = - 2 3 ∆V V ln f 1 -f . ( 18
)
The final expression of Ėtp then follows by equation ( 15): [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]]. ( 19)

3 Ėtp = - 1 σ M ∆V V (ln f ) Σ ′ ḟ [

Remarks

(1) Equation ( 19) predicts an infinite "slope" dE tp /df at the beginning of the transformation (f = 0). This prediction does not seem to be confirmed by experiments (see e.g. [START_REF] Desalos | Comportement dilatométrique et mécanique de l'austénite métastable d'un acier A533[END_REF], among many other works), and has been repeatedly criticized (see e.g. [START_REF] Taleb | A micromechanical modeling of the Greenwood-Johnson mechanism in transformation induced plasticity[END_REF] and Weisz-Paltrault ( 2017)), on the grounds that it is a consequence of the over-simplifying hypothesis of instantaneous complete plastification of the mother-phase, tied to the disregard of elasticity.

(2) [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF] have assessed Approximations (1) and ( 2) above through finite element micromechanical simulations. They found that whereas Approximation (1) is reasonable (see their Fig. 6), Approximation (2) gives a good value of σ ′ Ω M only during the first half of the transformation, the approximation σ ′ Ω M ≃ (1f )Σ ′ being in fact better (see their Fig. 7). With this improved approximation the expression (19) of Ėtp becomes [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF], improved]. ( 20)

Ėtp = - 1 σ M ∆V V (1 -f )(ln f ) Σ ′ ḟ [
3 Limit-analysis of a hollow sphere subjected to external and internal loadings

Preliminary remarks

Since [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]'s approach of [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism is based on disregard of elasticity, it seems natural to use the theory of limit-analysisin which elasticity plays no role either, see e.g. [START_REF] Leblond | Classical and sequential limitanalysis revisited[END_REF]'s review -for an alternative treatment.4 But this raises two difficulties.

• In [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s interpretation, the "driving force" of transformation plasticity is an "internal active mechanism" (the gradual growth of nuclei of some new phase with a different specific volume), in practice due to the variation of the temperature. But limit-analysis considers only "passive" materials devoid of such mechanisms.

• An ancillary difficulty is that the object of limit-analysis is to define the set of limitloads. Such loads are not arbitrary but lie on some hypersurface in their defining space. In contrast, when looking for an expression of the transformation plastic strain rate, we want the macroscopic stress to be arbitrary; how the two points of view can be reconciled is not a priori clear.

In this work the first, major difficulty will be obviated by excluding the growing core of daughter-phase from the volume considered. The internal mechanism just mentioned will then become an inner boundary condition (some prescribed displacement arising from the expansion of the embedded daughter-phase), perfectly accounted for by limit-analysis.

The same trick will be seen to also solve the second difficulty, albeit in a more subtle way.

The aim of the limit-analysis will be to derive the explicit equation of the set of limit-loads of the RVE considered, together with the associated overall flow rule, that is the expression of the various components of its overall deformation rate. How these results can be applied to the modelling of [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism of transformation plasticity will be explained in Section 4 below.

A final remark pertains to some important difference between the former treatment of the problem, presented in Subsection 2.5, and the new one to be discussed now. The old treatment relied on an assumption of small external stresses, since the average value

δǫeq δf Ω M
was evaluated at order 0 in Σ, that is for Σ = 0. In contrast the new treatment will not make any hypothesis on the magnitude of the applied stress tensor Σ. This is the keypoint that will permit to treat the case of external stresses of the order of the yield stress of the weaker, mother-phase.

Generalities

We thus consider (Fig. 3) a hollow sphere made exclusively of mother-phase. In line with the notations of Subsection 2.5, the outer and inner radii are denoted R and r f , respectively (with f given by equation ( 13)); the whole domain, Ω; the outer spherical shell of mother-phase, Ω M ; and the inner empty sphere, Ω D . For commodity in the calculations three sets of coordinates: Cartesian (x 1 , x 2 , x 3 ) with origin at the centre O of the sphere; cylindrical (ρ, φ, z) with z ≡ x 3 ; spherical (r, θ, φ), will be used, together with the naturally associated orthonormal bases (e 1 , e 2 , e 3 ); (e ρ , e φ , e z ); (e r , e θ , e φ ). On the outer boundary ∂Ω, a single unit normal vector oriented outwards, n ≡ e r , is considered. On the inner boundary ∂Ω D , two unit normal vectors are considered, n + ≡ e r oriented toward the exterior of Ω D , and n -≡ -e r oriented toward the exterior of Ω M .

Ω M n + _ n Σ Ω D x 3 . x 1 x 2 r f Σ n R O
Fig. 3. The spherical RVE considered here.

The outer and inner boundaries of the RVE Ω M , identical to ∂Ω and ∂Ω D , are subjected to conditions of homogeneous stress [START_REF] Mandel | Contribution théorique à l'étude de l'écrouissage et des lois de l'écoulement plastique[END_REF][START_REF] Hill | The essential structure of constitutive laws of metal composites and polycrystals[END_REF]:

     σ(x) . n(x) = Σ . n(x) on ∂Ω (r = R) σ(x) . n -(x) = -Σ . n -(x) on ∂Ω D (r = r f ) (21)
where Σ and Σ are symmetric second-rank tensors, the first representing the macroscopic stress applied onto the volume considered and the second some internal loading. The minus sign in equation ( 21) 2 is introduced conventionally so as to produce a more natural-looking expression of the virtual power of external forces, see equation ( 24) below.

It should be noted that equations ( 21) do not define the only possible choice of boundary conditions. It would be possible for instance to use, on the inner boundary, conditions of homogeneous boundary strain rate [START_REF] Mandel | Contribution théorique à l'étude de l'écrouissage et des lois de l'écoulement plastique[END_REF][START_REF] Hill | The essential structure of constitutive laws of metal composites and polycrystals[END_REF], compatible with the kinematic conditions imposed in the transformation plasticity problem by the expansion of the inner core of daughter-phase. The limit-analysis based on such boundary conditions would be somewhat more involved but ultimately lead to the same results. The choice of conditions of homogeneous boundary stresses is therefore made here for the sake of simplicity, without any detrimental consequences on the results.

For any displacement field u(x) defined over the domain of mother-phase Ω M and extended arbitrarily (but smoothly) over the void Ω D , we define some overall external and internal strain tensors E, E through the natural formulae

E ≡ ǫ(u) Ω ; E ≡ ǫ(u) Ω D , (22)
or equivalently by Green's theorem:

E ij = 1 vol(Ω) ∂Ω 1 2 (u i n j + u j n i ) dS ; E ij = 1 vol(Ω D ) ∂Ω D 1 2 u i n + j + u j n + i dS.
(23) Equations ( 23) make it clear that E and E are in fact independent of the arbitrary extension of the displacement field over Ω D .

With the boundary conditions (21), the virtual power P e of external forces exerted on the boundary of the RVE considered, that is the domain Ω M occupied by the sole motherphase, is given by

P e ≡ ∂Ω σ ij n j ui dS + ∂Ω D σ ij n - j ui dS = ∂Ω Σ ij n j ui dS + ∂Ω D Σ ij n + j ui dS = Σ ij ∂Ω 1 2 ( ui n j + uj n i ) dS + Σ ij ∂Ω D 1 2 ui n + j + uj n + i dS ,
that is by equations ( 23) and ( 13):

P e = vol(Ω)Σ ij Ėij + vol(Ω D )Σ ij Ėij = vol(Ω) Σ : Ė + f Σ : Ė . ( 24
)

Trial displacement fields and load parameters

The limit-analysis problem can in principle tolerate any macroscopic stress and strain tensors Σ, Σ, E, E. However in the problem of transformation plasticity of interest here, overall geometrical and material isotropy entails identity of the principal directions of all 4 tensors. Indeed the prescribed quantities in this problem consist of the external stress tensor Σ, which introduces 3 orthogonal planes of symmetry defined by its principal directions, plus the internal strain E, a multiple of the unit tensor (see equation ( 45) below) which does not modify the group of symmetries. The "responses" to these prescribed quantities, consisting of the internal stress tensor Σ plus the external strain tensor E, must respect the planes of symmetry defined by Σ, and therefore possess the same principal directions.

This means that we may choose the x 1 -, x 2 -and x 3 -directions parallel to the common principal directions of the tensors Σ, Σ, E, E; they are then all diagonal in the Cartesian basis (e 1 , e 2 , e 3 ), and their diagonal components may be denoted with a single index, Σ i ,

Σ i , E i , E i (i = 1, 2, 3).
This being said, the first issue to be addressed is that of the minimum number of incompressible trial displacement fields needed. The family of fields considered must account for all possible values of the kinematic parameters E 1 , E 2 , E 3 , E 1 , E 2 , E 3 . But these 6 parameters are tied by the relation

tr E = E 1 + E 2 + E 3 = f tr E = f E 1 + E 2 + E 3 (25)
resulting from incompressibility of the mother-phase. (This equation means that the volumetric strain of the RVE arises exclusively from the expansion of the core of daughterphase). Thus there are only 5 independent kinematic parameters, so that a minimum of 5 independent incompressible trial displacement fields is required.

We shall exploit here an analogy with the problem of a spherical RVE containing a concentric spherical traction-free void; this problem has been considered many times, starting with the pioneering work of [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I -Yield criteria and flow rules for porous ductile media[END_REF], for the derivation of models for porous plastic materials, in the context of prediction of ductile rupture. The more complex boundary conditions considered here, with the additional internal loading, are of no consequence upon the space of kinematically admissible, incompressible displacement fields adapted to the problem. We shall therefore use the elementary fields introduced by [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I -Yield criteria and flow rules for porous ductile media[END_REF], completed by the more complex [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion, and related problems[END_REF]-type fields proposed by [START_REF] Monchiet | A micromechanics-based modification of the Gurson criterion by using Eshelby-like velocity fields[END_REF], which read as follows:

                                       u (1) (x) = R 3 r 2 e r u (2) (x) = - x 1 2 e 1 - x 2 2 e 2 + x 3 e 3 = - ρ 2 e ρ + ze z u (3) (x) = 2 5f r 5 f r 4 - sin θ 2 e ρ + cos θ e z + 1 2f r 3 f r 2 1 - r 2 f r 2 3 cos 2 θ -1 e r u (4) (x) = x 1 e 1 -x 2 e 2 u (5) (x) = 2 5f r 5 f r 4 sin θ (cos φ e 1 -sin φ e 2 ) + 1 f r 3 f r 2 1 - r 2 f r 2 sin 2 θ cos(2φ) e r . (26)
The first three fields are axisymmetric: the first and second, proposed by [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I -Yield criteria and flow rules for porous ductile media[END_REF], correspond to a radial incompressible expansion and a uniform deviatoric straining of the spherical shell, respectively; the third, introduced by [START_REF] Monchiet | A micromechanics-based modification of the Gurson criterion by using Eshelby-like velocity fields[END_REF], is of Eshelby (1957)-type. The fourth and fifth fields are non-axisymmetric: again the fourth, proposed by [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I -Yield criteria and flow rules for porous ductile media[END_REF], corresponds to a uniform deviatoric straining of the shell, whereas the fifth, introduced by [START_REF] Monchiet | A micromechanics-based modification of the Gurson criterion by using Eshelby-like velocity fields[END_REF], is of Eshelby (1957)-type. 5 Using Cartesian, cylindrical or spherical coordinates to calculate the integrals of equations ( 23) providing the overall external and internal strain tensors corresponding to these 5 fields, 6 we get with obvious notations:

                                     E (1) 1 = E (1) 2 = E (1) 3 = 1 ; E (1) 1 = E (1) 2 = E (1) 3 = 1 f E (2) 1 = E (2) 2 = - 1 2 , E (2)
3 = 1 ; E

(2)

1 = E (2) 2 = - 1 2 , E (2)
3 = 1

E (3) 1 = E (3) 2 = - 1 5 , E (3) 3
= 2 5 ; E (3) 1 = E (3) 2 = - 1 5f , E (3) 3
= 2 5f E (4) 1 = 1 , E (4) 2 = -1 , E (4)
3 = 0 ; E (4) 1 = 1 , E (4) 2 = -1 , E (4)
3 = 0

E (5) 1 = 2 5 , E (5)
2 = - 2 5 , E (5)
3 = 0 ; E (5) 1 = 2 5f , E (5)
2 = - 2 5f , E (5)
3 = 0. ( 27
)
Note that all fields except u (1) have tr E (i) = 0; that is, the volumetric strain of the RVE is to be described through a single field, u (1) .

We now consider an incompressible trial displacement field of the form 23) rather than ( 22) permits to circumvent the problem of defining smooth extensions of the fields u (i) (i = 1, ..., 5) over the void domain Ω D ; a non-trivial task in view of the singularity of some of these fields at the origin.

u(x) ≡ 5 i=1 q i u (i) (x) (28
where the q i 's (i = 1, ..., 5) are kinematic parameters. (Note that although any prescribed values of the kinematic parameters E 1 , E 2 , E 3 , E 1 , E 2 , E 3 obeying relation ( 25) may be matched through some suitable choice of the q i 's, this does not mean that the fields u (i) (i = 1, ..., 5) define a basis of the space of general admissible fields; equation (28) just defines the projection of such fields onto the subspace generated by the fields u (i) (i = 1, ..., 5)).

The components of the corresponding overall external and internal strain tensors read, by equations ( 27):

                                             E 1 = q 1 - q 2 2 - q 3 5 + q 4 + 2q 5 5 E 2 = q 1 - q 2 2 - q 3 5 -q 4 - 2q 5 5 E 3 = q 1 + q 2 + 2q 3 5 E 1 = q 1 f - q 2 2 - q 3 5f + q 4 + 2q 5 5f E 2 = q 1 f - q 2 2 - q 3 5f -q 4 - 2q 5 5f E 3 = q 1 f + q 2 + 2q 3 5f . ( 29
)
The load parameters Q i (i = 1, ..., 5) "conjugate" to the kinematic parameters q i are defined via the expression of the virtual power of external forces, P e ≡ 5 i=1 Q i qi , applicable to arbitrary values of the virtual velocities qi . By equations ( 24) and ( 29), they are given by

                                     Q 1 ≡ vol(Ω) Σ 1 + Σ 2 + Σ 3 + Σ 1 + Σ 2 + Σ 3 Q 2 ≡ vol(Ω) Σ 3 - Σ 1 2 - Σ 2 2 + f Σ 3 - Σ 1 2 - Σ 2 2 Q 3 ≡ vol(Ω) . 2 5 Σ 3 - Σ 1 2 - Σ 2 2 + Σ 3 - Σ 1 2 - Σ 2 2 Q 4 ≡ vol(Ω) Σ 1 -Σ 2 + f Σ 1 -Σ 2 Q 5 ≡ vol(Ω) . 2 5 Σ 1 -Σ 2 + Σ 1 -Σ 2 . ( 30
)
Remark. Although the family of trial displacement fields considered is directly borrowed from the work of [START_REF] Monchiet | A micromechanics-based modification of the Gurson criterion by using Eshelby-like velocity fields[END_REF], the fields u (3) , u (5) of Eshelby (1957)-type play different roles in the previous work and here. For a traction-free void, as considered by [START_REF] Monchiet | A micromechanics-based modification of the Gurson criterion by using Eshelby-like velocity fields[END_REF], there are only 3 independent kinematic parameters, E 1 , E 2 , E 3 , so that no more than 3 displacement fields are strictly necessary; the fields u (3) , u (5) were added by [START_REF] Monchiet | A micromechanics-based modification of the Gurson criterion by using Eshelby-like velocity fields[END_REF] to [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I -Yield criteria and flow rules for porous ductile media[END_REF]'s fields u (1) , u (2) , u (4) merely as an improvement, to get a better approximation of the overall yield criterion and flow rule (at the expense of an additional minimization of the overall dissipation). With the internal loading considered here, the fields u (3) , u (5) become indispensable since there are 5 independent kinematic parameters and therefore a minimum of 5 fields is required.

Overall plastic dissipation

The local plastic dissipation in the mother-phase (rigid-ideal-plastic material obeying the von Mises criterion with yield stress σ M ) is defined as the product σ M ǫeq where ǫeq ≡ ( 2 3 ǫ : ǫ) 1/2 is the equivalent strain rate. For any incompressible trial velocity field u(x) defined by the rates qi (i = 1, ..., 5), identical to the time-derivative of a trial displacement field u(x) of type (28), the overall plastic dissipation D is then defined as

D( q1 , ..., q5 ) ≡ Ω M σ M ǫeq dΩ. ( 31
)
This dissipation is independent of the value of the tensor Σ characterizing the internal loading. It follows that its value is exactly the same as in the case considered by [START_REF] Monchiet | A micromechanics-based modification of the Gurson criterion by using Eshelby-like velocity fields[END_REF] of a traction-free void. We shall therefore content ourselves with providing their final, approximate result for this value, referring the interested reader to their paper for details of their approximations and calculations:

D( q1 , ..., q5 ) = vol(Ω) σ M 1/f 1 q 2 2 + 4 3 q 2 4 + 4 q 2 1 + 6g(f ) 25 q 2 3 + 4 3 q 2 5 x 2 1/2 dx x 2 (32)
where the change of variable x ≡ R 3 /r 3 has been used, and g(f ) is the positive function defined by the formula

g(f ) ≡ 1 -4f 1 -f 2/3 2 1 -f . (33)
(The integral in equation ( 32) may be calculated in closed form but the result will not be needed here).

Overall yield criterion

The fundamental inequality of the kinematic approach of limit-analysis reads [START_REF] Hill | On the state of stress in a plastic-rigid body at the yield point[END_REF])

5 i=1 Q i qi ≤ D( q1 , ..., q5 ) (34)
for all quintuplets of sustainable loads7 (Q 1 , ..., Q 5 ) and rates of kinematic parameters ( q1 , ..., q5 ). The set of limit-loads, that is the overall yield locus, is the boundary of the domain defined in the space of quintuplets (Q 1 , ..., Q 5 ) by inequalities ( 34) for all values of the quintuplet ( q1 , ..., q5 ); that is the envelope of the family of hyperplanes of equation 5 i=1 Q i qi = D( q1 , ..., q5 ) parametrized by the quintuplet ( q1 , ..., q5 ). The parametric equations of this envelope read

Q i = ∂D ∂ qi ( q1 , ..., q5 ) (i = 1, ..., 5) (35)
where the qi 's act as parameters; in these equations the quintuplets (Q 1 , ..., Q 5 ) and ( q1 , ..., q5 ) are no longer independent like in inequality (34), but tied through the macroscopic constitutive law -that is, the quintuplet of kinematic rates ( q1 , ..., q5 ) is that actually generated by the load (Q 1 , ..., Q 5 ).

In order to get the explicit equation of the overall yield locus, we must eliminate the parameters qi (i = 1, ..., 5) in equations ( 35). This may be done by using Gurson's lemma, stated and proved with maximum clarity and economy by Madou and Leblond (2012):

Gurson's lemma. Let I(α, β) denote the integral defined by

I(α, β) ≡ x 2 x 1 α 2 + β 2 x 2 dx x 2 (36)
where x 1 and x 2 are fixed positive bounds. Then the partial derivatives of I are connected through the following relation independent of α and β:

∂I ∂α 2 + 2 x 1 x 2 cosh ∂I ∂β - 1 x 2 1 - 1 x 2 2 = 0. ( 37
)
This lemma, applied here to equation ( 32) with

x 1 ≡ 1, x 2 ≡ 1 f , α ≡ q 2 2 + 4 3 q 2 4 , β ≡ 4 q 2 1 + 6g(f ) 25 q 2 3 + 4 3 q 2 5 , I(α, β) ≡ D vol(Ω) σ M , (38) yields ∂D/∂α vol(Ω) σ M 2 + 2f cosh ∂D/∂β vol(Ω) σ M -1 -f 2 = 0. ( 39
)
But the definitions (38) 3,4 of α and β imply that

                     ∂D ∂ q1 = 4 q1 β ∂D ∂β ∂D ∂ q3 = 6g(f ) q3 25β ∂D ∂β ∂D ∂ q5 = 8g(f ) q5 25β ∂D ∂β ;          ∂D ∂ q2 = q2 α ∂D ∂α ∂D ∂ q4 = 4 q4 3α
∂D ∂α , and it follows, using again these definitions, that

             1 4 ∂D ∂ q1 2 + 25 6g(f ) ∂D ∂ q3 2 + 25 8g(f ) ∂D ∂ q5 2 = ∂D ∂β 2 ∂D ∂ q2 2 + 3 4 ∂D ∂ q4 2 = ∂D ∂α 2 .
Inserting these expressions into equation (39), then using equation ( 35) and the definitions (30) of the load parameters Q i , we get the equation of the overall yield locus in the form

1 σ 2 M T 3 - T 1 2 - T 2 2 2 + 3 4σ 2 M (T 1 -T 2 ) 2 +2f cosh    1 σ M (S 1 + S 2 + S 3 ) 2 4 + 2 3g(f ) S 3 - S 1 2 - S 2 2 2 + (S 1 -S 2 ) 2 2g(f ) 1/2    -1 -f 2 = 0 (40)
where S 1 , S 2 , S 3 , T 1 , T 2 , T 3 are the diagonal components of the tensors S and T defined by

S ≡ Σ + Σ ; T ≡ Σ + f Σ. (41)
Re-expressing quantities in equation ( 40) in terms of the invariants of S and T, we get the final form of the overall yield criterion:

Φ(Σ, Σ, f ) ≡ T 2 eq σ 2 M + 2f cosh S H σ M -1 -f 2 = 0 , S H ≡ 1 4 (tr S) 2 + 2 3g(f ) S 2 eq 1/2 (42)
where X eq ≡ 3 2 X ′ : X ′ 1/2 denotes the von Mises norm of a given stress tensor X (X ′ is the deviator of X); Φ(Σ, Σ, f ) is the overall yield function.

In the absence of an internal loading (Σ = 0), S eq and T eq become identical to Σ eq , tr S to tr Σ and the criterion (42) reduces to that of [START_REF] Monchiet | A micromechanics-based modification of the Gurson criterion by using Eshelby-like velocity fields[END_REF] (their equation ( 41)). In the general case, however, it is new.

Overall flow rule

We follow here Hill (1951)'s classical approach, adapted to the present situation. Let E denote the space of pairs (X, X) of symmetric second-rank tensors, equipped with the scalar product • defined by (X, X) • (Y, Y) ≡ X : Y + X : Y. Consider, within E:

• a pair (Σ, Σ) of overall stress tensors defining a limit-load;

• the pair ( Ė, Ė) of overall external and internal strain rates generated by this limit-load;

• another pair (Σ * , Σ * ) of overall stress tensors defining an arbitrary sustainable load.

Consider also the following fields:

• the stress field σ(x) resulting from application of the limit-load (Σ, Σ);

• the strain rate field ǫ(x) generated by this stress field through the local plastic flow rule; • a stress field σ * (x) statically admissible with the pair (Σ * , Σ * ) and plastically admissible (respecting the inequality σ * eq ≤ σ M at every point) -the existence of such a field being guaranteed by the very definition of a sustainable load.

Then, by the expression (24) of the virtual power of external forces and the principle of virtual work,

vol(Ω) (Σ -Σ * ) : Ė + f (Σ -Σ * ) : Ė = Ω M (σ -σ * ) : ǫ dΩ.
The integrand here is non-negative as a consequence of the local normality rule. It follows that

(Σ -Σ * ) : Ė + (Σ -Σ * ) : f Ė = (Σ -Σ * , Σ -Σ * ) • ( Ė, f Ė) ≥ 0
for every sustainable load (Σ * , Σ * ).

It results from there, following a classical reasoning, that the pair ( Ė, f Ė) is positively collinear in the space E to the outer normal vector to the set of limit-loads, that is the overall yield locus of equation Φ(Σ, Σ, f ) = 0; thus there exists a non-negative scalar Λthe overall plastic multiplier -such that

Ė = Λ ∂Φ ∂Σ (Σ, Σ, f ) ; f Ė = Λ ∂Φ ∂Σ (Σ, Σ, f ). ( 43
)
Calculating the derivatives here, we get the final form of the two-part overall flow rule:

             Ė = Λ σ 2 M 3T ′ + f sinh(S H /σ M ) S H /σ M 1 2 (tr S) 1 + 2 g(f ) S ′ Ė = Λ σ 2 M 3T ′ + sinh(S H /σ M ) S H /σ M 1 2 (tr S) 1 + 2 g(f ) S ′ , Λ ≥ 0 (44)
where again a prime applied to a tensor denotes its deviatoric part.

Note that equations (44) are compatible with relation (25) connecting the traces of the outer and inner strain tensors.

4 Application to Greenwood-Johnson's mechanism of transformation plasticity

Principle and preliminaries

An elementary volume in a metal or alloy undergoing a phase transformation under stress is represented by the RVE defined in Subsection 3.2, subjected to a prescribed external stress Σ plus an internal strain rate Ė resulting from the expansion of the gradually growing core of daughter-phase. The elements of the problem are as follows:

• The data include the tensors Σ and Ė -the calculation of the second is presented just below. • The unknowns include the internal stress tensor Σ, the overall strain rate Ė and the plastic multiplier Λ. What is looked for is Ė; Σ and Λ are ancillary unknowns of no intrinsic interest.

• The equations available to determine the unknowns from the data are the overall yield criterion (42) and two-part flow rule (44).

Note that this summary elucidates the second difficulty mentioned in Subsection 3.1, in that it shows how an expression of the transformation plastic rate may be derived for arbitrary values of the external stress from limit-analysis, even though this theory concentrates only on special (limit-) loads. The keypoint is that the overall yield criterion ties the external and internal stress tensors together but does not constrain their values individually, so that the value of the former may still be chosen at will -the value of the latter will simply "adjust" through the criterion.

The first, preliminary task is to determine the value of the internal strain rate Ė resulting from the transformation. Consider the situation of Subsection 2.5 wherein the spherical layer lying between r f and r f + δr f transforms from the mother-to the daughter-phase within a small time-interval [t, t + δt], generating a radial increment of displacement δu + on its outer boundary. The diagonal components of the increment of the internal strain tensor are then obviously

δE 1 = δE 2 = δE 3 = δu + r f = ∆V V δr f r f = 1 3 ∆V V δf f
where equations ( 13) and ( 16) have been used; the value of the internal strain rate tensor follows:

Ė = 1 3 ∆V V ḟ f 1. (45)
A straightforward consequence of equation ( 45) is that the deviatoric part of the internal strain rate tensor Ė is zero. By the flow rule (44) 2 combined with the definitions (41) of the tensors S and T, this implies that

Σ ′ = - 3 + 2 g(f ) sinh(S H /σ M ) S H /σ M 3f + 2 g(f ) sinh(S H /σ M ) S H /σ M Σ ′ . (46)
A second preliminary task is to determine the relation between the transformation plastic strain rate Ėtp and the overall strain rate Ė resulting from the limit-analysis. This limitanalysis being carried out in the absence of elasticity and without any variations of Σ and T , equations ( 7) 1 in rate form plus (12) 1 yield Ė = Ėthm + Ėtp . But Ėthm is a multiple of the unit tensor by equation ( 7) 2 , and Ėtp is traceless by equation ( 12) 3 . It follows that Ėtp identifies with the deviatoric part of the overall strain rate Ė obtained in the limit-analysis:

8 Ėtp ≡ Ė′ = Λ σ 2 M 3T ′ + 2f g(f ) sinh(S H /σ M ) S H /σ M S ′ (47)
where the flow rule (44) 1 has been used.

8 And, of course, Ėthm identifies with the hydrostatic part of Ė; this is confirmed by equations ( 25) and ( 45) which yield the value 1 3 ∆V V ḟ 1 for this hydrostatic part.

Special case: small external stresses

In a first step, we shall consider the case of small applied stresses (Σ eq ≪ σ M ), in which a simple and explicit formula for the transformation plastic strain rate may be obtained; Ėtp will be formally calculated at order 1 in Σ ′ . Preliminary calculations of some quantities at order 0 will however be necessary.

Calculations at order 0. At this order, Σ ′ is considered as nil. It then follows from equation ( 46) that Σ ′ is also zero, so that by equation ( 42) 2 , S H = 1 2 |tr S|. The criterion (42) 1 then yields

2f cosh S H σ M -1 -f 2 = 0 ⇒ cosh S H σ M = 1 2 f + 1 f ⇒ S H σ M = -ln f
where we have discarded the negative solution S H σ M = ln f since S H is by definition nonnegative. From there follows that

sinh(S H /σ M ) S H /σ M = 1 -f 2 -2f ln f . ( 48
)
Combination of the flow rule (44) 2 and the expression (45) of Ė then yields

Λ σ 2 M 1 -f 2 -4f ln f tr S = 1 3 ∆V V ḟ f .
Since Λ is positive and in practice so is also ∆V V (see Fig. Λ

σ 2 M = 2 3σ M ∆V V ḟ 1 -f 2 . ( 49
)
Calculations at order 1. Equation ( 46) shows that Σ ′ is of order 1 in Σ ′ ; more precisely, using the value of sinh(S H /σ M ) S H /σ M at order 0, equation ( 48), we get

Σ ′ = - 1 -f 2 -3f g(f ) ln f 1 -f 2 -3f 2 g(f ) ln f Σ ′
and therefore by the definitions (41) of S and T,

S ′ = 3f (1 -f )g(f ) ln f 1 -f 2 -3f 2 g(f ) ln f Σ ′ ; T ′ = (1 -f ) 2 (1 + f ) 1 -f 2 -3f 2 g(f ) ln f Σ ′ .
We then see that in the expression (47) of Ėtp , we only need the values of sinh(S H /σ M ) S H /σ M and Λ σ 2 M at order 0; using equations ( 48) and ( 49), we finally get the desired value of Ėtp at order 1:

Ėtp = 2 σ M ∆V V (1 -f ) 2 1 -f 2 -3f 2 g(f ) ln f Σ ′ ḟ [Σ eq ≪ σ M ]. ( 50
)
Comments on this result will be made below, but it is worth noting immediately that the "slope" dE tp /df at the onset of the transformation is no longer infinite like in [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]'s work, but amounts to

dE tp df (f = 0) = 2 σ M ∆V V Σ ′ [Σ eq ≪ σ M ]. ( 51
)
This shows that the infinite slope obtained in the previous work was primarily due to the inadequacy of the "uncontrolled" Approximations ( 1) and ( 2) it relied upon, see Subsection 2.5. (The disregard of elasticity however also contributed to this incorrect result, see the works of [START_REF] Taleb | A micromechanical modeling of the Greenwood-Johnson mechanism in transformation induced plasticity[END_REF] and Weisz-Paltrault ( 2017)).

General case: arbitrary external stresses

We shall now see that for arbitrary external stresses (Σ eq comparable to σ M ), the transformation plastic strain rate Ėtp still admits an analytical expression, but now involving a quantity which must be evaluated numerically by solving a transcendental equation.

Let us define the following dimensionless quantities:

X ≡ Σ eq σ M ; X ≡ Σ eq σ M ; Y ≡ 1 3 tr S σ M ; Z ≡ S H σ M , (52)
the first of which is supposedly known whereas the other three are not.

Let us first show how the values of Y and Z may be deduced from that of X. Equation ( 46) implies that the tensors Σ ′ and Σ ′ are negatively collinear so that

Σ ′ = - Σ eq Σ eq Σ ′ = - X X Σ ′ . ( 53
)
It then follows from the definition of S, equation (41

) 1 , that S ′ = (1 -X/X)Σ ′ so that S 2 eq = (1 -X/X) 2 Σ 2 eq = σ 2 M (X -X) 2 . The definition (42) 2 of S H then implies that Z = 9 4 Y 2 + 2 3g(f ) (X -X) 2 1/2 ⇒ Y = 2 3 Z 2 - 2 3g(f ) (X -X) 2 1/2 (54)
(the positivity of Y is justified below). Also, it follows from the definition of T, equation (41) 2 , and equation (53

) that T ′ = (1 -f X/X)Σ ′ so that T 2 eq = (1 -f X/X) 2 Σ 2 eq = σ 2 M (X -f X) 2 . The criterion (42) then yields (X -f X) 2 + 2f cosh Z -1 -f 2 = 0 ⇒ Z = F (X -f X; f ) where F (x; f ) ≡ arg cosh 1 + f 2 -x 2 2f . ( 55
)
Thus, if the value of X is known, those of Z and Y follow from equations ( 55) and ( 54) 2 .

To now derive an equation on X, combine equations ( 46) and ( 53) to get

3 + 2 g(f ) sinh Z Z 3f + 2 g(f ) sinh Z Z = X X ⇒ 3(X -f X) + 2 g(f ) sinh[F (X -f X; f )] F (X -f X; f ) (X -X) = 0 ( 56
)
where equation ( 55) has been used. The transcendental equation ( 56) on X permits to deduce its value from that of X at least numerically.

Let us finally show how to calculate Ėtp once X, Y and Z are known. It follows from the deviatoric part of the flow rule (44) 2 and the fact that

Ė′ = 0 that 3T ′ + 2 g(f ) sinh Z Z S ′ = 0, so that the expression (47) of Ėtp simplifies into Ėtp = - Λ σ 2 M 2(1 -f ) g(f ) sinh Z Z S ′ .
Now combination of the hydrostatic part of the flow rule (44) 2 and the expression (45) of Ė implies that Λ

σ 2 M sinh Z 2Z tr S = 1 3 ∆V V ḟ f .
(Incidentally, since Λ and ∆V V are positive, this equation shows that tr S and therefore Y are positive, as anticipated above). The preceding expression of Ėtp then becomes

Ėtp = - 4(1 -f ) 3f g(f ) ∆V V ḟ S ′ tr S .
Using finally the relations tr S = 3Y σ M and S ′ = (1 -X/X)Σ ′ , we get from there the final expression of Ėtp :

Ėtp = 4 9σ M 1 -f f g(f ) ∆V V X/X -1 Y Σ ′ ḟ [General formula]. (57)
To summarize, the procedure to get the transformation plastic strain rate is as follows: get X by solving the transcendental equation ( 56); then get Z and Y from equations ( 55) and (54) 2 ; finally get Ėtp from equation (57).

Explicit formula for the beginning of the transformation

As a particular case of formula (57), an explicit, remarkably simple expression of the transformation plastic strain rate at the beginning of the transformation (f → 0 + ) may be derived, even for arbitrary external stresses. The keypoint here is that provided that the quantity f X is assumed to be of order O(f ) (which will be verified a posteriori), the value of Z may be calculated directly from equation ( 55) independently of that of X:

Z = arg cosh 1 -X 2 f (1 + O(f )) = ln 1 -X 2 f + O(f ), implying that sinh Z Z = 1 -X 2 2f ln 1-X 2 f (1 + O(f )) .
One then gets from equations (56) (upon division by X) and ( 54) 2 :

X X -1 = 3f ln 1-X 2 f 1 -X 2 (1 + O(f )) ; Y = 2 3 ln 1 -X 2 f + O(f ).
(The first equation shows that X = O(1) and therefore f X = O(f ), as announced). Equation ( 57) then finally yields

Ėtp = 2 σ M ∆V V Σ ′ 1 -Σ 2 eq /σ 2 M (1 + O(f )) ḟ [f → 0 + ]. ( 58
)
This formula evidences the amplifying effect of the factor

1 1-Σ 2 eq /σ 2
M upon the transformation plastic strain rate Ėtp , when the von Mises equivalent stress Σ eq becomes comparable to the yield stress σ M of the mother-phase.

A composite approximation

Use of the general formula (57) for Ėtp is somewhat hampered by the necessity of solving equation ( 56) numerically to get the value of X. It is therefore tempting to propose the following simple "composite approximation" of Ėtp , based on a heuristic mix of formulae (50), applicable for Σ eq ≪ σ M but arbitrary f , and (58), applicable for arbitrary Σ eq but f ≪ 1:

Ėtp = 2 σ M ∆V V (1 -f ) 2 1 -f 2 -3f 2 g(f ) ln f Σ ′ 1 -Σ 2 eq /σ 2 M ḟ [Approximation]. ( 59
)
5 Comparison with other theories, experiments, and micromechanical simulations

Elements of comparison -Generalities

The comparisons presented in this section are based on the following theoretical, experimental and numerical elements:

• [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]'s theoretical formula (19) for Ėtp and its supposedly improved variant (20); our general formula (57) and its approximation (59); • Desalos (1981)'s old, but high-quality experimental study of transformation plasticity in the A533 steel, completed more recently with respect to multiaxial aspects by [START_REF] Coret | Experimental study of the phase transformation plasticity of 16MND5 low carbon steel under multiaxial loading[END_REF][START_REF] Coret | Experimental study of the phase transformation plasticity of 16MND5 low carbon steel induced by proportional and nonproportional biaxial loading paths[END_REF]); • new finite element micromechanical simulations of [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism, performed with the commercial programme SYSWELD developed by ESI-Group (SYSWELD, 2017).

Although it is not the object of this paper to present and discuss these new micromechanical simulations in detail, a few comments on them are in order here. In principle, they are analogous to many previous ones (see the pioneering work of [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF] and those of [START_REF] Ganghoffer | Finite element calculation of the micromechanics of a diffusional transformation[END_REF]; [START_REF] Abrassart | Influence des transformations martensitiques sur les propriétés mécaniques des alliages du système Fe-Ni-Cr-C. Ph.D. thesis[END_REF][START_REF] Barbe | Numerical modelling of the plasticity induced during diffusive transformation. An ensemble averaging approach for the case of random arrays of nuclei[END_REF]; [START_REF] Barbe | A numerical modelling of 3D polycrystal-to-polycrystal diffusive phase transformations involving crystal plasticity[END_REF]; [START_REF] Otsuka | FFT-based modelling of transformation plasticity in polycrystalline materials during diffusive phase transformation[END_REF], to cite just a few of its many successors). However they differ from them through consideration of a RVE of spherical shape, exactly identical to that used in the theoretical homogenization process; the aim being to provide the "cleanest" possible assessment of the model predictions by eliminating all sources of discrepancies tied to differences in the RVEs considered. A spherical volume made of elastic-plastic material is thus meshed in concentric spherical layers, which are "transformed" one after another from the centre to the external surface by artificially modifying the values of their specific volume and yield stress from these of the mother-phase to those of the daughter-phase. This volume is loaded externally9 through conditions of homogeneous boundary stress (HBStress) or homogeneous boundary strain (HBStrain) -the aim being to complete the theoretical analysis based exclusively on HBStress conditions, by studying the effect of boundary conditions.

The physical constants used correspond to the γ-and α-phases of the A533 steel employed in the experimental studies of [START_REF] Desalos | Comportement dilatométrique et mécanique de l'austénite métastable d'un acier A533[END_REF] and [START_REF] Coret | Experimental study of the phase transformation plasticity of 16MND5 low carbon steel under multiaxial loading[END_REF][START_REF] Coret | Experimental study of the phase transformation plasticity of 16MND5 low carbon steel induced by proportional and nonproportional biaxial loading paths[END_REF], and are as follows: Young's modulus and Poisson's ratio, E = 182, 000 MPa and ν = 0.3 (identical values for both phases); yield stress, σ M = 145 MPa for the mother-phase, σ D = 950 MPa for the daughter-phase; relative change of specific volume from the motherto the daughter-phase, ∆V V = 0.0252.

In the cases considered for comparison in Subsections 5.2 and 5.3, the RVE is subjected to a uniaxial and constant stress denoted Σ, and the resulting transformation plastic strain in the direction of the load is denoted E tp (f ). (The other components are of no interest since they are either zero or trivially tied to E tp (f ) through incompressibility).

Comparison of evolutions of the transformation plastic strain

In a first step we focus on the evolution of the transformation plastic strain as the transformation proceeds, characterized by the ratio E tp (f )/E tp (1) which increases from 0 to 1. Figure 4 shows this ratio as a function of the volume fraction f of the daughter-phase. Subfigure 4(a) compares the predictions of [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]'s original formula (19); its supposedly improved variant (20); our general formula (57); its approximation (59); and Desalos (1981)'s phenomenological formula E tp (f )/E tp (1) ≃ f (2-f ), which he found applicable to all his experimental results for the A533 steel irrespective of the value of the stress applied. The predictions of equations ( 19), ( 20) and ( 59) are independent of the value of the von Mises equivalent stress Σ eq , but those of equation ( 57) are not; for this equation the curve shown corresponds to the value Σ eq ≡ Σ = 100 MPa (to be compared to the yield stress of the mother-phase, σ M = 145 MPa). Several points are noteworthy.

• All theoretical curves, except that corresponding to the supposedly improved variant (20) of [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]'s original formula (19), provide reasonable representations of [START_REF] Desalos | Comportement dilatométrique et mécanique de l'austénite métastable d'un acier A533[END_REF]'s experimental results. However in all cases the increase of the transformation plastic strain they predict is a bit fast during the first half of the transformation. • The infinite slope dE tp /df at the onset of the transformation (f = 0) predicted by both [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]'s original formula (19) and its supposedly improved variant (20), is in clear contradiction with [START_REF] Desalos | Comportement dilatométrique et mécanique de l'austénite métastable d'un acier A533[END_REF]'s experimental results.

• In contrast our new formulae (57) and ( 59) both predict a finite slope at f = 0, in better agreement with Desalos (1981)'s results. Nevertheless the predicted slopes are still a bit too large. The discrepancy may be due to the total disregard of elasticity in the theoretical treatment, which leads to an overestimation of the plastic strains in the deforming mother-phase; see the works of [START_REF] Taleb | A micromechanical modeling of the Greenwood-Johnson mechanism in transformation induced plasticity[END_REF] and Weisz-Paltrault (2017) on this topic. • Use, instead of [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]'s original formula (19), of its supposedly improved variant (20), in fact results in a degradation of the agreement of model predictions and experimental results. This observation is a bit surprising since formula ( 20) is based on a better estimate of the mean value of the stress deviator in the mother-phase, and underlines the limitations of [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]'s original approach based on the specific Approximations ( 1) and ( 2), see Subsection 2.5. • There is little difference between the predictions of our general formula (57) and its simplified variant (59), in spite of the fact that the value of the ratio Σ eq /σ M is high (0.69).

Subfigure 4(b) compares the predictions of our general formula (57) to the results of micromechanical simulations obtained with HBStress and HBStrain conditions, this time for a low stress, Σ = 20 MPa. (The curve corresponding to [START_REF] Desalos | Comportement dilatométrique et mécanique de l'austénite métastable d'un acier A533[END_REF]'s heuristic formula mentioned above, applicable to all values of the stress he considered, is also again provided for reference).

• There is a non-negligible gap between the numerical results obtained with HBStress and HBStrain conditions. This emphasizes the influence of boundary conditions, and evidences the limitations of an approach based on such a simple and small RVE as a spherical volume of mother-phase containing a single growing core of daughter-phase. • The theoretical predictions agree extremely well with these numerical results obtained with HBStress conditions, much less so with those obtained with HBStrain conditions. • Desalos (1981)'s experimental results serving as a reference, the numerical simulation based on HBStrain conditions predicts an evolution of the transformation plastic strain which is too fast during the first half of the transformation, and too slow during the second half.

Comparison of final values of the transformation plastic strain

We now study the "amplitude" of transformation plasticity, characterized by the value of the transformation plastic strain after complete transformation, E tp (1). Figure 5 shows this quantity as a function of the overall stress applied Σ.

Subfigure 5(a) compares the predictions of [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]'s original formula (19); its variant (20); our general formula (57); its approximation (59); and Desalos (1981)'s experimental results for the A533 steel up to a stress of 70 MPa, which he found to be well reproduced by the heuristic formula E tp (1) ≃ 10 -4 Σ (with Σ in MPa). • All theoretical curves provide acceptable representations of [START_REF] Desalos | Comportement dilatométrique et mécanique de l'austénite métastable d'un acier A533[END_REF]'s experimental results. In particular, the predictions of [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]'s original formula ( 19) and its variant (20) are of comparable quality. • For large stresses, there is a large difference between model predictions obtained with our general formula (57) and its approximation (59). But Desalos (1981)'s experimental results limited to moderate stress levels do not permit to decide which is the better one.

Subfigure 5(b) compares the predictions of our general formula (57) to the results of micromechanical simulations obtained with HBStress and HBStrain conditions [START_REF] Desalos | Comportement dilatométrique et mécanique de l'austénite métastable d'un acier A533[END_REF]'s experimental results being again provided for reference).

• The numerical results obtained with HBStress and HBStrain conditions are again markedly different. But the situation is opposite to that for the ratio E tp (f )/E tp (1), see Subsection 5.2 above: the results corresponding to HBStrain conditions are here better than those corresponding to HBStress conditions, Desalos (1981)'s experimen-tal results being taken as a reference. HBStress conditions clearly seem to lead to an overestimation of the amplitude of transformation plasticity. • The predictions of our general formula (57) agree quite well with the numerical results obtained with HBStrain conditions;10 they reproduce in particular the tendency of numerical values of E tp (1) to increase nonlinearly with the stress applied. • Numerical simulations permit to settle the question of the compared quality of our general formula (57) and its simplified variant (59): the former formula is definitely better, the latter leading to a clear overestimation of the transformation plastic strain after complete transformation.

Comparison with multiaxial experiments and micromechanical simulations

Another interesting aspect deserving to be checked versus experiments and micromechanical simulations is the prediction of our general formula (57) -in line with that of [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]'s earlier one ( 19) -of collinearity of the transformation plastic strain rate and the overall stress deviator. It will just be mentioned here that the said collinearity has been approximately verified experimentally by [START_REF] Coret | Experimental study of the phase transformation plasticity of 16MND5 low carbon steel under multiaxial loading[END_REF][START_REF] Coret | Experimental study of the phase transformation plasticity of 16MND5 low carbon steel induced by proportional and nonproportional biaxial loading paths[END_REF] for the A533 steel, and is also approximately confirmed by extra finite element micromechanical simulations, analogous to those discussed above but with various multiaxial loadings instead of uniaxial ones.

Synthesis and perspectives

The aim of this paper was to propose a new theoretical treatment of [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism of transformation plasticity of metals and alloys, based on limit-analysis. The new approach rests upon the same basic elements as [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]'s classical one, including in particular an approximate solution to the model problem of growth of a spherical core of daughter-phase within a concentric spherical volume of mother-phase subjected to a general 3D loading. But use of the powerful variational framework of the kinematic approach of limit-analysis makes it both more general and versatile.

In a first step, we have developed an approximate limit-analysis of a hollow sphere subjected to both external and internal loadings through classical conditions of homogeneous boundary stress [START_REF] Mandel | Contribution théorique à l'étude de l'écrouissage et des lois de l'écoulement plastique[END_REF][START_REF] Hill | The essential structure of constitutive laws of metal composites and polycrystals[END_REF]. The internal loading is intended to represent the effect, upon the external shell of mother-phase, of the expansion of the core of daughter-phase due to the volumetric part of the transformation strain. The treatment is based on an extension of a work of [START_REF] Monchiet | A micromechanics-based modification of the Gurson criterion by using Eshelby-like velocity fields[END_REF] on plastic porous materials -containing traction-free voids -which itself stood as an extension of the classical work of [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I -Yield criteria and flow rules for porous ductile media[END_REF] supplementing this author's elementary trial velocity fields with more complex, Eshelby (1957)-type ones. The output consists of some approximate overall yield locus and associated plastic flow rule obeying the normality property, extending those found by [START_REF] Monchiet | A micromechanics-based modification of the Gurson criterion by using Eshelby-like velocity fields[END_REF] through incorporation of some additional internal loading.

In a second step, we have shown how to apply these results to the treatment of [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism of transformation plasticity. An analytical formula for the transformation plastic strain rate has been found for fully general 3D loading conditions; this formula unfortunately contains an auxiliary unknown which must be determined by numerically solving a transcendental equation. However it reduces to simpler ones free of such complexities, in two distinct special cases: (i) small external stresses; (ii) beginning of the transformation (whatever the magnitude of the external stress tensor). All these theoretical results have been compared, with globally satisfying results, to experiments performed on the A533 steel [START_REF] Desalos | Comportement dilatométrique et mécanique de l'austénite métastable d'un acier A533[END_REF][START_REF] Coret | Experimental study of the phase transformation plasticity of 16MND5 low carbon steel under multiaxial loading[END_REF][START_REF] Coret | Experimental study of the phase transformation plasticity of 16MND5 low carbon steel induced by proportional and nonproportional biaxial loading paths[END_REF], and also fresh finite element micromechanical simulations of the mechanism investigated, based on exactly the same spherical RVE as the theoretical analysis.

There are several advantages to the new approach. A minor one consists in dispensing with some ad hoc technical hypotheses made in [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF]'s classical treatment, among which one of highly questionable validity. A more important one is that the value of the transformation plastic strain rate at the onset of the transformation is no longer infinite like in the previous treatment -a non-physical prediction which did not fail to raise numerous criticisms. A third one lies in the possibility to consider external stresses of arbitrary magnitude instead of small ones only; the new theory predicts a nonlinear increase of the transformation plastic strain with the stress applied, in line with experimental observations [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF].

However the major advantage of the new approach probably lies in the new perspectives it opens, thanks to the power and efficiency of the kinematic method of limit-analysis.

• Refining the approximate overall yield criterion and plastic flow rule by including extra trial velocity fields, for instance, would not raise major difficulties. • In this work the hypothesis was made, in line with [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s ideas, that internal stresses generating microplasticity in the weaker mother-phase essentially arise from the difference of specific volume between the phases, that is the volumetric part of the transformation strain. However in reality this transformation strain is not a multiple of the unit tensor, and it is perfectly conceivable that its deviatoric part also contributes to the microplasticity of the mother-phase. This effect could be accounted for by modifying the expression of the overall strain rate imposed on the inner boundary of the domain of mother-phase, without making the limit-analysis significantly more complex.11 • Another interesting extension would consist of considering growth of a spheroidal, instead of spherical, core of daughter-phase, using [START_REF] Monchiet | Macroscopic yield criteria for ductile materials containing spheroidal voids: An Eshelby-like velocity fields approach[END_REF]'s extension of their earlier work on plastic materials containing spherical voids [START_REF] Monchiet | Macroscopic yield criteria for ductile materials containing spheroidal voids: An Eshelby-like velocity fields approach[END_REF] to such a pore geometry. The spheroidal geometry would permit to represent, in an approximate way, growing nuclei of daughter-phase of elongated shape; such nuclei have been observed in rolled plates and are known to result in a strong anisotropy of transformation plasticity (see e.g. [START_REF] Desalos | Comportement dilatométrique et mécanique de l'austénite métastable d'un acier A533[END_REF], among other authors) -for instance, the transformation plastic strain is no longer zero in the absence of external stress.

In addition, the notable gap found between the results of finite element micromechanical simulations performed with conditions of homogeneous boundary strain and stress, underlines the influence of boundary conditions for such a small and simplistic RVE as a spherical volume containing a single growing core of daughter-phase; and suggests to perform more realistic simulations on much larger RVEs containing many cores of daughter-phase, subjected to rigorous periodic boundary conditions. In addition to settling the issue of the effect of boundary conditions, these simulations would permit to study the respective influences of nucleation and growth of nuclei of daughter-phase, unlike those discussed here which automatically ruled out nucleation of multiple nuclei through consideration of a single one. Such simulations will be performed in the near future using Moulinec and Suquet (1998)'s FFT-based method of numerical homogenization, much better adapted to calculations of this type than the standard finite element method.

A Appendix : additional considerations on the thermal strains of individual phases and the overall thermometallurgical strain

As a typical example, Fig.

A.1 show the dilatometric diagram (free strain vs. temperature) of the A533 steel heated at 30 • Cs -1 and cooled at -2 • Cs -1 , determined experimentally by [START_REF] Desalos | Comportement dilatométrique et mécanique de l'austénite métastable d'un acier A533[END_REF]. The diagram evidences both transformations from the α-phase (ferrite, bainite or martensite) to the γ-phase (austenite) during heating, and from the γ-phase to the α-phase during cooling; but our focus in the present paper is on the latter transformation -the effect of transformation plasticity during heating being erased by the subsequent vanishing of the stresses at high temperatures. Hence the mother-and daughter-phases are considered to be the γ-and α-phases respectively. Note that the observed strains are defined in reference to the daughter (α) phase at a temperature of 20 • C. The thermal expansions of the two phases, with different coefficients α M and α D , are conspicuous here; and the temperature-dependent relative difference of specific volume ∆V V (T ) between them, given by equation ( 4), is apparent in the vertical gap between the two straight lines corresponding to the pure phases. During the transformations the two phases coexist so that the overall strain observed lies between the thermal strains of individual phases at the temperature considered. Now by definition the dilatometric diagram is obtained in the absence of any external load so that the overall elastic and plastic strains E e , E p are zero; under such conditions the overall strain coincides with the thermo-metallurgical strain E thm defined by equation ( 7) 2 . The geometrical interpretation of the simple "linear mixture rule" giving this strain in terms of the thermal strains of individual phases is that the volume fraction f of the daughterphase may be obtained from the dilatometric diagram by a simple "lever rule", f ≡ QR A final remark is that as a consequence of equation ( 7) 2 , the rate Ėthm of the thermometallurgical strain naturally splits into a "thermal part" proportional to Ṫ and a "transformation part" proportional to ḟ :

Ėthm = [(1 -f )α M 1 + f α D 1] Ṫ + ǫ th D (T ) -ǫ th M (T ) ḟ .
But it must be stressed that the thermometallurgical strain E thm itself cannot be decomposed in a similar way, as some authors have wrongly claimed; that is, it is not a simple sum of a function of T and a function of f , as is obvious from equation ( 7) 2 .

B Appendix : on the relative importance of [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]Johnson (1965)'s and[START_REF] Magee | Transformation kinetics, microplasticity and ageing of martensite in Fe-31 Ni[END_REF]'s mechanisms and their experimental separation

In some cases Magee (1966)'s mechanism is important and may even completely dominate over that of [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]. The most obvious case is that of shape memory alloys, in which the difference of specific volume between the phases is very small so that [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism is virtually absent. As a result, the thermomechanical behaviour of such materials during phase transformations is governed by the pseudo-elastic transformation strain; the origin of the "memory effects" observed lies in the reversible nature of this strain.

Another example is that of "TRIP steels". Although the acronym TRIP stands for "TRansformation Induced Plasticity", the situation it refers to is rather different from that considered here. TRIP steels are characterized by the presence of a significant amount of untransformed austenite at room temperature. When such steels are subjected to an increasing stress at this temperature, this "retained" austenite is transformed into α-phase with a significant accompanying deformation, and this generates an important increase of ductility. Since the stress applied has a decisive impact upon the kinetics of the transformation,12 it is highly probable that it also has a large influence upon the orientation of the transformation strain [START_REF] Magee | Transformation kinetics, microplasticity and ageing of martensite in Fe-31 Ni[END_REF]'s mechanism). This large impact of the stress applied upon the transformation marks a significant difference with the case considered here of transformations essentially governed by the decrease of the temperature, rather than by application of a stress.

In the case considered here of transformations occurring during the cooling period of thermomechanical processes, the situation is less clear. The classical metallurgical view is that [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism should dominate for diffusion-controlled transformations, and that of [START_REF] Magee | Transformation kinetics, microplasticity and ageing of martensite in Fe-31 Ni[END_REF] for martensitic ones. To go beyond these qualitative ideas, one may try to experimentally distinguish between the two mechanisms by using the different physical natures -plastic for [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism, pseudo-elastic for that of [START_REF] Magee | Transformation kinetics, microplasticity and ageing of martensite in Fe-31 Ni[END_REF] -of the microscopic strains assumed to be responsible for transformation plasticity. One possibility is to subject a stressed specimen to repeated thermal cycles. The transformation plastic strain must be perfectly cumulative with the number of cycles if it arises from [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism, because nothing prevents the accumulation of standard plastic strains; but it must "saturate" at some stage if it is due to Magee (1966)'s mechanism, because a maximum value of the overall strain is obtained when the transformation strains of the various transforming regions are all collinear. The experiment has been performed several times on different metals and alloys [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF][START_REF] Desalos | Comportement dilatométrique et mécanique de l'austénite métastable d'un acier A533[END_REF][START_REF] Taleb | New investigations on transformation induced plasticity and its interaction with classical plasticity[END_REF] with the conclusion, for the materials considered, of perfect cumulation of the transformation plastic strain with the number of cycles, and therefore of domination of [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism.

Another possibility is to exploit memory effects expected to arise, for Magee (1966)'s mechanism, from the reversibility of the transformation strain. This can be done by applying some stress on the material during cooling, then eliminating this stress and reheating. If [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism prevails, the transformation plastic strain obtained at the end of cooling must not be recovered during re-heating since standard plasticity is irreversible; but if Magee (1966)'s mechanism dominates, this strain must be at least partially recoverable. The experiment was performed at least once, by [START_REF] Gigou | Plasticité de transformation : cas de la transformation martensitique[END_REF], with the result that the recoverable part of the transformation plastic strain did not amount to more than 10% of this strain; the ensuing conclusion was again that Greenwood and Johnson (1965)'s mechanism was dominant in the alloy considered.

The general conclusion is that for the materials and types of thermomechanical histories considered in the present paper, although [START_REF] Magee | Transformation kinetics, microplasticity and ageing of martensite in Fe-31 Ni[END_REF]'s mechanism cannot be ruled out a priori without a dedicated experimental study, [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF]'s mechanism often dominates.

  write the local flow rule (for the von Mises criterion) in the classical form δǫ p δf =

)5

  The third field is obtained from Monchiet et al. (2011)'s expressions by taking, with their notations, d * 11 = d * 22 = -1 2f , d * 33 = 1 f , d * 12 = d * 13 = d * 23 = 0; the fifth, by taking d * 11 = -d * 22 = 1 f , d * 33 = d * 12 = d * 13 = d * 23 = 0. 6 The use of equations (

  A.1 of Appendix A), this equation entails that tr S > 0 and therefore, by what precedes, tr S = 2S H = -2 σ M ln f ; the value of the plastic multiplier follows:

  Theory and micromechanical simulations.

Fig. 4 .

 4 Fig. 4. Comparison of evolutions of the transformation plastic strain.

  Theory and micromechanical simulations.

Fig. 5 .

 5 Fig. 5. Comparison of transformation plastic strains after complete transformation.

Fig

  Fig. A.1. The dilatometric diagram of the A533 steel (after Desalos (1981)).

  of 800 • C in Fig. A.1.

Even data pertaining to the yield stress of the weaker phase at the temperatures of interest,

The rates Ṫ and ḟ are formally considered as independent here. In reality these rates are tied through the kinetics of the transformation and latent heat effects; but it is not the object of this paper to discuss such aspects.

Since various formulae for Ėtp will be given, an indication of the authors or context of application of each one will be provided between brackets.

In essence, the idea of employing such a theory could be found, in a sketchy form, in Abrassart (1972)'s thesis. Unfortunately rough approximations made in the choice of typical geometry and trial displacement fields led to large errors in the final result, the transformation plastic strain rate being underestimated by a factor of the order of 3.

This expression designates loads that may be applied to the structure without inducing plastic collapse.

There is no internal boundary -and therefore no internal loading -since the core of daughterphase is included in the mesh.

This is maybe not as surprising as it may seem. Indeed although the limit-analysis presented in Section 3 was, strictly speaking, developed for HBStress conditions, it is probably also appropriate, in some approximate sense, for HBStrain conditions insofar as the trial displacement fields used satisfy the latter conditions in an average sense, if not exactly.

Such an extension would however raise the difficult issue of accounting for the selfaccommodation of the transforming regions, that is the partial or complete compensation of the deviatoric parts of the transformation strains from one region to another.

This is a consequence of the highly unstable character of retained austenite at room temperature: a small increase of "driving force" is sufficient to trigger the transformation.