Constitutive modelling of mechanically induced martensitic transformations: Prediction of transformation surfaces
Abstract
Purpose
The purpose of this work is to apply a recently proposed constitutive model for mechanically induced martensitic transformations to the prediction of transformation loci. Additionally, this study aims to elucidate if a stressassisted criterion can account for transformations in the socalled straininduced regime.
Design/methodology/approach
The model is derived by generalising the stressbased criterion of Patel and Cohen (1953), relying on lattice information obtained using the Phenomenological Theory of Martensite Crystallography. Transformation multipliers (cf. plastic multipliers) are introduced, from which the martensite volume fraction evolution ensues. The associated transformation functions provide a variant selection mechanism. Austenite plasticity follows a classical single crystal formulation, to account for transformations in the straininduced regime. The resulting model is incorporated into a fully implicit RVEbased computational homogenisation finite element code.
Findings
Results show good agreement with experimental data for a metastable austenitic stainless steel. In particular, the transformation locus is well reproduced, even in a material with considerable slip plasticity at the martensite onset, corroborating the hypothesis that an energybased criterion can account for transformations in both stressassisted and straininduced regimes.
Originality/value
A recently developed constitutive model for mechanically induced martensitic transformations is further assessed and validated. Its formulation is fundamentally based on a physical metallurgical mechanism and derived in a thermodynamically consistent way, inheriting a consistent mechanical dissipation. This model draws on a reduced number of phenomenological elements and is a step towards the fully predictive modelling of materials that exhibit such phenomena.
Keywords
Citation
de Bortoli, D., Adziman, F., de Souza Neto, E. and Pires, F. (2018), "Constitutive modelling of mechanically induced martensitic transformations", Engineering Computations, Vol. 35 No. 2, pp. 772799. https://doi.org/10.1108/EC0320170087
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:Emerald Publishing Limited
Copyright © 2018, Emerald Publishing Limited
1. Introduction
Mechanically induced martensitic phase transformations play a crucial role in the overall mechanical behaviour of many industrially important materials, such as metastable austenitic stainless steels and TRansformation Induced Plasticity (TRIP) steels (Roters et al., 2010). From a modelling perspective, they pose significant challenges stemming from a complex interplay between a variety of physical mechanisms at the crystal lattice scale. These include, for instance, volumetric expansions across multiple transformation systems and their interactions with crystallographic slip in both the austenitic and martensitic phases.
Due to their inherent microstructural complexity, a phenomenological approach to the constitutive modelling of these materials often falls short of appropriately predicting their mechanical behaviour. The development of robust models relying on a reduced number of phenomenological elements remains largely an open task for this class of materials. In this context, multiscale models can help elucidating the relationship between microscopic properties and the corresponding macroscopic material behaviour, enabling the simulation of engineeringscale problems that account for the underlying microstructure in a predictive fashion.
Much research has been dedicated to predicting the crystallographical features of martensitic transformations; the currently most accepted theory is known as the Phenomenological Theory of Martensite Crystallography (PTMC), developed in independently by Wechsler et al. (1953) and Bowles and Mackenzie (1954). While it has been extended since its inception[1], the theory has nevertheless stood the test of time and still is the base of many widely used constitutive models. The PTMC is thus named because it does not aim to explain the mechanisms behind the transformation; instead, it merely predicts the observed crystallographic features of the martensitic transformations in many materials: orientation relationships, habit planes, shape deformation and other geometrical features. The theory was later refined by Ball and James (1987), who obtained the same predictions from an energy minimisation perspective.
Patel and Cohen (1953) observed that mechanical loadings have an influence on the transformation starting temperature. Their proposed explanation for this phenomenon hinged on the work done between applied stresses and transformation displacements. From that, they derived a simple transformation criterion – limited to uniaxial stress states and small strains – based on the mechanical contribution to the total thermodynamical driving force, in what came to be known as a stressassisted transformation.
It has also been suggested that a second, distinct mechanism manifests in certain conditions (mainly depending on temperature): the straininduced transformation (Olson and Cohen, 1972). Here, the transformation happens at stresses higher than the austenite yield stress, leading Olson and Cohen to propose that slip activity in the austenite generates martensite nucleation sites as a result of shear band intersections. At lower temperatures, where transformation happens before yielding, stressassisted transformations are dominant; conversely, at higher temperatures, straininduced transformations are more significant (Olson and Azrin, 1978). Some popular constitutive models derive from these observations (Olson and Cohen, 1975; Stringfellow et al., 1992).
Although the two types of transformation produce martensites with different morphologies and characteristics (Maxwell et al., 1974; Snell et al., 1977), there is no consensus in the literature on whether they are indeed due to two different mechanisms, or whether all mechanically induced transformations can be explained in terms of a single stressbased criterion (Tamura, 1982; Chatterjee and Bhadeshia, 2007; Kundu, 2014). Many authors proposed transformation models based on the latter assumption that positively reproduce experimental data for TRIP steels and metastable austenitic stainless steels (Tamura, 1982; Chatterjee and Bhadeshia, 2007; Perdahcioğlu et al., 2008a, 2008b; Geijselaers and Perdahcioğlu, 2009).
In addition to those mentioned above, there is a large number of constitutive models for materials undergoing mechanically induced nonthermoelastic martensitic transformations, encompassing a broad variety of approaches, ranging from phenomenological models that postulate the existence of internal variables (Leblond et al., 1989; Leblond, 1989; Fischer, 1990; Bhattacharya and Weng, 1994; Govindjee and Miehe, 2001; Kubler et al., 2011) to those of a micromechanical nature (Marketz and Fischer, 1995; Fischer and Reisner, 1998; Reisner et al., 1998; Cherkaoui et al., 1998, 2000; Cherkaoui and Berveiller, 2000; Ganghoffer and Simonsson, 1998; Idesman et al., 1999, 2000; Levitas et al., 1998a, 1998b; Levitas et al., 2002; Lani et al., 2007). Many of the more recent models use a computational homogenisation framework, such as those of Iwamoto (2004), Suiker and Turteltaub (2005), Turteltaub and Suiker (2005, 2006), Tjahjanto et al. (2008), Hallberg et al. (2007), Kouznetsova and Geers (2008), Sun et al. (2008), Lee et al. (2010), Yadegari et al. (2012) and Perdahcioğlu and Geijselaers (2012). Most of these, however, are formulated in a small strains context, an assumption of questionable validity due to large local transformation deformations, even though the resulting macroscopic deformation is generally small. Additionally, many models use averaging techniques assuming the collective behaviour of the microscopic transformation variants, ignoring some of the finescale interactive complexity between them. Other frequently used ingredients are predefined functional forms for the evolution of the martensite volume fraction, which involve parameters that bear limited physical significance and need to be calibrated from experiments. In this context, a transformation model relying primarily on readily measurable properties of the parent and product lattices (that is, those resulting from wellestablished frameworks such as the PTMC), avoiding the introduction of phenomenological elements, would be a valuable addition to this collection. The model analysed here has been initially formulated by Adziman (2014) and de Bortoli (2017), and its assessment is the main object of this contribution.
This article is organised as follows: in Section 2, an overview of the main experimental observations motivating the model are presented, along with the resulting formulation. In Section 3, some notes are made about the model’s algorithmic treatment in the context of an implicit finite element discretisation. In Section 4, a metastable austenitic stainless steel for which experimental data is available is analysed, with particular attention dedicated to determining its transformation locus. Finally, Section 5 presents a summary of the main achievements of this contribution, along with perspectives for future developments.
1.1 Notation
For the most part, the notation used is standard in modern continuum mechanics, where summation (Einstein) convention is assumed unless otherwise noted. Vectors are denoted by italic boldface lowercase letters such as v, while their uppercase counterparts are used for secondorder tensors such as T. Scalars and scalarvalued functions are usually written as lightface italic letters such as s. I = δ_{ij} is the secondorder identity tensor. The scalar product of two vectors is denoted by u · v = u_{i}v_{i}, whereas the double contraction of two secondorder tensors is written as S : T = S_{ij}T_{ij}. The tensor product of u and v is denoted by u ⊗ v. The operator dev[·] is the deviatoric projection: given a symmetric secondorder tensor S,
2. Formulation
2.1 Mechanically induced martensitic transformations
This work focuses on modelling irreversible stressassisted/straininduced austenitetomartensite phase transformations that occur in ferrous alloys under isothermal mechanical loading. This encompasses, for instance, both fully austenitic and multiphase steels partly composed of metastable austenite, such as TRIP steels. These ferrous alloys generally have facecentred cubic (FCC) austenite phases that transform to bodycentred tetragonal (BCT) martensite. For these materials, the martensite tetragonality is directly correlated to the alloy’s carbon content, so that in low carbon steels, BCT martensite can be approximated as bodycentred cubic (BCC) (Nishiyama, 1978).
The phenomena of importance in these transformations take place at the crystal lattice scale, where the movement of atoms dictates the relation between parent (austenite) and product (martensite) lattice properties. Using this latticescale information, a continuum model is constructed on the crystal scale. Its main driving assumptions are based on the following important experimental observations:
Plasticity can occur in the parent austenite phase before the martensitic transformation onset, depending on the alloy and on conditions such as testing temperature, as shown in Figure 1. Below the temperature M_{s}, martensite forms with no external loading; through applied stress, it can be raised until the limit M_{d}, also known as the martensite deformation temperature. The previously described phenomena have been observed in metastable austenitic stainless steels (Perdahcioğlu et al., 2008a, 2008b) and TRIP steels (Chatterjee and Bhadeshia, 2007). It appears that modelling such plasticity is of critical importance to capture the complex interactions between crystallographic slip and the martensitic transformation. Additionally, modelling the inelastic behaviour of the austenite may shed some light on the question of whether stressassisted and straininduced martensitic transformations are indeed two distinct mechanisms, or they both can be explained in a unified formulation following an energybased transformation criterion.
The geometric relation between the austenite and martensite lattices can be satisfactorily predicted using the PTMC, resulting in a set of habit planes and transformation deformation vectors. These define the volumetric and shear strains involved in the transformation and their direction. The PTMC predicts that, for given parent and product lattices, there are multiple transformation systems[2] – 24 in the case of an FCCtoBCT/BCC transformation.
A critical energy required to initiate the transformation is assumed to exist. At a given temperature, this energy barrier is considered a material property, depending on both parent and product phases relative chemical freeenergies. This is illustrated in Figure 2, where the chemical freeenergy of the austenite (γ) and martensite (α′) phases as a function of temperature is shown. The scalar M_{s} is the temperature at which transformation starts spontaneously, while T_{0} is the temperature at which both phases are in equilibrium. Above T_{0}, the austenite has lower chemical freeenergy than the martensite – being therefore more stable – so transformation is not possible. Between M_{s} and T_{0}, the chemical freeenergy difference between the phases is not enough for spontaneous transformation; external mechanical loads can, however, supply the remaining necessary energy. Once again, as clearly seen in the figure, the energy necessary to trigger a transformation is a function of the temperature: ΔG_{mech}_{T}, T ∈ [M_{s}, T_{0}]. At a given temperature, ΔG_{mech} is equal to the stress power due to external loadings at which transformation starts. This is a widely accepted idea in the literature (Bhadeshia et al., 2009; Tamura, 1982), with many proposed models based on it [for instance, Kouznetsova and Geers (2008); Perdahcioğlu and Geijselaers (2012)].
Given the existence of this critical transformation energy, a set of transformation functions (cf. yield functions in the slip plasticity case) is postulated, generalising for arbitrary stress states what was initially proposed by Patel and Cohen (1953). At a given material point, the selection of active set of transformation systems is also performed by using transformation functions: only the most favourably oriented variants with respect to the local stress state transform.
At a given material point, once transformation starts in any system, plastic slip activity is considered to cease in the underlying crystal lattice; when the transformation finishes, the resulting fully martensitic material behaves elastically again.
2.2 Kinematics
Following standard practice in crystal plasticity modelling, the total deformation gradient F at a given material point is multiplicatively decomposed to account for individual contributions of the deformation mechanisms under consideration[3]:
Here, F^{e}, F^{tr} and F^{pa} are, respectively, the elastic, martensitic transformation and austenite plasticity deformation gradients.
2.2.1 Austenite plastic slip.
In the austenite single crystal, plastic slip happens along preferred crystallographic directions. Each slip system α is defined by unit vectors m^{α} and s^{α} denoting, respectively, the normal vector to the slip plane and the slip direction. As the slip direction is contained in the slip plane, m^{α} · s^{α} = 0 meaning these deformations are isochoric. The usual flow rule from crystal plasticity is assumed (Asaro and Needleman, 1985):
In the previous equation, the plastic shear rates
2.2.2 Elastic behaviour.
The reversible part of the crystal deformation is modelled using a regularised neoHookean hyperelastic potential, resulting in the following expression for the Kirchhoff stress τ:
2.2.3 Martensitic transformation kinematics.
The crystallographic theory of martensite predicts, for given parent and product lattices, a set of N_{tr} transformation systems; these are characterised by a habit plane with unit normal vector m^{i}, along with shear directions s^{i}, for i = 1, 2,…, N_{tr}. The number of transformation systems N_{tr} depends on the respective symmetries of the parent and product lattices, while the system vectors depend on the relation between their lattice parameters (Wechsler et al., 1953; Ball and James, 1987).
The deformation due to the martensitic transformation on a system i, depicted in Figure 3, consists of a habit plane shear of magnitude ξ in the direction s^{i}, along with an expansion of magnitude δ in the direction m^{i}:
2.2.4 Transformation multipliers.
At a given material point, the transformation is considered to potentially happen in multiple systems concurrently; its evolution can thus be modelled by introducing transformation multipliers
These multipliers can be interpreted as local volume fractions of martensite transformed along each system. The total martensite and austenite volume fractions are then given, respectively, by:
Note that the transformation irreversibility is reflected by the fact that its multipliers are monotonically increasing functions of time:
2.3 Thermodynamical considerations
2.3.1 Freeenergy function.
The mechanical freeenergy density ψ is postulated as independent from both crystallographic slip and martensitic transformation status, being thus a function of the elastic deformation gradient only: ψ = ψ(F^{e}). Using the multiplicative decomposition from equation (1), its rate of change during the martensitic transformation evolution is given by:
2.3.2 Dissipation rate.
Combining the above with the Clausius–Duhem inequality yields:
2.3.3 Transformation functions; mechanical dissipation consistency.
As is clear from Figure 2, for a given temperature between M_{s} and T_{0}, the activation of the martensitic transformation requires a critical mechanical energy ΔG_{mech} (per unit reference volume). A constitutive model that dissipates precisely this amount of energy over the course of the martensitic transformation can be devised by postulating the existence of N_{tr} transformation functions (cf. yield functions in plasticity) of the type:
In the above, T^{i} is the projection of the transformation conjugate stress tensor into the transformation system i : T^{i} = T : (d^{i} ⊗ m^{i}). These projections can be interpreted as the martensitic transformation analogous counterparts to the resolved Schmid stresses in slip plasticity.
The transformation flow rule follows from equation (12):
The consistency of the dissipation follows from equations (18) and (20):
Integrating this dissipation rate through the course of the transformation, starting from a fully austenitic state (γ_{m} = 0) and ending with pure martensite (γ_{m} = 1), results in:
It is therefore evident that the total mechanical energy dissipated during transformation is rigorously equal to the energy density required for transformation, ΔG_{mech}.
2.3.4 Relation to the Mohr–Coulomb yield criterion.
As will be clear momentarily, the above transformation functions are connected to the classical Mohr–Coulomb yield function:
Consider now the martensitic transformation functions (20). Expressing the stress tensor T on the orthonormal basis associated to the habit plane of transformation system i, {s^{i}, m^{i}, t^{i}}, where t^{i} = s^{i} × m^{i}, yields:
Clearly, T_{simi} and T_{mimi} are shear and normal internal tractions acting in the habit plane, respectively. The transformation starts when the work due to these tractions reaches the energy barrier ΔG_{mech}. For the Mohr–Coulomb criterion, yield starts when the critical combination (25) is reached on any plane at a material point, whereas for transformation criterion, the combination must occur on a transformation habit plane. For a material with randomly oriented austenite crystals, the transformation surface can thus be expected to converge to a Mohr–Coulombtype locus with an increase in the number of grains, as any orientation can be made arbitrarily close to a habit plane.
Additionally, comparing equations (25) and (26), it is apparent that the parameters of the equivalent Mohr–Coulomb yield surface are:
Thus, ϕ is the angle between the transformation shear direction s^{i} and its total deformation vector d^{i} ≡ ξs^{i} + δm^{i} . The cohesion parameter c can be interpreted, analogously to the Mohr–Coulomb model, as the habit plane shear stress that is necessary for transformation to start in the absence of normal stresses to the habit plane [as can be seen by setting T_{mimi} = 0 in equation (26)].
It is interesting to note that this kind of transformation surface has been reported in shapememory alloys undergoing mechanically induced martensitic transformations [for instance, see Orgéas and Favier (1998)]. The materials tested by Orgéas and Favier display almost no pressuresensitivity in their transformation surfaces. This can be understood from the fact that the martensitic transformations in shapememory alloys involve practically no dilatation (δ ≈ 0). Thus, ϕ ≈ 0° and the resulting Mohr–Coulomb surface is weakly pressuredependent.
3. Algorithmic treatment
The constitutive equations above are integrated using a fully implicit backwardEuler method. For the austenite plasticity, an integrator based on the exponential returnmapping is used as it is exactly volumepreserving, satisfying the plastic deformation incompressibility for any step size (Miehe, 1996). Full details of the associated linearised equations and algorithmic procedures are presented in de Bortoli (2017) and Adziman (2014).
In the above rateindependent model, the set of active systems (i.e. systems for which slip/transformation is currently active) needs to be known for the stress update to be performed. This problem has no trivial solution, due to the high number of potentially active systems[8]. Namely, given the necessary linear dependencies between the multiple slip/transformation systems, there are in general various combinations of multipliers that produce the same incremental plastic/transformation deformation gradient and resulting stress tensor. Additionally, Newtontype algorithms have convergence difficulties when the active set determination is done concurrently with the regular iterations. Algorithms that perform the search of active systems are still an open area of investigation in crystal plasticity, having been extensively studied primarily in the context of slip deformation (Borja and Wren, 1993; Anand and Kothari, 1996; Akpama et al., 2016).
A common way to avoid the above issues relies on viscoplastic regularisations of the original problem (Asaro and Needleman, 1985). Thus, a ratedependent formulation is used, even though the materials of interest in this work are not modelled in conditions where actual ratedependent phenomena (like creep) are of importance. The rateindependent formulation can be recovered in the limit of vanishing ratesensitivity parameters. This approach has the disadvantage, however, of leading to numerical difficulties due to the resulting stiffness of the system of returnmapping equations (Steinmann and Stein, 1996).
To that end, a sliprate law proposed by Perić (1993) is adopted for both the evolution of slip plastic multipliers:
Material parameters μ_{pa} and μ_{tr} are viscosityrelated (with time dimensions), while є_{pa} and є_{tr} are nondimensional ratesensitivity parameters. The rateindependent limit of the above expressions is reached as either μ_{pa} → 0, μ_{tr} → 0 or є_{pa} → 0,є_{tr} → 0. Here, the proper choice of these parameters is a tradeoff between obtaining sufficiently rateindependent solutions and computational efficiency. Note that the presence of the absolute value and sign functions in equation (28) is due to the fact that crystallographic slip can happen in either direction along a particular slip system. The same, however, does not apply to the martensitic transformation, leading to the differences between equations (28) and (29).
Another noteworthy algorithmic aspect is the assumption governing the coupling of transformation and plasticity. Namely, once the transformation criterion is satisfied at a given material point, the austenite plasticity in it ceases and only the transformation can progress. This simplifies considerably the model’s computational treatment, as the coupling between the two phenomena does not have to be dealt with explicitly in the stress integration procedure. Nevertheless, this assumption might lead to problems due to the sudden change in the material’s stiffness, especially if it has already undergone a large amount of plastic slip. In a polycrystalline material, this effect’s importance is expected to be diluted in the homogenised material response.
4. Results
The martensitic transformation constitutive model is now applied to the analysis of an ASTM A564 alloy (also known as Sandvik Nanoflex), a metastable austenitic stainless steel for which Perdahcioğlu, Geijselaers and Huétink published extensive experimental data (Perdahcioğlu et al., 2008a, 2008b; Geijselaers and Perdahcioğlu, 2009; Perdahcioğlu and Geijselaers, 2012). Its chemical composition is given in Table I.
Their experiments include multiple tests ranging from simple shear to plane strain tension, showing clear evidence that this alloy undergoes mechanically induced martensitic transformations (Geijselaers and Perdahcioğlu, 2009). Additionally, this material displays considerable plastic deformation before the martensitic transformation onset, at equivalent strains of the order of 5 per cent, mostly due to slip (Perdahcioğlu and Geijselaers, 2012). Under these conditions, it provides an opportunity to test how a stressassisted transformation criterion behaves in the straininduced regime. Of particular interest is the fact that Perdahcioğlu, Geijselaers and Huétink found that the volume of martensite could be broadly considered a function of the mechanical driving force only.
For this alloy, the parent phase has an FCC lattice and thus the usual octahedral slip systems of Table II are considered. The product phase can be considered approximately BCC, due to its low carbon content. The lattice parameters for the phases are:
The 24 transformation variants corresponding to these families are shown in Table III. The transformation dilatation and shear – equal for all variants – is given by:
It is worth noting the strain magnitude: the shear component, in particular, is very significant, putting into question the small strains assumption used by many constitutive models found in the literature.
4.1 Mesh and material parameters
To analyse this material in a computational homogenisation finite element code, representative volume elements (RVEs) are instantiated using Neper, a polycrystal generation and meshing library (Quey et al., 2011). As this material does not possess a particular texture, the polycrystal grains are chosen to be randomly oriented in the FCC cubic symmetry group. Due to the incompressibility of plastic slip, 10node quadratic tetrahedral elements with 4 integration points are used in order to minimise spurious volumetric locking issues in the finite element formulation. To assess the appropriate number of grains necessary to capture the isotropic limit of the material behaviour, two RVEs are created consisting of 100 and 150 crystals using meshes with approximately the same numbers of elements and nodes. The polycrystal geometries and associated meshes are shown in Figure 4.
In addition to the aforementioned crystallographic directions for both slip and martensitic transformation, elastic and transformation material parameters are taken from Perdahcioğlu and Geijselaers (2012) and given in Table IV. The same authors also report stressstrain curves for a stable austenite phase in a simple shear test; this data is used to calibrate the hardening behaviour of the austenitic phase, assumed to follow a Nadai–Ludwik law:
Perdahcioğlu and Geijselaers also calibrated one such hardening curve to this material. Their constitutive model for the austenite is, however, an isotropic plasticity model whose hardening curve is given in terms of effective Cauchy stress. As such, their reported parameters are of no immediate use to the calibration of a critical Schmid stress hardening curve, as the relation between both is not trivial. With that in mind, a new calibration is performed with the martensitic transformation model. To do that, an artificially high critical energy value is temporarily set so that the transformation criterion is not satisfied and only austenite plasticity persists. A simple inverse iterative procedure is then followed until a satisfactory fit is achieved, resulting in the parameters displayed in Table IV. It is important to note once again that, as this alloy’s plasticity is of a rateindependent nature, the choice of the viscoplastic parameters in Table IV is motivated by a simple accuracycomputational time tradeoff: the values reported here are found to make the response sufficiently close to rateindependent while maintaining the problem computationally tractable in terms of solution time.
In the numerical solution of this problem, the prescribed deformation gradient is a 50 per cent simple shear, resulting in an equivalent strain ε_{eq} ≈ 0.28, where:
This deformation is applied in 1 s, resulting in an average strain rate of approximately 0.28 s^{−1} in the RVE.
This simple shear test is analysed using different homogenisation boundary conditions: uniform strain (Taylor), linear boundary displacements, periodic boundary displacement fluctuations and uniform boundary tractions (de Souza Neto and Feijóo, 2010). As the polycrystal meshes do not have onetoone nodal correspondence between opposing RVE faces, the periodic boundary condition is not trivially enforced. This is nevertheless the boundary condition of most interest, as it converges faster than the others to the representative properties of the homogenised medium (Terada et al., 2000), so a mortarbased formulation proposed by Reis and Andrade Pires (2014) is used here.
The resulting homogenised stressstrain curves are shown in Figure 5 for both meshes and all boundary conditions. From these results, it can be seen that the homogenised material behaviour is practically the same for both RVEs, confirming that 150 is a large enough number of grains to capture the isotropic material behaviour. Thus, in subsequent analyses, only the 150 grain RVE will be used, as the computational cost associated to both meshes is approximately the same (as they have similar numbers of degrees of freedom), and no appreciable difference can be expected from adding more crystallites. Figures 6 and 7 show the accumulated plastic slip distribution at the final deformation stage for both RVEs and all boundary conditions. It can be readily noticed that although both RVEs behave similarly from a homogenised perspective, their internal variable and displacement fluctuation distributions are completely different.
4.2 Homogenised stress–strain curves
Here, the full martensitic transformation material model is tested in the same simple shear example. With the calibrated hardening curve for the austenite at hand, the RVE is loaded once again in a shear deformation of 50 per cent with the martensitic transformation enabled. The resulting homogenised stress–strain curves are shown in Figure 8, along with experimental data from Perdahcioğlu and Geijselaers (2012). It can be seen that the model shows good agreement with the experimental results, capturing the observed hardening effect due to the transformation. However, an overly stiff response is observed once the transformation begins; this can be interpreted as a result of the simple coupling approach between transformation and austenite slip plasticity since, at a given material point, the latter stops once the former begins. Additionally, as the deformations become large, numerical convergence difficulties ensue due to the stiffness of the regularised model equations.
4.3 Transformation surface
To produce a transformation surface visualisation, the RVE is loaded with a set of macroscopic deformation gradients F representing different loading paths. For each deformation path, the homogenised Cauchy stress at which transformation starts at any material point is recorded, providing a point of the transformation surface. Rather than working with a hypersurface in sixdimensional stress space, for which a visualisation would be impossible, it is instead preferable to use the threedimensional principal Cauchy stress space – also known as Haigh–Westergaard space – with coordinates σ_{1}, σ_{2} and σ_{3}. Even in this case, instead of providing a full threedimensional depiction of the surface, its projections in the deviatoric stress plane[9], at different levels of hydrostatic pressure p = (σ_{1} + σ_{2} + σ_{3})/3, provide a more convenient visualisation.
4.3.1 Deviatoric plane cross section.
To cover all directions in the deviatoric stress plane, a loading programme ranging from triaxial extension (two equal principal stresses with the third one being higher) to simple shear (one principal stress equal to zero, and the other two equal but opposite in sign) and finishing at triaxial compression (analogous to triaxial extension but with the third eigenvalue lower than the other two) has to be devised. These stress states are labelled, respectively, TXE, SHR and TXC in Figure 10 and correspond to Lode angles θ of + 30°, 0° and −30°, with:
As the transformation criterion is expected to be pressuredependent and the computational code in which these results are obtained is straindriven – that is, in the case of a homogenisation analysis, the deformation gradient is the main input variable and the homogenised stress tensor the main output – it is not possible to directly impose a given stress state on the RVE. In particular, given the pressureindependent nature of the austenite slip plasticity constitutive model, even if care is taken to impose a set of deformation gradients with equal volumetric component (i.e. same determinant), the resulting homogenised hydrostatic pressure is not the same for each loading path. Thus, it is not possible to directly obtain a cross section of the transformation surface at a given hydrostatic pressure level using the full transformation and plasticity model.
A simple way to circumvent that limitation involves the suppression of the austenite plastic deformation, so that all grains in the polycrystal behave elastically until the transformation starts. In this case, the homogenised hydrostatic pressure is found to be approximately equal for all imposed macroscopic deformation gradients with a given determinant. One drawback of this approach is due to the isotropic hyperelastic potential adopted in this model: all grains deform identically until the transformation starts, as the crystallographical orientation has no effect until that point. This means that the different homogenisation boundary conditions yield the same result; with that in mind, results in this section will only be presented for the periodic boundary condition.
To impose the load programme on the RVE, a oneparameter family of deformation gradients F(α) is defined as a function load parameter α ∈ [–1, 1]. Ideally, the resulting Lode angle should vary linearly with α, so that a minimal number of analysis points will yield a meaningful transformation surface cross section. One such family can be defined by:
To predict the Lode angles resulting from the prescription of this deformation gradient family for α ∈ [–1, 1], its value is calculated analytically assuming a linear elastic material deforming under small strains, as a first approximation. In this case, the principal stresses are directly proportional to the principal strains, so the Lode angle can be easily determined. The results are displayed in Figure 9, clearly showing that the full range of Lode angles is covered. However, it can be seen by restricting the value of α to the interval [–0.163, 0.163], the full range of Lode angles θ ∈ [–30°, 30°] is also covered with no overlap and in almost linear fashion. The limits of this interval (α ≈ ±0.163) are determined by finding the values of α for which two of the principal stresses are equal, with the third one being nonzero.
With the loading programme defined, the RVE is deformed along each prescribed F until the transformation starts at any material point in the mesh; at that moment, the principal Cauchy stresses σ_{1}, σ_{2}, σ_{3} are recorded. Assuming that these are ordered such that σ_{1} ≥ σ_{2} ≥ σ_{3}, their values are used to compute the deviatoric plane coordinates of a point in the transformation surface[10]. Varying α in the predefined range, a sextant of the transformation surface’s deviatoric plane projection is covered. The other five are then obtained by appropriate permutations of σ_{1}, σ_{2} and σ_{3}, resulting in Figure 10. Data reported by Geijselaers and Perdahcioğlu (2009) are also shown, from which it is clear that the martensitic transformation model shows very good agreement with the experiments. The deviatoric plane projection of the equivalent Mohr–Coulomb locus is also shown at all hydrostatic pressures analysed, showing once again excellent agreement with the numerical results.
To study the pressuresensitivity of the transformation criterion, the same analyses are run two more times using nonisochoric deformation gradients. These are obtained from the same set F(α) with an added volumetric component, which is expansive in the first case (det[F] > 1) and compressive in the second (det[F] < 1). This yields the other two sets of points shown in Figure 10: one with hydrostatic pressure p ≈ 160 MPa and another with p ≈ –180 MPa. As would be expected given the volumetric expansion component of the transformation in this material, positive (tensile) hydrostatic pressures favour the transformation, whereas negative (compressive) hydrostatic pressures inhibit its onset.
4.3.2 Meridional profile.
Another way of highlighting the pressure dependency of the surface is through its meridional profile: by holding the Lode angle constant and varying the hydrostatic pressure, a crosssection of the surface along the hydrostatic axis (i.e. the axis along direction (1, 1, 1) in principal stress space) is obtained. The results of one such series of analyses, for Lode angle θ = 30°, are shown in Figure 11, where Haigh–Westergaard coordinates are used to maintain isomorphism with principal stress space (i.e. lengths and angles are preserved between both representations).
Using equations (27) and (32), the Mohr–Coulomblike surface equivalent to this material is predicted to have the following parameters:
The validity of this equivalence can be further checked using known properties of the Mohr–Coulomb yield surface, such as the fact the surface apex is located at a hydrostatic pressure value of:
5. Conclusions
In this work, a recently proposed largestrain, multiscale model for mechanically induced martensitic phase transformations is analysed. The model relies on the PTMC to obtain transformation properties such as shape change and habit plane orientations, also postulating a thermodynamically consistent generalisation of the stressassisted transformation criterion of Patel and Cohen (1953). It is shown that the model rigorously dissipates the mechanical energy barrier during the transformation course, in accord with experimental observations. The associated transformation functions also act as a variant selection mechanism: systems more favourably aligned with the local stress state are preferred for transformation. Particular attention is paid to the relation between this model’s transformation criterion and the classical Mohr–Coulomb yield function. In fact, the transformation model is shown to be of a pressuredependent nature, as a consequence of its nonisochoric character.
The constitutive model also includes the effects of austenite slip plasticity, incorporated using a finitestrain single crystal plasticity model. Even though the viscous properties are negligible for the alloys studied at the temperatures of interest, a viscoplastic formulation is used – for both austenite plasticity and martensitic transformation –to regularise the stress integration algorithm, preventing the difficulties associated to the selection of a set of active systems.
The mechanical behaviour of a metastable austenitic stainless steel is then analysed using this constitutive model. Given the extensive experimental data available in the literature, the chosen material provides a valuable test case for the constitutive model, as its metastable austenitic phase exhibits significant slip plasticity before the onset of the mechanically induced martensitic transformation. In a simple shear test, the overall effect of the martensitic transformation is captured by the model, but an overly stiff response is observed at higher strains. This effect is most likely due to the computational strategy adopted to handle the coupling between plasticity and transformation. A more robust formulation to address the simultaneous evolution of slip activity and transformation at intermediate martensite volume fractions is a topic of interest for further investigation. Nevertheless, the constitutive model shows particularly promising results in reproducing experimentally obtained transformation loci, providing further evidence that an energybased transformation criterion can account for both the stressassisted and the straininduced transformation regimes.
Figures
Figure 8.
Homogenised shear stressstrain curves for a simple shear test, showing both the current model’s predictions and experimental data from Perdahcioğlu and Geijselaers (2012)
Chemical composition of the metastable austenitic stainless steel ASTM A564, in Wt.%
C + N  Cr  Ni  Mo  Cu  Ti  Al  Si 

<0.05  12.0  9.1  4.0  2.0  0.9  0.4  <0.5 
Octahedral slip systems in FCC single crystals
α  m^{α}  s^{α}  α  m^{α}  s^{α} 

1 


7 


2 


8 


3 


9 


4 


10 


5 


11 


6 


12 


Transformation system vectors for the ASTM a564 alloy.
i  m^{i}  d^{i} 

1  [m_{1}, m̄_{2}, m_{3}]  [d̄_{1}, d_{2}, d_{3}] 
2  [m̄_{1}, m_{2}, m_{3}]  [d_{1}, d̄_{2}, d_{3}] 
3  [m_{1}, m_{2}, m_{3}]  [d̄_{1}, d̄_{2}, d_{3}] 
4  [m̄_{1}, m̄_{2}, m_{3}]  [d_{1}, d_{2}, d_{3}] 
5  [m̄_{2}, m_{1}, m_{3}]  [d_{2}, d̄_{1}, d_{3}] 
6  [m_{2}, m̄_{1}, m_{3}]  [d̄_{2}, d_{1}, d_{3}] 
7  [m_{2}, m_{1}, m_{3}]  [d̄_{2}, d̄_{1}, d_{3}] 
8  [m̄_{2}, m̄_{1}, m_{3}]  [d_{2}, d_{1}, d_{3}] 
9  [m_{1}, m_{3}, m̄_{2}]  [d̄_{1}, d_{3}, d_{2}] 
10  [m̄_{1}, m_{3}, m_{2}]  [d_{1}, d_{3}, d̄_{2}] 
11  [m_{1}, m_{3}, m_{2}]  [d̄_{1}, d_{3}, d̄_{2}] 
12  [m̄_{1}, m_{3}, m̄_{2}]  [d_{1}, d_{3}, d_{2}] 
13  [m̄_{2}, m_{3}, m_{1}]  [d_{2}, d_{3}, d̄_{1}] 
14  [m_{2}, m_{3}, m̄_{1}]  [d̄_{2}, d_{3}, d_{1}] 
15  [m_{2}, m_{3}, m_{1}]  [d̄_{2}, d_{3}, d̄_{1}] 
16  [m̄_{2}, m_{3}, m̄_{1}]  [d_{2}, d_{3}, d_{1}] 
17  [m_{3}, m_{1}, m̄_{2}]  [d_{3}, d̄_{1}, d_{2}] 
18  [m_{3}, m̄_{1}, m_{2}]  [d_{3}, d_{1}, d̄_{2}] 
19  [m_{3}, m_{1}, m_{2}]  [d_{3}, d̄_{1}, d̄_{2}] 
20  [m_{3}, m̄_{1}, m̄_{2}]  [d_{3}, d_{1}, d_{2}] 
21  [m_{3}, m̄_{2}, m_{1}]  [d_{3}, d_{2}, d̄_{2}] 
22  [m_{3}, m_{2}, m̄_{1}]  [d_{3}, d̄_{2}, d_{1}] 
23  [m_{3}, m_{2}, m_{1}]  [d_{3}, d̄_{2}, d̄_{1}] 
24  [m_{3}, m̄_{2}, m̄_{1}]  [d_{3}, d_{2}, d_{1}] 
Numerical values: m_{1} ≈ 0.608; m_{2} ≈ 0.178; m_{3} ≈ 0.774; d_{1} ≈ 0.156; d_{2} ≈ 0.046; d_{3} ≈ 0.159
Parameters used in the analyses
Material property  Value  Equation no.  

Elastic constants  E  210 GPa  8 
ν  0.3  
Austenite plasticity  τ_{y}_{0}  100 MPa  33 
K  195 MPa  
γ_{pa0}  0.01  
m  0.6  
μ_{pa}  0.2 s  28  
є_{pa}  0.2  
Martensitic transformation  ΔG_{mech}  56 MPa  20 
μ_{tr}  0.2 s  29  
є_{tr}  0.2 
Notes
For a detailed historical account of competing theories formulated both before and after the PTMC, see Kelly (2012).
In the materials science literature, the terms transformation system and variant are often used interchangeably, unlike their usage in group theory [as in, for instance, Hane and Shield (1998)].
See Reina and Conti (2014) for a rigorous micromechanical justification of the multiplicative split in the context of slip plasticity.
It is implied that the summation above is done only on active slip systems (i.e. those where there currently is slip activity), as otherwise
Summation of the repeated index is not implied in the last equation.
Schmid’s law’s applicability to FCC crystals is well established, in contrast to the BCC case, where nonSchmid effects are often important (Dao and Asaro, 1993). Modified versions of Schmid’s law have been proposed for this purpose in the literature [for example, see Yalcinkaya et al. (2008)], but are beyond the scope of this work.
Once more, summation of the repeated index is not implied in the last equation.
For instance, the 12 slip systems in FCC crystals and 24 martensitic transformation variants in the FCCtoBCC case.
The deviatoric stress plane is the set of principal Cauchy stresses that satisfy σ_{1} + σ_{2} + σ_{3} = 0.
Since it is the isotropic limit of the material behaviour that is of interest, the orientation used to describe the macroscopic stresses has no significance.
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Acknowledgements
The first and second authors gratefully acknowledge the Zienkiewicz Research Scholarships provided by Swansea University’s College of Engineering during the course of much of this work.