Analysis of a moving mask approximation for martensitic transformations
aa r X i v : . [ m a t h . A P ] N ov Analysis of a moving mask approximation for martensitictransformations
Francesco Della Porta
Mathematical Institute, University of OxfordOxford OX2 6GG, UK [email protected] ∗ November 21, 2018
Abstract
In this work we introduce a moving mask approximation to describe the dynamics of austeniteto martensite phase transitions at a continuum level. In this framework, we prove a new type ofHadamard jump condition, from which we deduce that the deformation gradient must be of theform + a ⊗ n a.e. in the martensite phase. This is useful to better understand the complexmicrostructures and the formation of curved interfaces between phases in new ultra-low hysteresisalloys such as Zn Au Cu , and provides a selection mechanism for physically-relevant energy-minimising microstructures. In particular, we use the new type of Hadamard jump conditionto deduce a rigidity theorem for the two well problem. The latter provides more insight on thecofactor conditions, particular conditions of supercompatibility between phases believed to influencereversibility of martensitic transformations. The aim of this work is to study from a mathematical point of view the complex microstruc-tures arising during the austenite to martensite phase transition in ultra-low hysteresis alloys such asZn Au Cu (see [31]). Austenite to martensite transitions are solid to solid phase transitions, inwhich the underlying crystal lattice of an alloy experiences a change of shape as temperature is movedacross a certain critical temperature θ T . When the temperature is above θ T , the alloy has a uniquecrystalline structure, called austenite, which is energetically preferable; when the temperature is low-ered below θ T , the energetically preferable state for the crystal is no longer austenite but martensite,which usually has more then one variant. Often, a change of crystalline structure implies a change inthe macroscopic properties of the material, which can thus be controlled by changing the temperatureof the sample. A serious obstacle to practical applications of shape-memory and other such materials ∗ Current institution: Max-Planck Institute for Mathematics in the Sciences, Leipzig, Germany,
Acknowledgements:
This work was supported by the Engineering and Physical Sciences Research Council[EP/L015811/1]. The author would like to thank John Ball for his helpful suggestions and feedback which greatlyimproved this work, as well as Richard James, Giacomo Canevari and Xian Chen for the useful discussions.
1s reversibility of the transformation. Indeed, after a small number of cycles, one can usually observe ashift in the transition temperature and in the latent heat. Furthermore, the formation of micro-cracksduring the phase transition often leads to early failure by rupture.An important step towards understanding the factors influencing reversibility can be found in[12]. There, the authors study particular conditions of geometric compatibility between martensiticvariants called cofactor conditions, that were first introduced in [6]. Among these conditions, there isthe requirement that the middle eigenvalue of the lattice transformation matrices is equal to 1, whichwas previously shown (see [33]) to influence reversibility. In [12], the authors prove that under thecofactor conditions no elastic transition layer is needed to make simple laminates compatible withaustenite, and point out that this fact might have important consequences on the reversibility of thephase transitions. Indeed, the authors observe that transition layers are intuitively both a cause ofthermal hysteresis, and of the formation of dislocations and nucleation of micro-cracks, that, aftermany cycles, induce loss of good reversibility properties.The recent fabrication announced in [31] of Zn Au Cu , the first material closely satisfying thecofactor conditions (the relative error is of order 10 − ), partially confirms this conjecture. Indeed, thismaterial exhibits ultra-low hysteresis and does not seem to incur any loss of reversibility after morethan 16 ,
000 thermal cycles. We refer the reader interested in ultra-low hysteresis alloys also to [14],where the fabrication of a new material undergoing 10 cycles with very little fatigue is announced. Adiscussion on the relation between the cofactor conditions and ultra-low hysteresis alloys can be foundin [26, 27].As remarked in [31], it is intriguing and unusual that martensitic microstructures in Zn Au Cu are drastically different in consecutive thermally induced transformation cycles, this being partiallymotivated by the fact that the cofactor conditions are close to being satisfied by both some type I andsome type II twins, which can all form zero energy interfaces with austenite.The aim of this paper is to further study these microstructures, and to identify a common char-acterization for all of them. To this end we start from the following observation: from the dynamicalpoint of view, it looks as if in every thermally induced phase transformation cycle in Zn Au Cu there was a mask moving across the domain covering and uncovering martensite microstructures. Thisis equivalent to saying that the martensitic microstructures do not change after the phase transitionhas happened, which seems to be a particularly legitimate approximation in materials satisfying thecofactor conditions, where no interface layer is needed between phases. But this hypothesis makessense also in many other materials as long as one considers macroscopic deformations.As a first step, we give a mathematical characterisation of the moving mask approximation and weframe it in the context of nonlinear elasticity, where phase changes are interpreted as elastic deforma-tions. In particular, microstructures satisfying the moving mask approximation are special solutionsto a simplified model for the dynamics of martensitic transformations which was introduced in [15].The model in [15] is derived from the equations for the conservation of energy and momentum inthe context of dynamics for nonlinear elasticity, and describes the evolution of the phase interface asa moving shock wave (see Section 3). Then, by deriving a new type of Hadamard jump condition(Section 4), we prove that every martensitic microstructure satisfying the moving mask assumptionand some further technical hypotheses must be of the form ∇ y ( x ) = + a ( x ) ⊗ n ( x ) , a.e. , (1.1)where n ( x ) is, up to a change of sign, the phase interface normal at x when the point x is on the2nterface. The above result does not follow directly from the assumption that the deformation gradientis unchanged after the phase transition, and is not a direct consequence of previously known Hadamardjump conditions if y is just Lipschitz as in our case. We refer the interested reader to Section 2.1 fora brief review on known versions of the Hadamard jump conditions and on why they do not imply(1.1). Subsequent experiments (see [13]) have measured k cof ( ∇ y − ) k in a sample of Zn Au Cu .The measured values are of the order of 10 − , which seem to be small enough to partially confirm thevalidity of (1.1) and of the moving mask approximation. Conversely, from the moving mask hypothesesand our new Hadamard jump condition, we can reconstruct the position of the austenite-martensiteinterface during the phase transition from a martensitic deformation gradient.Under suitable hypotheses, we prove also that ∇ · a = 0. This result relies on a Hadamard jumpcondition for strains in BV (Ω) that is proved in Section 5 and generalizes that in [18]. As a consequenceof the fact that ∇ y = + a ⊗ n almost everywhere in the martensitic microstructures, we can provea rigidity theorem for compound twins and a result that allows for a better understanding of thenature of curved austenite-martensite interfaces for type I twins. Under some further assumption, wecan extend the rigidity result to a general two well problem not satisfying the cofactor conditions.This result explains the importance of satisfying the cofactor conditions in order to have non-constantaverage deformation gradients of the form (1.1), which are obtained by finely mixing two martensiticvariants.The dynamics of the phase transition in Zn Au Cu are very complex, and far from beingcompletely understood. As in the static case, a major obstacle towards a good understanding ofthe phenomenon remains the lack of a characterization of the quasiconvex hull of the set of possibledeformation gradients. Nonetheless, our results provide an interesting set of tools that can be used tounderstand further the complex microstructures arising in martensitic phase transitions. Indeed, ourmoving mask hypothesis can be seen as a selection mechanism for physicaly relevant energy minimisingmicrostructures arising in thermally induced martensitic transformations.Further investigation on why martensitic microstructures are so different in different thermal cyclesin Zn Au Cu is carried out in [16]. Indeed, in [16] we show that this material satisfies somefurther conditions of compatibility (on top of the cofactor conditions) that makes the set of possiblemacroscopic deformation gradients of the form (1.1) unusually large.The plan for the paper is the following: in Section 2.1 we give a brief overview of the nonlinearelasticity model, and introduce concepts which will be useful for our analysis, namely twinning, thecofactor conditions and k-rectifiable sets. In Section 3 we recall results from [15] and in this contextwe introduce the mathematical definition of moving mask approximation. In Section 4 we prove adynamic variant of the Hadamard jump condition for Lipschitz functions and curved interfaces. Asexplained above, the results rely on the hypothesis that the deformation gradient remains constantin time at a point of the domain, once the phase transition has occurred. The last two sections aredevoted to proving the rigidity results, and some results on moving austenite-martensite interfacesand on possible microstructures that can be explained using our model.3 Preliminaries
In order to describe austenite to martensite phase transitions in crystalline solids, one of themost successful mathematical continuum models is nonlinear elasticity, which has proved to capturemany aspects of the physical phenomena such as the formation of twins (see [6]) and to be usefulin understanding related behaviour such as the shape-memory effect (see [10]), and, more recently,hysteresis (see [33]). In this and the next subsection, we give a brief overview of the theory followingclosely [9, 12]. For more details we refer the reader to [6, 8, 11].The nonlinear elasticity model is based on the idea of looking at changes in the crystal lattice aselastic deformations in the continuum mechanics framework. Following [6], we hence assume that thedeformations minimize a free energy of the type E ( y , θ ) = Z Ω φ ( ∇ y ( x ) , θ ) d x . (2.2)Here, θ denotes the temperature of the crystal. Three different regimes are distinguishable dependingon this parameter: θ < θ T and θ > θ T , where respectively martensite and austenite phases minimizethe energy, and θ = θ T where these are energetically equivalent. In (2.2), the bounded Lipschitzdomain (open and connected) Ω stands for the reference configuration of undistorted austenite at θ = θ T and y ( x ) denotes the position of the particle x ∈ Ω after the deformation. Finally, φ is thefree-energy density, depending on the temperature θ and the deformation gradient ∇ y , satisfying thefollowing properties: • D := { F ∈ R × : det F > } , φ ( · , θ ) : D → R is a function bounded below by a constant depending on θ for each θ > • φ ( · , θ ) satisfies frame-indifference, i.e., for all F ∈ D and all rotations R ∈ SO (3), φ ( RF , θ ) = φ ( F , θ ). This property reflects the invariance of the free-energy density under rotations; • φ has cubic symmetry, i.e., φ ( FQ , θ ) = φ ( F , θ ) for all F ∈ D and all rotations Q in the symmetrygroup of austenite P , the group of rotations sending a cube into itself (see [11] for more details); • denoting by K θ the set of minima for the free-energy density at temperature θ , i.e., K θ := { F ∈D : F ∈ argmin( φ ( G, θ )) } , K θ = α ( θ ) SO (3) , θ > θ T SO (3) ∪ S Ni =1 SO (3) U i ( θ T ) , θ = θ T S Ni =1 SO (3) U i ( θ ) , θ < θ T . (2.3)Here, α ( θ ) is a scalar dilatation coefficient satisfying α ( θ T ) = 1, while U i ( θ ) ∈ R × Sym + are the N positive definite symmetric matrices corresponding to the transformation from austenite tothe N variants of martensite at temperature θ . Here and below R × Sym + represents the set of3 × K θ when θ < θ T , and neglect the dependence on θ of the U i . We remark that foreach U i , U j there exists R ∈ P such that R T U j R = U i , so that U i , U j share the same eigenvalues.4ased on both experimental evidence and the mathematical complexity of other cases, most resultsin the literature are related to planar austenite-martensite interfaces { x ∈ R : x · n = k } , with normal n , at θ = θ T . In this case, under suitable conditions on the lattice deformation and for some n ∈ R ,it is possible to construct a sequence y j such that ∇ y j → SO (3) in measure for x · n < k (2.4) ∇ y j → N [ i =1 SO (3) U i in measure for x · n > k. (2.5)Denoting by L the three-dimensional Lebesgue measure, we notice that (2.4)-(2.5) implylim j →∞ L { x ∈ Ω : ∇ y j ( x ) / ∈ K θ T } = 0 , and, under some further hypotheses on φ , y j is a minimizing sequence for E ( · , θ T ) (see [6] for moredetails). Furthermore, since y j can be constructed so as to be bounded in W , ∞ (Ω , R ), there existsa subsequence y k j and y ∈ W , ∞ (Ω , R ) such that y k j converges to y weakly* in the same space.However, the energy functional is not quasiconvex and, in general, the minimum is not attained inthe classical sense. Therefore, ∇ y is not a minimizer for E ( · , θ T ), but just of its relaxation, E qc ( · , θ T ).From a physical point of view, ∇ y represents the deformation gradient in the sample at a macroscopicscale, an average of the fine microstructures with gradients in K := N [ i =1 SO (3) U i . It is important to remark that, in general, macroscopic deformation gradients ∇ y are not elements of K a.e. in Ω. Instead, we have ∇ y ∈ K qc a.e. in Ω, where K qc := ( M ∈ R × (cid:12)(cid:12)(cid:12) f ( M ) ≤ max K f , for all continuousquasiconvex f : R × → R ) , is the quasiconvex hull of the set K (see [29]). Characterizing the set of possible macroscopic defor-mations K qc is very important in order to fully understand the nonlinear elasticity model. On theother hand, the set of constant macroscopic gradients B which can form an interface with austenite,having constant gradient A , is in general smaller then the whole of K qc . Indeed, a Lipschitz functionwhose gradient is equal to A , B a.e. in Ω, with A , B ∈ R × must satisfy a generalized version of theHadamard jump condition proved in [6]: Proposition 2.1 ([6, Prop. 1]) . Let Ω ∈ R be open and connected. Assume y ∈ W , ∞ (Ω , R ) satisfies ∇ y ( x ) = A , a.e. x ∈ Ω A ; ∇ y ( x ) = B , a.e. x ∈ Ω B , where A , B ∈ R × and Ω A , Ω B are disjoint measurable sets such that Ω A ∪ Ω B = Ω , L (Ω A ) > , L (Ω B ) > . Then, A − B = a ⊗ n , a , n ∈ R , | n | = 1 . B to be rank-one connected to austenite, having constant gradient A , across every interface between the two. Fur-thermore, it implies that, in this case, every phase interface is planar. For these reasons, Proposition2.1 is the background for many results for compatibility between phases.However, this type of result fails to be true for more general gradients ∇ y ∈ L ∞ (Ω; R × ). As amatter of fact, as shown in [6], it is possible to construct a Lipschitz function z ∈ W , ∞ (Ω , R ) which isconstant in the set Ω ∩ { x : x · n > c } for some c ∈ R , n ∈ R , and whose gradient ∇ y ∈ { F , F , F , F } in Ω ∩ { x : x · n < c } for some matrices F i , such that ∇ y is not rank-one almost everywhere inΩ ∩ { x : x · n < c } . Indeed a fractal behaviour of ∇ y close to x · n = c , finely mixing martensiticvariants near the interface, allows one to achieve compatibility between incompatible gradients. Thatis, compatibility is achieved on the average. Possible approaches to recovering the Hadamard jumpcondition in an average sense can be found in [3], or in Remark 5.1 below. Another generalization ofProposition 2.1 was proved in [9] by assuming y ∈ W , ∞ (Ω , R ) to be C both in Ω A and Ω B , withΩ A , Ω B two open disjoint subdomains of Ω, separated by a piecewise C , possibly curved, 2-dimensionalinterface Γ such that Ω = Ω A ∪ Ω B ∪ Γ. In the case of martensitic transformations, ∇ y = in Ω A ,while in Ω B ∇ y represents a continuously varying macroscopic deformation gradient correspondingto a continuously varying martensitic microstructure. This result can be extended to y ∈ H (Ω , R )with ∇ y ∈ BV (Ω; R × ) as done in the two dimensional setting in [18], or more generally in Lemma5.1 below. However, the deep result of [25] states that, in the case where y ∈ W , ∞ ( R , R ), and y is constant in { x · n > c } , the polyconvex hull of the set (cid:8) ∇ y ( x ) : x ∈ { x · n < c } (cid:9) , which contains (cid:8) ∇ y ( x ) : x ∈ { x · n < c } (cid:9) qc , might not contain a matrix which is rank-one connected to , the defor-mation gradient in { x · n > c } .On the other hand, in order to fully capture the complex microstructures observed in Zn Au Cu ,we are interested in macroscopic deformation gradients that are just in L ∞ (Ω , R × ). Therefore, inSection 4 we generalize Proposition 2.1 to non-constant deformation gradients in L ∞ (Ω; R × ) andto curved interfaces. However, given the above mentioned counterexamples of [6, 25], we need tochange perspective and introduce some further hypotheses. This is done by recalling the idea of amoving mask explained in the introduction, which is mathematically framed in Section 3, and wherethe deformation gradient at a certain point x changes only once during phase transition, i.e., whenthe martensite-austenite interface passes through x . As explained in the previous section, the existence of a constant macroscopic martensitic deforma-tion compatible with austenite, is related to the existence of a matrix F ∈ K qc such that F = + a ⊗ n .Conditions on the deformation parameters under which such matrices exist have been first investigatedin [6] in the case of two wells, i.e., N = 2, and then generalized in [4, 5]. The case where N = 2 is themost widely studied, as it is the only one for which an explicit characterization of K qc is known, andturns out to be a fundamental tool to explain a wide range of experimental observations. Therefore,we now focus on the possibility of a pair of martensitic variants forming interfaces with austenite. Thenotation and results of this section follow closely those in [12].Let us first recall that given two different variants of martensite, represented by U , U ∈ R × Sym + ,there exists a rotation R ∈ SO (3) satisfying U = RU R T . A first useful result is the following:6 roposition 2.2 ([12, Prop. 12]) . Let U , U ∈ R × Sym + with U = U . Suppose further that they arecompatible in the sense that there is a matrix ˆ R ∈ SO (3) such that ˆ RU − U = b ⊗ m , (2.6) b , m ∈ R . Then there is a unit vector ˆ e ∈ R such that U = ( − + 2ˆ e ⊗ ˆ e ) U ( − + 2ˆ e ⊗ ˆ e ) . (2.7) Conversely, if (2.7) is satisfied, then there exist ˆ R ∈ SO (3) , b , m ∈ R such that (2.6) holds. Equation (2.6) is called the compatibility condition for two variants of martensite; the solutionsto this equation can be classified into three categories: compound, type I and type II twins. It ispossible to prove that (see e.g., [11]), once U and U are given and (2.7) holds, the compatibilitycondition always has two solutions (ˆ R I , b I ⊗ m I ) and (ˆ R II , b II ⊗ m II ). The solutions can be expressedas follows:type I m I = ˆ e , b I = 2 (cid:16) U − ˆ e | U − ˆ e | − U ˆ e (cid:17) , (2.8)type II m II = 2 (cid:16) ˆ e − U ˆ e | U ˆ e | (cid:17) , b II = U ˆ e , (2.9)where ˆ e is as in (2.7). If ˆ e satisfying (2.7) is unique up to change of sign, the two solutions (2.8) and(2.9) of (2.6) are called type I and type II twins respectively. In case there exist two different non-parallel unit vectors satisfying (2.7), the resulting pair of solutions (2.8)–(2.9) are called compoundtwins. Nonetheless, it is possible to prove (see e.g., [11]) that in the case of compound twins, giventwo different unit vectors satisfying (2.7), namely ˆ e and ˆ e , then b I ⊗ m I := 2 (cid:16) U − ˆ e | U − ˆ e | − U ˆ e (cid:17) ⊗ ˆ e = U ˆ e ⊗ (cid:16) ˆ e − U ˆ e | U ˆ e | (cid:17) =: b II ⊗ m II . Therefore, there are just two solutions to (2.7), even in the case of compound twins, each of whichcan be considered as both a type I and a type II twin. Below, however, when we refer to type I ortype II solutions of (2.6) we assume implicitly that they are not compound solutions. Furthermore,we sometimes abuse of notation and write that U , U form a compound twin if the solutions of thetwinning equations (2.6) are compound twins. The following characterization of compound twins isused below: Proposition 2.3 ([12, Prop. 1]) . Let U and U be two different variants of martensite and ˆ e a unitvector such that (2.7) is satisfied. Then there exists a second unit vector ˆ e not parallel to ˆ e satisfying (2.7) if and only if ˆ e is perpendicular to an eigenvector of U . When this condition is verified, ˆ e isunique up to change of sign and is perpendicular to both ˆ e and that eigenvector. Let us now consider a simple laminate, i.e., a constant macroscopic gradient ∇ y equal a.e. to λ ˆ RU + (1 − λ ) U for some λ ∈ (0 ,
1) and some rank-one connected RU , U ∈ K . Following [6, 12] wefocus on the possibility for such ∇ y to be compatible with austenite. By Proposition 2.1, a necessarycondition is that SO (3) has a rank-one connection with λ ˆ RU +(1 − λ ) U . The existence of ( R , λ, a ⊗ n )solving R (cid:2) λ ˆ RU + (1 − λ ) U (cid:3) − = R (cid:2) λ ( U + b ⊗ m ) + (1 − λ ) U (cid:3) − = a ⊗ n , (2.10)7hat is a twinned laminate compatible with austenite, was first studied in [32] and later in [6]. Latticedeformations and parameters of materials that are usually considered in the literature lead to twinswith exactly four solutions to equation (2.10). Nonetheless, in some cases the number of solutions canbe just zero, one or two, and, under some particular condition on the lattice parameters, as in thecase of the material discovered in [31], (2.10) is satisfied for all λ ∈ [0 , Theorem 2.1 ([12, Thm. 2]) . Let U , U ∈ R × Sym + be distinct and such that there exist ˆ R ∈ SO (3) and b , m ∈ R satisfying ˆ RU = U + b ⊗ m . Then, (2.10) has a solution R ∈ SO (3) , a , n ∈ R for each λ ∈ [0 , if and only if the following cofactor conditions hold: (CC1) The middle eigenvalue λ of U satisfies λ = 1 , (CC2) b · U cof ( U − ) m = 0 , (CC3) tr U − det U − | b | | m | − ≥ . In the last part of this section, we report some results from [12] related to the cofactor conditionsin type I/II twins.
Theorem 2.2 ([12, Thm. 7]) . Let U , U ∈ R × Sym + be distinct and such that ˆ R ∈ SO (3) , b I , m I ∈ R is a type I solution to (2.6) . Suppose further that U , b I , m I satisfy the cofactor conditions. Then,there exist R ∈ SO (3) , a ∈ R , n , n ∈ S and ξ = 0 such that R U = + a ⊗ n , R ( U + b I ⊗ m I ) = + a ⊗ ξ n . (2.11) Furthermore, R [ U + λ b I ⊗ m I ] = + a ⊗ (cid:0) λξ n + (1 − λ ) n (cid:1) , for all λ ∈ [0 , . (2.12) Theorem 2.3 ([12, Thm. 8]) . Let U , U ∈ R × Sym + be distinct and such that ˆ R ∈ SO (3) , b II , m II ∈ R is a type II solution to (2.6) . Suppose further that U , b II , m II satisfy the cofactor conditions. Then,there exist R ∈ SO (3) , a , a ∈ R , n ∈ S and ξ = 0 such that R U = + a ⊗ n , R ( U + b II ⊗ m II ) = + ξ a ⊗ n . (2.13) Furthermore, R [ U + λ b II ⊗ m II ] = + (cid:0) λξ a + (1 − λ ) a (cid:1) ⊗ n , for all λ ∈ [0 , . (2.14) k -rectifiable sets In this section we recall some standard results on Lipschitz functions and k -rectifiable sets from[2, 21, 28] (see also [1] for properties of level sets of Lipschitz functions). We denote by H k the k -dimensional Hausdorff measure, and write H k E for its restriction to an H k measurable subset E . C c ( R d ) stands for the space of continuous functions with compact support in R d , while B d ( x , r )denotes the d -dimensional ball centred at x , with radius r and of volume ω d r d . We start with thefollowing definitions: 8 efinition 2.1. A Lipschitz k -graph G is a set of points in R d with d > k such that there exists anopen and connected set ω ⊂ R k , a Lipschitz map ψ : ω → R d − k , and a rotation Q ∈ SO ( d ) satisfying G := (cid:8) Q x , x = ( x ′ , ψ ( x ′ )) , x ′ ∈ ω (cid:9) . Definition 2.2.
Let E ⊂ R d be an H k -measurable set satisfying H k ( E ) < ∞ . We say that E is k -rectifiable if there exist countably many Lipschitz mappings f i : R k → R d such that H k (cid:16) E \ ∞ [ i =1 f i ( R k ) (cid:17) = 0 . An equivalent characterization for such sets is given by the following result:
Proposition 2.4 ([2, Prop. 2.76]) . Any H k -measurable set E is countably H k -rectifiable if and onlyif there exist countably many Lipschitz k -graphs G i ⊂ R N , such that H k (cid:16) E \ ∞ [ i =1 G i (cid:17) = 0 . In what follows, a particular case of [21, Theorem 3.2.22] is also used:
Theorem 2.4.
Let Ω ⊂ R be open, bounded and connected, Z ⊂ R be a -rectifiable set and f : Ω → Z a Lipschitz function. Define the -dimensional Jacobian of f by: J f := sX i,j ( ∇ f ) ij . Then: • for L almost every x ∈ Ω , either J f ( x ) = 0 , or the image of ∇ f ( x ) is a -dimensional vectorspace, i.e., rank ∇ f ( x ) ≤ , L -almost everywhere in Ω ; • for H almost all ξ ∈ Z , f − ( ξ ) is -rectifiable; • for every integrable g : Ω → [ −∞ , ∞ ] Z Ω g ( x ) J f ( x ) d x = Z Z Z f − ( ξ ) ∩ Ω g ( s ) d H ( s ) d H ( ξ ) (2.15) The aim of this section is to give a precise definition of the moving mask assumption (see Definition3.1 below), and to frame it in the context of dynamics for nonlinear continuum mechanics. This isdone by recalling first the simplified model derived in [15] to describe the evolution of martensitictransformation in the context of nonlinear continuum mechanics. In this framework, we introducesome hypotheses approximating experimental observation. These hypotheses are made precise in Def-inition 3.1. We remark that the model in [15] is used here just to frame the moving mask assumption,and that the rest of the paper relies on Definition 3.1 only, which could be hence taken by the readeras a standalone assumption. 9n [15] we introduced a continuum model for the evolution of martensitic transformations. Afterpassing to the limit in which the elastic constants tend to infinity and the interface energy densitytends to zero, we deduced that the deformation gradients and the temperature field generate in thelimit a Young measure ν x ,t (see as a reference [29, 30]) and a function θ satisfying in a suitable sense ρ θ t − d ∆ θ = − θ T ∂∂t Z R × η ( A ) d ν x ,t ( A ) , a.e. in Ω × (0 , T ) , (3.16)supp ν x ,t ⊂ SO (3) ∪ K, a.e. in Ω × (0 , T ) , (3.17)complemented with some initial and boundary conditions. Here, ρ is the density of the body, d is adiffusivity coefficient which is supposed to be constant, and η is a smooth function such that η ( F ) = 0 , for all F ∈ SO (3) , η ( F ) = − αθ T , for all F ∈ K, for some constant α > R R × η ( A ) d ν x ,t ( A ), and θ and ν x ,t . Nonetheless, we aim to characterise solutions independently ofthe constitutive relation. In order to do this we introduce some hypotheses on the solutions, that arebased on experimental observation and that together we call the moving mask approximation, definedprecisely in Definition 3.1 below, using the following ingredients: • the phases are separated, that is there exist open sets Ω A ( t ) , Ω M ( t ) ⊂ Ω such thatΩ A ( t ) ∩ Ω M ( t ) = ∅ , L (cid:0) Ω \ (Ω A ( t ) ∪ Ω M ( t )) (cid:1) = 0 , a.e. t ∈ (0 , T ) , and ν x ,t ( SO (3)) = 1 , a.e. x ∈ Ω A ( t ), a.e. t ∈ (0 , T ) ,ν x ,t ( K ) = 1 , a.e. x ∈ Ω M ( t ), a.e. t ∈ (0 , T ) . The domain can hence be divided for almost every t ∈ (0 , T ) into two regions, the region withmartensite Ω M ( t ), and the region with austenite Ω A ( t ). Thus, Z R × η ( A ) d ν x ,t ( A ) = − αθ T χ Ω M ( x , t ) , (3.18)where χ Ω M ( x , t ) is the characteristic function of Ω M ( t ). The austenite-martensite phase bound-ary is sharp in this case. In terms of macroscopic deformation gradients this reads ∇ y ( x , t ) ∈ K qc , a.e. in Ω M ( t ), a.e. t ∈ (0 , T ) , ∇ y ( x , t ) ∈ SO (3) , a.e. in Ω A ( t ), a.e. t ∈ (0 , T ) , • during the phase transition, the macroscopic deformation gradient remains equal to a constantrotation in the austenite region. This is the case, for example, when the austenite region isconnected; • the phase interface moves continuously. More precisely, for almost every point x in the domain,there exists a time when x is contained in the phase interface (see also [15, Remark 5.2]);10 the microstructures do not change after the transformation has happened. This assumptionmakes particular sense in the context of materials satisfying the cofactor conditions, whereaustenite and finely twinned martensite can be exactly compatible across interfaces, and evenmore in Zn Au Cu where the phase transition has very low thermal hysteresis and thermalexpansion is hence negligible.As remarked in the introduction, this construction reflects the idea of a moving mask that uncoversa martensitic microstructure, as can be seen in the video of [31]. Mathematically we can define themoving mask approximation as follows: Definition 3.1.
We say that ∇ y ∈ L ∞ (Ω; R × ) satisfies the moving mask approximation if • for each t ∈ [0 , T ] there exist Ω M ( t ) , Ω A ( t ) ⊂ Ω disjoint and open, such that L (cid:0) Ω \ (Ω A ( t ) ∪ Ω M ( t )) (cid:1) = 0; • either Ω A ( t ) ⊂ Ω A ( t ) , for all ≤ t ≤ t ≤ T , or Ω A ( t ) ⊂ Ω A ( t ) , for all ≤ t ≤ t ≤ T ; • for a.e. x ∈ Ω there exists t = t ( x ) ∈ [0 , T ] such that x ∈ Ω A ( t ) ∩ Ω M ( t ) ; • there exists Q ∈ SO (3) such that for every t ∈ [0 , T ] the map y M ( · , t ) satisfying ∇ y M ( x , t ) = ( ∇ y ( x ) , a.e. in Ω M ( t ) Q , a.e. in Ω A ( t ) is in W , ∞ (Ω; R ) . Remark 3.1.
We note that, in the case Ω M ( s ) ⊂ Ω M ( t ) for each s, t ∈ [0 , T ] with s < t , we have \ t ∈ [0 ,T ] Ω A ( t ) = ∅ , [ t ∈ [0 ,T ] Ω M ( t ) = Ω . Remark 3.2.
If we assume (3.18), then the formula for differentiation of integrals on time dependentdomains implies that D ∂∂t Z R × η ( A ) d ν x ,t ( A ) , ψ E = h ˙ χ Ω M ( ∇ y ) , ψ i = ddt Z Ω M ψ d x = Z Γ( t ) ( v · n ) ψ d H , ∀ ψ ∈ C ∞ (Ω) , (3.19)provided Ω A ( t ) , Ω M ( t ) and v · n are smooth enough (see e.g., [22]). Here Γ( t ) := Ω \ (Ω A ∪ Ω M )( t )is a surface separating Ω A ( t ) from Ω M ( t ), n denotes the outer normal to Ω M and v ( s ) is the velocityof the interface at the point s ∈ Γ( t ) at time t . By h· , ·i we denoted the duality pairing between adistribution and a test function. A version of (3.19) in the case of some solutions to (3.16)–(3.17)satisfying the moving mask assumptions is given by Corollary 4.4 below.11 Generalized Hadamard conditions
In this section we restrict our attention to deformation gradients ∇ y that satisfy the moving maskapproximation as stated in Definition 3.1. In order to say something more about solutions under theseassumptions, we prove below a variant of the Hadamard jump condition reflecting this hypothesis.In what follows, we restrict, without loss of generality, to the case Ω M ( s ) ⊂ Ω M ( t ) for every s < t .As before, below Ω ⊂ R is an open bounded connected set with Lipschitz boundary. For simplicity,rather than working with the deformation map y , in this section we mostly work with the displacementmap z := y − Q x , where Q is as in Definition 3.1.We start by proving the result when the phase interfaces are planar. This situation describes, forexample, the propagation of a simple martensitic laminate in the austenite phase. Proposition 4.1.
Let Γ( t ) be a family of parallel planes perpendicular to n ∈ S , Γ( t ) := (cid:8) x ∈ R : x · n = h ( t ) (cid:9) , for some non-decreasing function h ∈ C ([0 , T ]) satisfying h (0) = inf x ∈ Ω x · n , h ( T ) = sup x ∈ Ω x · n . For t ∈ [0 , T ] define Ω M ( t ) := Ω ∩ (cid:8) x · n < h ( t ) (cid:9) , Ω A ( t ) := Ω ∩ (cid:8) x · n > h ( t ) (cid:9) . Let z ∈ W , ∞ (Ω; R ) be such that Z = Z ( x , t ) satisfying ∇ Z ( x , t ) = ( ∇ z ( x ) , a.e. in Ω M ( t ) , a.e. in Ω A ( t ) (4.20) is in W , ∞ (Ω; R ) for a.e. t ∈ (0 , T ) . Then,1. there exists a ∈ L ∞ (Ω; R ) such that ∇ z ( x ) = a ( x ) ⊗ n , a.e. x ∈ Ω .
2. if Ω ∩ Γ( t ) is connected for every t ∈ (0 , T ) then z = f ( x · n ) for some f ∈ W , ∞ ((0 , T ); R ) .Proof. By rotating the system of coordinates we can assume without loss of generality that n = e .Let us consider the set B ⊂ Ω of points where z is differentiable, and the set B of points x ∈ Ω suchthat there exists t ∗ ∈ (0 , T ) for which x ∈ Γ( t ∗ ) and Z ( · , t ∗ ) ∈ W , ∞ (Ω; R ). By continuity of h we havethat L (cid:0) Ω \ ( B ∩ B ) (cid:1) = 0. Let us thus consider a generic point ˆ x ∈ B ∩ B , and notice that, since Ωis open, there exists r > B (ˆ x , r ) ⊂ Ω. By (4.20), Z ( · , t ∗ ) must be constant in eachconnected component of Ω A ( t ). In particular, as Z ( · , t ∗ ) ∈ W , ∞ (Ω , R ) is continuous, it must be aconstant on Γ( t ∗ ) ∩ B (ˆ x , r ). At the same time, continuity of z and Z ( · , t ∗ ) implies also z ( x ) = Z ( x , t ∗ )for every x ∈ Γ( t ∗ ) ∩ B (ˆ x , r ). Therefore, the function z ( x ) must be constant on Γ( t ∗ ) ∩ B (ˆ x , r ). Thisimplies, ∂ z ∂x i (ˆ x ) = 0 , i = 1 , . x ∈ B ∩ B yields the first statement. On the other hand, if Ω ∩ Γ( t ) is connectedfor a.e. t ∈ (0 , T ), then z ( x ) is constant on Γ( t ∗ ) ∩ Ω for a.e. t ∈ (0 , T ) and hence z = z ( x ). Thisconcludes the proof. Remark 4.1.
We could replace the hypothesis concerning the connectedness of Ω A ( t ) by assumingthat Z ( x , t ) is equal to a constant c ( t ) ∈ R in Ω A ( t ) for a.e. t ∈ [0 , T ]. Both these assumptions areautomatically satisfied if Ω is convex. Remark 4.2.
In the case z = f ( x · n ), and Γ( t ) is a single plane, the phase interfaces must coincide withthe level sets of f . Therefore, given an experimentally measured martensitic macroscopic deformationgradient, under the assumption that it satisfies the moving mask approximation, and is of the form + a ( x · n ) ⊗ n , for some a ∈ R , n ∈ S , we can reconstruct the position of austenite-martensite phaseinterfaces, by taking the level sets of f ( x · n ) = R x · n a ( s ) d s. Furthermore, in the case z = f ( x · n ),and Γ( t ) is a single plane, the discontinuities in the macroscopic deformation gradient can occur onlyacross the planes x · n = constant . This is, for example, the case for type II twins satisfying thecofactor conditions, for which we refer the reader to Proposition 6.2.Proposition 4.1 can be partially generalized to the case where Γ( t ) is a family of curved interfaces.As a first step, we need to introduce the concept of moving interfaces for our problem, generalizingthe previous requirements on planar such interfaces. Definition 4.1.
We say that Γ( t ) ⊂ Ω is a family of moving interfaces in Ω if:(i) there exist two families of open disjoint sets Ω M ( t ) , Ω A ( t ) ⊂ Ω and a bounded open interval I T := [0 , T ] such that for every t in I T , Ω = Ω M ( t ) ∪ Ω A ( t ) ∪ Γ( t ) and Γ( t ) ∩ Ω M ( t ) = Γ( t ) ∩ Ω A ( t ) = ∅ . Furthermore, Ω M ( t ) is non-decreasing in t, i.e., Ω M ( t ) ⊂ Ω M ( s ) , Ω A ( s ) ⊂ Ω A ( t ) , ∀ t < s ∈ I T ; (ii) the set B := x ∈ Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ∈ Γ( t ∗ ) , t ∗ ∈ I T and there exist U x ⊂ Ω openand connected, x ∈ U x , and a Lipschitz -graph G x differentiable at x such that G x ∩ U x ⊂ Γ( t ∗ ) ∩ U x ⊂ Ω M ( t ∗ ) ∩ Ω A ( t ∗ ) ∩ U x is measurable and L (cid:0) Ω \ B (cid:1) = 0 .Points in B are called regular points for Γ( t ) . At this point we can also introduce the concept of a regular moving mask approximation:
Definition 4.2.
We say that y ∈ W , ∞ (Ω; R ) satisfies a regular moving mask approximation if itsatisfies the moving mask approximation and Γ( t ) = Ω \ (Ω A ( t ) ∪ Ω M ( t )) is a family of moving interfaces in Ω , where Ω A , Ω M are as in Definition a x b Γ( t )Γ( t )Ω M ( t )Ω A ( t )Figure 1: Points which are not regular: x a is in a smooth k -graph contained in Γ( t ), but is notseparating Ω A from Ω M . Γ( t ) does not coincide with a Lipschitz function differentiable at x b . Remark 4.3.
The requirement Γ( t ∗ ) ∩ U ⊂ Ω A ( t ∗ ) ∩ Ω M ( t ∗ ) ∩ U in Definition 4.1(ii) is mainly toguarantee that the set where an interface is cutting either Ω A or Ω M and not separating one from theother is small (see e.g., the point x a in Figure 1). In this way, families of moving interfaces satisfyingthe separation condition may also describe further nucleations in the interior of Ω A during the phasetransition.Below, we say that a curve c : [ t , t ] → R , for some t , t ∈ R , is simple if c ( s ) = c ( t ) for each s, t ∈ [ t , t ]. The following theorem generalizes Proposition 4.1 to curved interfaces: Theorem 4.1.
Let Γ( t ) be a family of moving interfaces in Ω . Assume z ∈ W , ∞ (Ω; R ) is such thatthe function Z = Z ( x , t ) satisfying ( ∇ z ( x ) , a.e. in Ω M ( t ) , a.e. in Ω A ( t ) (4.21) with Ω A ( t ) , Ω M ( t ) as in Definition , is in W , ∞ (Ω; R ) for every t ∈ I T . Then, there exist a ∈ L ∞ (Ω; R ) , n ∈ L ∞ (Ω; S ) such that ∇ z ( x ) = a ( x ) ⊗ n ( x ) , a.e. x ∈ Ω . (4.22) Conversely, let z ∈ W , ∞ (Ω; R ) be such that (4.22) is satisfied and z (Ω) is contained in the image ofan absolutely continuous simple curve c : I T → R of finite length. Then, if | a | > a.e. in Ω , thereexists a family of moving interfaces in Ω , and a Z = Z ( x , t ) in W , ∞ (Ω; R ) satisfying (4.21) for every t ∈ I T . Furthermore, L (cid:0) Ω \ (Ω A ( t ) ∪ Ω M ( t )) (cid:1) = 0 for every t ∈ I T . Remark 4.4.
The assumptions on the image of z in Theorem 4.1 are motivated by the followingobservation: if z ∈ C (Ω; R ), and ∇ z is rank-one everywhere in Ω, then the constant rank theoremimplies that, around every x ∈ Ω, the image of z is a simple absolutely continuous curve. However,the set z (Ω) can a priori show branching and other complex structures even in the regular case (e.g.,if Ω is non-convex). For the sake of clarity of the proof, in this paper we restrict ourselves to theeasier case where z (Ω) is a simple absolutely continuous curve. Nonetheless, a statement similar tothe second implication in Theorem 4.1 can be proved for maps z : Ω → R whose image satisfies • z (Ω) is 1 − rectifiable; • for H − a.e. ξ ∈ z (Ω) there exist an open ball D ξ ⊂ R such that D ξ ∩ z (Ω) is a simple curveof finite length which is absolutely continuous.14 emark 4.5. Assume that the moving mask assumption holds, and that we can reconstruct z fromexperimental observations. Then, provided the image of z satisfies the stated assumptions, the secondpart of Theorem 4.1 gives a useful tool to reconstruct phase interfaces during the phase transformation.For the proof of Theorem 4.1 we need the following Lemma: Lemma 4.1.
Let f ∈ L ∞ (Ω; R × ) be such that f ( x ) = ( b ⊗ m )( x ) for a.e. x ∈ Ω . Then, there exist a , n ∈ L ∞ (Ω; R ) such that f ( x ) = a ( x ) ⊗ n ( x ) and | n ( x ) | = 1 for a.e. x ∈ Ω .Proof. This is just a matter of measurability of a , n . As f is measurable, so is f T f = ( | b | m ⊗ m ), sois its trace tr( f T f ) = | b | | m | and so is the function g := ( | b | − | m | − f T f , if | b | | m | = 0,0 , otherwise . Therefore, we define Ω := { x ∈ Ω : g ( x ) = 0 } and n ( x ) = (cid:0) ( g ( x )) , g ( x )( g ( x )) − , g ( x )( g ( x )) − (cid:1) T , for almost every x ∈ Ω . This is actually possible because g ii ≥ i = 1 , ,
3. Define alsoΩ := { x ∈ Ω \ Ω : g = 0 } , Ω := { x ∈ Ω \ (Ω ∪ Ω ) : g = 0 } , and define n in Ω , Ω respectively by n = (cid:0) g ( g ) − , ( g ) , g ( g ) − (cid:1) T , n = (cid:0) g ( g ) − , g ( g ) − , ( g ) (cid:1) T . Therefore, choosing n arbitrarily and such that | n | = 1 in the set where g = , we have constructed n ∈ L ∞ (Ω; R ) as desired. Defining a := f n , we thus conclude the proof. Proof of Theorem 4.1.
We first prove (4.22).Let N z be the set where z is not differentiable and remark that, by the hypotheses, N := N z ∪ (Ω \ B )is an L -negligible set. Let x ∈ Ω \ N and take U x to be a neighbourhood of x as in Definition 4.1(ii). By taking a smaller connected neighbourhood of x , which we still denote by U x , we can assumethat G x ∩ U x is connected. We first claim that z is constant on G x ∩ U x . Indeed, as ∇ Z ( x , t ∗ ) = 0a.e. in Ω A ( t ∗ ), the continuity of Z ( x , t ∗ ) implies that Z ( x , t ∗ ) must be constant on every connectedcomponent of Ω A ( t ∗ ). Since Definition 4.1 (ii) implies G x ∩ U x ⊂ Ω A ( t ∗ ), we must have Z ( · , t ∗ ) = ˆ c for some ˆ c ∈ R on G x ∩ U x . On the other hand, continuity of z , Z ( · , t ∗ ) together with (4.21) implythat on every connected component of Ω M ( t ∗ ) z = Z ( · , t ∗ ) + ¯ c for some ¯ c ∈ R depending on theconnected component. Therefore, as by Definition 4.1 (ii) G x ∩ U x ⊂ Ω M ( t ∗ ), the fact that Z ( · , t ∗ )is constant on G x ∩ U x implies that so must be z .Now, as G x is a Lipschitz 2-graph, we can find a Lipschitz change of coordinates ψ : U x → V such that ψ ( G x ∩ U x ) = (cid:8) x ∈ R : x · n ( x ) = c Γ (cid:9) ∩ V, V ⊂ R , c Γ ∈ R and where n ( x ) is the normal vector to G x at x pointingoutwards from Ω M ( t ∗ ). Let us denote ψ ( x ) = ¯ x for every x ∈ U x . We define ¯ z as ¯ z (¯ x ) = z ( ψ − (¯ x )),and assuming without loss of generality that n ( x ) = e , we get that¯ z (¯ x ) = ¯ z (¯ x + s e i ) = ˆ c , i = 1 , , for each s such that ¯ x + s e i ∈ V. This is due to the fact that z ( x ) = ˆ c for every x ∈ U x ∩ G x .Therefore, ∂ ¯ z ∂ ¯ x i (¯ x ) = 0 , i = 1 , . On the other hand, as x ∈ Ω \ N , we have ∇ x z ( x ) = ∇ ¯ x ¯ z ( ψ ( x )) ∇ x ψ ( x ) . Since x is a regular point, ψ can be chosen to be differentiable in x , and therefore ¯ x is a point ofdifferentiability for ¯ z . Therefore, there exists a ∈ R such that ∇ ¯ x ¯ z (¯ x ) = a ⊗ n ( x ) . By putting together the last two identities and using the fact that N is negligible we finally deduce(4.22). Measurability of a , n follows from Lemma 4.1.We now prove the second statement. We first remark that since c is absolutely continuous andof finite length, it belongs also to W , ( I T ; R ) and there exists d ∈ W , ∞ ( I ∗ T ; R ) for some interval I ∗ T ⊂ R such that c ( I T ) = d ( I ∗ T ) (see e.g., [1] and references therein). Therefore c ( I T ) is 1-rectifiable.By Theorem 2.4, Γ( t ) := z − ( c ( t )) ∩ Ω are 2-rectifiable surfaces for almost every t , and z is equal toa constant on them. DefiningΩ M ( t ) := (cid:8) x ∈ Ω : ∃ s ∈ [0 , t ) such that x ∈ z − ( c ( s )) (cid:9) , Ω A ( t ) := (cid:8) x ∈ Ω : x / ∈ z − ( c ( s )) , ∀ s ∈ [0 , t ] (cid:9) , (4.23)it is easy to see that Definition 4.1(i) is satisfied, provided we can show that Ω A ( t ) , Ω M ( t ) are open.To this end, let us fix t ∗ ∈ I T , ˆ x ∈ Ω M ( t ∗ ), and let us denote by s ˆ x ∈ [0 , t ∗ ) the point such that z (ˆ x ) = c ( s ˆ x ). As z is Lipschitz, we can define R := k∇ z k − L ∞ | c ( t ∗ ) − z (ˆ x ) | , so that | z ( x ) − z (ˆ x ) | ≤ k∇ z k L ∞ | x − ˆ x | ≤ | c ( t ∗ ) − z (ˆ x ) | = 12 | c ( t ∗ ) − c ( s ˆ x ) | , (4.24)for all x ∈ B R (ˆ x ). Suppose now that in B R (ˆ x ) there exists a point x such that z ( x ) = c ( t ) forsome t ≥ t . Then the segment connecting x to ˆ x is still contained in B R (ˆ x ), and its image through z must be a connected part of the image of c . But as c is a simple curve, this implies that thereexists x ∈ B R (ˆ x ) such that z ( x ) = c ( t ∗ ), which is in contradiction with (4.24). Therefore, for every t ∗ ∈ I T , ˆ x ∈ Ω M ( t ∗ ) there exists an open ball centred at ˆ x contained in Ω M ( t ∗ ), and therefore Ω M ( t ∗ )is open. The same argument can be used to show that also Ω A ( t ) is open for each t . Clearly, Γ( t ) issequentially closed in Ω and Ω M ( t ) ∪ Γ( t ) , Ω A ( t ) ∪ Γ( t ) are closed in Ω as well. In this way we havealso shown that Z ( x, t ) defined as Z ( x, t ) = ( z ( x ) , in Ω M ( t ) c ( t ) , in Ω A ( t )16s in W , ∞ (Ω; R ) for every t ∈ I T .Now, since Γ( t ) is 2-rectifiable for almost every t , in order to show that L (cid:0) Ω \ (Ω A ( t ) ∪ Ω M ( t )) (cid:1) = 0for every t ∈ I T it is sufficient to prove that C := (cid:8) x ∈ Ω : x ∈ Γ( t ) , t ∈ I T , Γ( t ) is not 2-rectifiable (cid:9) has null L measure. By Theorem 2.4, C is the preimage through z , which is continuous, of a set ofmeasure zero, and is hence measurable. Now, we notice that by choosing g to be the indicator functionon C in the coarea formula (2.15), and identifying Z with the support of c , we have0 ≤ Z Ω g ( x ) | a | d x = Z Z Z z − ( ξ ) g ( s ) d H ( s ) d H ( ξ ) = 0 (4.25)as, by Theorem 2.4, this can just happen for a set of measure zero in Z . This, together with the factthat | a | > L ( C ) = 0.The rest of the proof is devoted to prove that Definition 4.1 (ii) is satisfied. To this aim, we firstclaim that for every point x ∈ D , with D := (cid:8) x ∈ Ω : ∇ z ( x ) exists, and ∇ z ( x ) = (cid:9) , there exist a Lipschitz 2-graph G x which is differentiable at x , an open neighbourhood U x and a t ∗ ∈ I T satisfying G x ∩ U x ⊂ Γ( t ∗ ) ∩ U x . In order to do that, we would need a generalised version ofthe constant rank theorem. However we were not able to find a version of it in the literature suitableto our application. We hence strongly exploit the structure of the image of z and a weak versionof the implicit function theorem. Here and below, given a vector v ∈ R , we denote by v i its i − thcomponent. Let us consider a generic ˆ x ∈ D and suppose, without loss of generality, that a (ˆ x ) = 0and that n (ˆ x ) = e . In this case, a version of the implicit function theorem as the one in [24, Thm.E] gives the existence of a connected neighbourhood N of (ˆ x , ˆ x ), and of a function ψ : N → R ,such that ψ (ˆ x , ˆ x ) = ˆ x , and z ( x , x , ψ ( x , x )) = z (ˆ x ) for every ( x , x ) ∈ N . Furthermore ψ isdifferentiable in (ˆ x , ˆ x ) and hence continuous and Lipschitz in N , and ∇ ψ (ˆ x , ˆ x ) = .Fixed ε = | a (ˆ x ) | , the fact that z is differentiable in ˆ x implies the existence of δ > z (ˆ x + ρ e ) − z (ˆ x ) = ρa + rρ, ∀| ρ | < δ, and where | r | < ε . Therefore, z (ˆ x + ρ e ) > z (ˆ x ) , if a (ˆ x ) δ > a (ˆ x ) ρ > ,z (ˆ x + ρ e ) < z (ˆ x ) , if − a (ˆ x ) δ < a (ˆ x ) ρ < , (4.26)for all | ρ | < δ . This implies the existence of h > c ( t ∗ + h ) , c ( t ∗ − h ) in z (Ω) such that c ( t ∗ + h ) > c ( t ∗ ) > c ( t ∗ − h ) . t ∗ ∈ I T is such that z (ˆ x ) = c ( t ∗ ). Furthermore, since z is Lipschitz, the dependence of c ( t ( x )) := z ( x ) is continuous. This together with (4.26) and the fact that c is simple, imply thatthe unique path connecting c ( t ∗ ± h ) to c ( t ∗ ) must be such that c ( t ∗ + s ) > c ( t ∗ ) > c ( t ∗ − s )either for every s ∈ (0 , h ) or for every s ∈ ( − h, x , x ) ∈ N such that z ( x , x , ψ ( x , x )) = c ( t ∗ ). By continuity of c ( t ( x )) there exist (˜ x , ˜ x ) ∈ N such that z (˜ x , ˜ x , ψ (˜ x , ˜ x )) = c ( t ∗ + s ) for some s with 0 < | s | < h . Thus, at the same time we shouldhave c ( t ∗ + s ) = c ( t ∗ ) because we are on a level set for z , and c ( t ∗ + s ) = c ( t ∗ ), which leads toa contradiction. We hence showed that c is constant implies also that c , c are constants, that is z ( x , x , ψ ( x , x )) = z (ˆ x ) = c ( t ∗ ) for every ( x , x ) ∈ N . This concludes the proof of the claim.It remains to prove that for all x ∈ D , it holds Γ( t ∗ ) ∩ U x ⊂ Ω L ( t ∗ ) ∩ Ω R ( t ∗ ) ∩ U x , where again t ∗ ∈ I T is such that x ∈ Γ( t ∗ ) . Suppose first that there exists a neighbourhood U of x ∈ Γ( t ∗ ) forsome t ∗ ∈ I T such that Ω A ( t ∗ ) ∩ U = ∅ or Ω M ( t ∗ ) ∩ U = ∅ . (4.27)This is z ( x ) = c ( t ∗ ), and z ( x ) = c ( s ( x )) with s ( x ) > t ∗ or s ( x ) < t ∗ for every x ∈ U . We wantto prove that either z is not differentiable in x , or ∇ z ( x ) = . Suppose not, then there exists β j ∈ R \ { } and a unit vector v j such that ∇ z j ( x ) · v j = β j for some j = 1 , ,
3. Observe also thatthe differentiability of z implies the existence of δ j ∈ (0 ,
1) such that z j ( x + α v j ) − z j ( x ) − αβ j = αr j ( α v j ) , ∀ α : | α | < δ j , (4.28)for some continuous functions r j bounded in modulus by β j . This implies that z j ( x + α v j ) − z j ( x )has the same sign as αβ j . Therefore, as c is simple, there exists an interval ( t δ j , t ∗ ) (or ( t ∗ , t δ j ))where c j ( t ) − c j ( t ∗ ) is both strictly positive and strictly negative for every t ∈ ( t δ j , t ∗ ) (or in ( t ∗ , t δ j )),thus leading to a contradiction. Therefore, if z is differentiable at x ∈ Γ( t ∗ ) and ∇ z ( x ) = , then x ∈ Ω A ( t ∗ ) ∩ Ω M ( t ∗ ).Now, by the coarea formula (2.15) with g chosen to be the characteristic function of D , we noticethat 0 = Z Z Z f − ( ξ ) ∩ Ω g ( s ) d H ( s ) d H ( ξ ) , and we deduce that z is differentiable with ∇ z = for H -almost every x ∈ Γ( t ) for almost every t ∈ I T . Let us call J T the subset of I T such that x ∈ D H -almost everywhere in Γ( t ) for every t ∈ J T . By arguing as above for the set C , the coarea formula implies that the set of x ∈ Ω suchthat z ( x ) ∈ c ( I T \ J T ) has measure zero. We can hence focus without loss of generality on Γ( t ∗ )for some t ∗ ∈ J T . Suppose now that there does not exist a neighbourhood of x s ∈ Γ( t ∗ ) such thatΓ( t ∗ ) ∩ U ⊂ Ω A ( t ∗ ) ∩ Ω M ( t ∗ ). In this case, as Γ( t ∗ ) is closed in Ω, there exists a neighbourhood U s of x s satisfying (4.27). However, as ∇ z exists and is non null H almost everywhere on Γ( t ∗ ), there exists x a ∈ Γ( t ∗ ) ∩ U s , x a ∈ D , and which must hence be in Ω A ( t ∗ ) ∩ Ω M ( t ∗ ), thus leading to a contradiction.We have therefore proved that the constructed family of moving interfaces Γ( t ) satisfies the conditionin Definition 4.1 (ii), which concludes the proof.The following corollaries are straightforward consequences of the above theorem:18 orollary 4.1. Let y ∈ W , ∞ (Ω; R ) satisfy a regular moving mask approximation. Then, there exist a ∈ L ∞ (Ω; R ) , n ∈ L ∞ (Ω; S ) such that ∇ y = Q + a ( x ) ⊗ n ( x ) , a.e. x ∈ Ω . (4.29) Conversely, if y ∈ W , ∞ (Ω; R ) satisfies (4.29) for some Q ∈ SO (3) , a ∈ L ∞ (Ω; R ) , n ∈ L ∞ (Ω; S ) and z (Ω) , with z ( x ) = y ( x ) − Q x , is contained in an absolutely continuous simple curve of finitelength, then y satisfies a regular moving mask approximation. Corollary 4.2.
Let
T > , Γ( t ) be a family of moving interfaces and Ω A ( t ) , Ω M ( t ) be connected forevery t ∈ (0 , T ) . Then, Z ∈ W , ∞ (Ω , R ) for each t ∈ (0 , T ) satisfying (4.21) is equal to Z ( x, t ) = ( z ( x ) + c ( t ) , in Ω M ( t ) , c ( t ) , in Ω A ( t ) , for some z ∈ W , ∞ (Ω , R ) such that ∇ z = a ⊗ n almost everywhere.Proof. The statement follows directly from Theorem 4.1, the fact that Z ( x , t ) is continuous in x foreach t and the hypothesis that Ω A ( t ) , Ω M ( t ) are connected.Corollary 4.2 hence implies that, under the above hypotheses, for each t ∈ I T , the interface Γ( t ) is asubset of { x ∈ Ω : z ( x ) = c ( t ) − c ( t ) } , that is of a level set of z . This also means that the image of z isa one-dimensional curve. However Γ( t ) does not need to coincide with the family of moving interfacesconstructed in the proof of Theorem 4.1, even if c − c is absolutely continuous, simple and of finitelength. Indeed, in the proof of Theorem 4.1 the phase interfaces must be constructed as subsets oflevel sets for z , but the construction of Ω A ( t ) , Ω M ( t ) is arbitrary and could be done differently. Forexample one could swap the definition of Ω A ( t ) and Ω M ( t ) in (4.23), or replace s ∈ [0 , t ) and s ∈ [0 , t ]in the definition of Ω M and Ω A respectively with | s − t | < t and s ∈ I T \ [ t − t, t + t ] for some t ∈ I T , thus getting a different family of moving interfaces.The next corollary gives some information about the interface velocity. We define the normalvelocity of Γ( t ∗ ) at time t ∗ ∈ I T and at x ∈ Γ( t ∗ ), namely ( v · n )( x , t ∗ ), as ˙ γ ( t ∗ ) · n ( x , t ∗ ), where n ( x , t ∗ ) is the unit normal to Γ( t ∗ ) at x ∈ Γ( t ∗ ), and γ ( t ) is a generic absolutely continuous pathdifferentiable at t ∗ such that γ ( t ) ∈ Γ( t ) for each t ∈ I T and γ ( t ∗ ) = x . Clearly ( v · n )( x , t ∗ ) is welldefined if its value is independent of the choice of γ among the admissible paths, and if n ( x , t ∗ ) is welldefined. Corollary 4.3.
Let z ∈ W , ∞ (Ω , R ) satisfy (4.22) , | a | > a.e. in Ω , and z (Ω) be contained in theimage of an absolutely continuous simple curve c : I T → R of finite length. Assume further that Γ( t ) is the family of moving interfaces constructed in the proof of Theorem . Then, the normal velocityof Γ( t ) at a point x ∈ Γ( t ) , denoted ( v · n )( x , t ) , satisfies a ( x )( v · n )( x , t ) = ˙ c ( t ) , a.e. t ∈ (0 , T ) , H -a.e. x ∈ Γ( t ) . (4.30) Proof.
By the coarea formula (2.15) with g chosen to be the characteristic function of the set where z is not differentiable and | a | >
0, we notice that0 = Z T Z z − ( c ( t )) ∩ Ω g ( s ) d H ( s ) | ˙ c ( t ) | d t.
19s the argument in the integral is non negative, we deduce that z is differentiable and | a | > H -almost every x ∈ Γ( t ) for almost every t ∈ I T . As showed in the proof of Theorem 4.1 n is welldefined for all these x . Let us consider γ ( t ), an absolutely continuous path in Ω such that γ ( t ) ∈ Γ( t )for every t ∈ I T . We have z ( γ ( t )) = c ( t ) , ∀ t ∈ ( t , t ) , for some 0 ≤ t < t ≤ T . Taking the time derivative of this identity we get˙ c ( t ) = ∇ z ( γ ( t )) ˙ γ ( t ) = a ( γ ( t )) (cid:0) ˙ γ ( t ) · n ( γ ( t ) , t ) (cid:1) = a ( x ) (cid:0) ˙ γ ( t ) · n ( x , t ) (cid:1) , which is the claimed result, as (cid:0) ˙ γ ( t ) · n ( x , t ) (cid:1) is independent of γ chosen for a.e. t ∈ I T , a.e. x ∈ Γ( t ) . Remark 4.6.
An important consequence of the above corollary is that a ( x ) | a ( x ) | is constant H almosteverywhere on Γ( t ) for almost every t . At the same time, there might be jumps in | a ( x ) | along a singleinterface and jumps for a ( x ) | a ( x ) | across interfaces. Remark 4.7.
If we assume the determinant of ∇ y = + ∇ z = + a ⊗ n to be a positive constant D almost everywhere in Ω, than we can deduce that on almost all interfaces | a ( x ) | can jump if and onlyif there is a jump in n ( x ). Indeed, this is a direct consequence of the following two facts: the first isthat the direction of a ( x ) is fixed on almost all interfaces, the second is that, det( ∇ y ) = D a.e. in Ωimplies a · n = D − a · n = D − H -almost everywhere on Γ( t ) for almost all t .A different perspective on the velocity of Γ( t ) is given by Corollary 4.4.
Let z ∈ W , ∞ (Ω , R ) satisfy (4.22) , | a | > a.e. in Ω , and z (Ω) be contained in theimage of an absolutely continuous simple curve c : I T → R of finite length. Assume further that Γ( t ) is the family of moving interfaces constructed in the proof of Theorem . Then, h ˙ χ Ω M , ξ i = | ˙ c ( t ) | Z Γ( t ) ξ ( s ) | a ( s ) | d H ( s ) , ∀ ξ ∈ C (Ω) , a.e. t ∈ (0 , T ) . Proof.
We first notice that, by the coarea formula, Z Ω χ Ω M ( x , t ) ξ ( x ) d x = Z T Z z − ( c ( τ )) ∩ Ω A ( t ) ξ ( s ) | a ( s ) | d H ( s ) | ˙ c ( τ ) | d τ = Z t Z z − ( c ( τ )) ∩ Ω ξ ( s ) | a ( s ) | d H ( s ) | ˙ c ( τ ) | d τ. Therefore, ddt Z Ω χ Ω M ( x , t ) ξ ( x ) d x = | ˙ c ( t ) | Z z − ( c ( t )) ∩ Ω ξ ( s ) | a ( s ) | d H ( s ) . which is the claimed result. 20 Basic properties of microstructures
According to the results of the previous sections, we can restrict our attention to deformationgradients satisfying for every t ∈ (0 , T ) ( ∇ y ( x , t ) = + a ( x ) ⊗ n ( x ) , a.e. x ∈ Ω M ( t ) , ∇ y ( x , t ) = , a.e. x ∈ Ω A ( t ) , for some a ( x ) ∈ L ∞ (Ω , R ) , n ( x ) ∈ L ∞ (Ω , S ) such that n ( x ) is normal to the austenite-martensiteinterface in x at a certain time t ∗ ∈ (0 , t ). Here we assumed without loss of generality to have Q = in Definition 3.1. In this way the martensitic macroscopic deformation gradient is a function of themoving, possibly curved, austenite-martensite interface during phase transition. In light of the aboveconsiderations, we assume that martensitic microstructures arising from austenite to martensite phasetransitions are described by deformation gradients ∇ y of the form ∇ y ( x ) = + a ( x ) ⊗ n ( x ) , ∇ y ∈ K qc , a.e. x ∈ Ω . (H1)As the determinant is constant in K , it follows thatdet ∇ y ( x ) = D a.e. x ∈ Ω , (H2)for some constant D >
0. (H1) and (H2) imply a ( x ) · n ( x ) = D − , (5.31)for a.e. x ∈ Ω, and ∇ × ( a i ( x ) n ( x )) = for each i = 1 , , , (5.32)in a weak sense.In conclusion, in what follows we look at martensite microstructures as a part of the domain where(H1) and (H2) hold. We first begin with an estimate for the norm of a ( x ): Proposition 5.1.
Let λ max and λ min be respectively the biggest and the smallest eigenvalues of themartensite deformation matrices U i ∈ K . Let also y ∈ W , ∞ (Ω; R ) satisfy (H1) , (H2) . Then, wehave | D − | ≤ | a | ≤ λ max − λ min , a.e. in Ω . Proof.
The first inequality follows trivially from Cauchy-Schwarz and the fact that a · n = D −
1. Inorder to get the other one, we observe that by the polar decomposition theorem we have that ∇ y ( x ) = R ( x ) F ( x ) , for almost every x ∈ Ω, where R ( x ) ∈ SO (3) and F ( x ) is symmetric positive definite. The argumentbelow holds for almost every x ∈ Ω. By arguing as in [5] one can deduce that, in order to have arank-one connection with the identity matrix, the eigenvalues of F , namely σ min ≤ σ mid ≤ σ max , mustsatisfy σ mid = 1 , σ min σ max = D . F = D = σ min σ max , tr( F T F ) = 1 + σ min + σ max . (5.33)On the other hand, ∇ y T ∇ y = F T F = + a ⊗ n + n ⊗ a + | a | n ⊗ n , which yields tr( F T F ) = 3 + 2 a · n + | a | = 1 + 2 D + | a | . (5.34)Therefore, by putting together (5.34) and (5.33) we deduce0 ≤ | a | = ( σ max − σ min ) ≤ ( λ max − λ min ) . Here we also made use of the following relation between eigenvalues proved in [5]: λ min ≤ σ min ≤ ≤ σ max ≤ λ max . Another interesting property regards the divergence of n ⊗ a : Proposition 5.2.
Let z ∈ W , ∞ (Ω; R ) be such that ∇ z ( x ) = a ( x ) ⊗ n ( x ) , a · n = D − ∈ R , a.e. in Ω . Then, ∇ · ( n ⊗ a ) = in the sense of distributions. Furthermore, if n ∈ W , (Ω; R ) with | n | = 1 a.e.in Ω , then ∇ · a = 0 in the sense of distributions.Proof. On the one hand, we have ∇ ( ∇ · z ) = ∇ ( D −
1) = . On the other hand ∇ ( ∇ · z ) = ∇ · ( ∇ z ) T = ∇ · ( n ⊗ a ) . Here, both identities should be understood in the sense of distributions. By putting them together wehence get the first statement. As a consequence, if n ∈ W , (Ω; R ) such that | n | = 1 a.e. in Ω, wecan write, Z Ω a · ∇ ϕ d x = Z Ω n T n ⊗ a · ∇ ϕ d x = Z Ω n ⊗ a : ∇ ( n ϕ ) d x − Z Ω ϕ n ⊗ a : ∇ n d x = − Z Ω ϕ a · ∇| n | d x = 0 , for every ϕ ∈ C ∞ c (Ω). 22 a = e n = e a = − e n = − e Figure 2: Picture of two parallel interfaces moving in opposite directions and meeting at a certaininterface where a , n are discontinuous. In this case, ∇ · ( n ⊗ a ) = , but ∇ · a = 0.In general, it is not true that ∇ · a = 0. Indeed, let us fix e ∈ R with | e | = 1, and consider z tobe of the form z ( x ) = e ( x · e ). Let us also fix c ∈ R such that x · e = c for some x ∈ Ω and define a = n = ( e , x · e < c, − e , x · e > c ;In this case, clearly a · n = 1 a.e. in Ω, and ∇ · ( n ⊗ a ) = 0 in the distributional sense. However, a ∈ BV (Ω; R ) satisfies ∇ a = − e ⊗ e H { x : x · e = c } , and, as | e | = 1 by hypothesis, ∇ · a = 0 in the distributional sense.Keeping in mind the counterexample above, we extend the validity of the identity ∇ · a = 0 in thefollowing Corollary 5.1. In this result we use the space of special functions with bounded variation onΩ, namely SBV (Ω). If ϕ ∈ SBV (Ω), its gradient is the sum of two Radon measures, one absolutelycontinuous with respect to the Lebesgue measure L , and the other concentrated on a 2-rectifiableset S ϕ , usually called the jump set. We denote by ϕ − , ϕ + the trace of ϕ on the two sides of the jumpset S ϕ respectively. We refer the interested reader to [2] and [20] for more details on this space. Corollary 5.1.
Let z ∈ W , ∞ (Ω; R ) be such that ∇ z ( x ) = a ( x ) ⊗ n ( x ) , a · n = D − ∈ R , a.e. in Ω . (5.35) Let a , n ∈ SBV (Ω; R ) ∩ L ∞ (Ω; R ) , | n | = 1 a.e. in Ω and let n − ( x ) = − n + ( x ) , if D = 1 , (5.36) a − ( x ) = − a + ( x ) ⇒ n − ( x ) = − n + ( x ) , if D = 1 , (5.37) for H − a.e. x ∈ S n ∩ S a . Then, ∇ · a = 0 in the sense of distributions. The following lemma is needed for the proof of Corollary 5.1:
Lemma 5.1.
Let ϕ ∈ H (Ω , R ) , with ∇ ϕ ∈ BV (Ω; R × ) . Then, there exists b ( x ) ∈ L ( S ∇ ϕ ; R ) such that ( ∇ ϕ ) + ( x ) − ( ∇ ϕ ) − ( x ) = b ( x ) ⊗ m ( x ) , H -a.e. x ∈ S ∇ ϕ , with m ( x ) being the normal to S ∇ ϕ in x . roof. The proof of this type of result is usually done via a blow up argument and exploits continuity.This may be possible here, but we use a slightly different proof. Let ϕ ∈ H (Ω; R ) and let A be aLipschitz open subset of Ω. Since ∇ × ∇ ϕ = 0 in the sense of distribution, we have that the followingintegration by parts formula holds (see [23, Ch. 2, (2.18)]) Z A ∇ ϕ i · ∇ × ψ d x = Z ∂A ( ∇ ϕ i × m ) · ψ d H , ∀ ψ ∈ H (Ω; R ) , (5.38)with m being the outpointing normal to A . We remark that, as stated in [23, Ch. 2, Thm 2.5],( ∇ ϕ i × m ) is a well defined object in H − ( ∂A ), and the integral on the right hand side should beinterpreted as h∇ ϕ i × m , ψ i H − ,H . Let now S be a Lipschitz 2-graph contained in Ω with normal m and let U i be a countable set of open neighbourhoods such that U i ⊂ Ω, U i \ S has exactly twoconnected components, namely U + i and U − i , and H ( S \ [ U i ) = 0 . We now define ( ∇ ϕ i × m ) ± to be the objects of H − ( S ) satisfying (5.38) respectively for A = U ± i .From (5.38) we deduce0 = Z U i ∇ ϕ i · ∇ × ψ d x = Z U + i ∇ ϕ i · ∇ × ψ d x + Z U − i ∇ ϕ i · ∇ × ψ d x = Z S ∩ U i (cid:0) ( ∇ ϕ i × m ) + − ( ∇ ϕ i × m ) − (cid:1) · ψ d H , for every ψ ∈ C ∞ c ( U i ; R ). By repeating this argument on all U i , this implies k ( ∇ ϕ i × m ) + − ( ∇ ϕ i × m ) − k H − ( S ) = 0 , i = 1 , , , (5.39)which is a weak Hadamard jump condition for H (Ω; R ) on Lipschitz 2-graphs. Furthermore, if ∇ ϕ ∈ SBV (Ω), we know that the set where it is discontinuous, namely S ∇ ϕ is 2-rectifiable (see e.g.,[2]). Therefore, by Proposition (2.4) we can cover S ∇ ϕ with countably many Lipschitz graphs where(5.39) holds. On the other hand, from [2, Thm 3.77] we know that the trace of ∇ ϕ i is well defined foralmost every point of S ∇ ϕ . Collecting these two facts we thus deduce the desired result. Remark 5.1.
Equation (5.39) is a very weak version of the Hadamard jump condition on Lipschitzsurfaces Γ with normal m for functions y ∈ H (Ω; R ). Indeed, we can only make sense to thetangential trace ∇ y × m of y on Γ as an object of H − (Γ), and (5.39) states that, across Γ, ∇ y × m must not jump as an object of H − (Γ), which is kind of an average sense. Proof.
It follows from the definition of jump points of a BV function (see e.g., [2, 20]) that (5.35)and | n | = 1 hold H –almost everywhere on S a ∪ S n . Therefore, under our hypotheses (5.36)–(5.37) a ⊗ n ∈ SBV (Ω) ∩ L ∞ (Ω) and S a ⊗ n = S a ∪ S n up to an H –negligible set. Here, a , n are chosento be the precise representatives for a , n . Furthermore, Lemma 5.1 implies that a Hadamard jumpcondition must hold across S a ⊗ n , so that a + ⊗ n + − a − ⊗ n − = b ⊗ m , H -a.e. on S a ⊗ n , (5.40)for some b ∈ L ∞ ( S a ⊗ n ; R ) and with m being the normal to S a ⊗ n . In case D = 1, this, together with(5.36) and | n | = 1, imply that the only possible scenarios are the following on S a ∪ S n :24a) n + = n − = m , in which case b = a + − a − ;(b) a + = ξ a − , in which case m k ξ n + − n − and b k a + k a − ,for some ξ ∈ L ∞ ( S n ). Taking the trace of (5.40) implies also b · m = 0. As a consequence, by (a)–(b)( a + − a − ) · m = 0 H -a.e. on S a ∪ S n , that is, the divergence of a has no singular part. In case D = 1,(5.37) allows also to have n + = − n − = m in S a , when a + = − a − . In which case ( a + − a − ) · m = 0follows just by the fact that a · n = 0.It just remains to check that the part of ∇ · a which is absolutely continuous with respect to L is null as well. To this aim, we first observe that, by the chain rule for BV functions (see e.g., [2,Example 3.97]) = ∇ · ( n ⊗ a ) = (cid:0) n ( ∇ a · a ) + ∇ a na (cid:1) L + (cid:0) n + ⊗ a + − n − ⊗ a − (cid:1) m H S a ⊗ n , (5.41)where we denoted by ∇ a the absolutely continous part of the gradient. Given (a)–(b) above, underour hypotheses we have (cid:0) n + ⊗ a + − n − ⊗ a − (cid:1) m = H − a.e. on S a ⊗ n . Furthermore, as | n | = 1 a.e.,we have = ∇| n | = n T ∇ a n L . Therefore, multiplying (5.41) by n and exploiting the fact that | n | = 1 a.e., we thus get ∇ a · a = 0 , a.e. in Ω . Therefore, as a ∈ SBV (Ω), for every φ ∈ C c (Ω) we have − Z Ω ∇ ϕ · a d x = Z S a ϕ ( a + − a − ) · m d H + Z Ω ϕ ∇ a · a d x = 0which concludes the proof.We now focus on compound twins. Thanks to Proposition 2.3 we know that if two martensite variants U , U ∈ R × Sym + form a compound twin, then there exists µ > v ∈ S such that U v = U v = µ v .In this case, [17, Theorem 2.5.1] states: Theorem 5.1.
Let U , . . . , U n ∈ R × Sym + , such that det U i = D > , and such that U i v = µ v for some µ > , v ∈ S , for each i = 1 , . . . , n . Let also H = n [ i =1 SO (3) U i . Then, there exists l ∈ N , w , . . . , w l such that H qc = ( F ∈ R × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det F = D , F T F v = µ v and | F w i | ≤ max j =1 ,...,n | U j w i | for each i = 1 , . . . , l . ) Therefore, in this simple case which includes compound twins, we can actually construct thequasiconvex hull of the set. An interesting result is given by the following lemma:25 emma 5.2.
Let U , . . . , U n ∈ R × Sym + , such that det U i = D > , and such that U i v = µ v for some µ > , v ∈ S , for each i = 1 , . . . , n . Let also H = n [ i =1 SO (3) U i , and µ = 1 . Then every map y ∈ W , ∞ (Ω; R ) satisfying ∇ y ( x ) ∈ H qc and ∇ y ( x ) = + a ( x ) ⊗ n ( x ) for a.e. x in Ω is such that a ⊗ n is constant, and | a | = | µ | − | D − µ | . Remark 5.2.
It comes up in the proof of Lemma 5.2 that the following condition1 ≥ µ (1 − µ )( D − µ ) > a ∈ R , n ∈ S such that + a ⊗ n ∈ H qc . We refer thereader to [5] for some stricter necessary conditions for this to hold.
Proof.
Thanks to a change of coordinates, we can suppose without loss of generality that v = e . Inthis case, every F ∈ H qc satisfies F T F = α γ γ β
00 0 µ (5.43)and αβ − γ = D µ − . (5.44)We are first interested in solving + a ⊗ n ∈ H qc for a ∈ R , n ∈ S . By (5.43), the first step is tosolve the following nonlinear system of equations for the components of a and n : a n + | a | n = α a n + | a | n = βa n + a n + | a | n n = γa n + a n + | a | n n = 0 a n + a n + | a | n n = 01 + 2 a n + | a | n = µ (5.45)subject to the constraint (5.44). As a first step we compute αβ − γ using system (5.45). Afterrearranging terms αβ − γ = 2( a n + a n ) + 1 + | a | ( n + n ) − ( a n − a n ) . Since | n | = 1, using last equation of (5.45) follows that | a | ( n + n ) = | a | (1 − n ) = | a | + 1 + 2 a n − µ , which leads to αβ − γ = 2( a · n ) + 2 + | a | − µ − ( a n − a n ) . αβ − γ = 2 D + | a | − µ − ( a n − a n ) . (5.46)Thus, it is immediately seen that (5.46) together with the first of (5.44) implies( a n − a n ) = 2 D + | a | − µ − D µ = | a | − µ ( D − µ ) . (5.47)On the other hand, exploiting the last equation of (5.45) in the fourth and fifth one we have a n − a n = − n n ( µ −
1) (5.48) a n − a n = n n ( µ − . (5.49)We note that it is legitimate to divide by n since the last identity in (5.45) together with µ = 1implies n = 0. By putting together (5.47)-(5.49) we thus get a × n = − n n ( µ − n n ( µ − ± q | a | − µ ( D − µ ) . (5.50)Since ( a × n ) · n = 0 we have n r | a | − µ ( D − µ ) = 0 , (5.51)which, as n = 0, leads to | a | = µ ( D − µ ) . In this case the norm of a is hence forced to be constant.We restrict ourselves without loss of generality to the case D = µ .We now want to write a in terms of the orthogonal vectors ( n , n ⊥ , n × n ⊥ ), where n ⊥ := n − n , n × n ⊥ = n n n n n − . (5.52)It is important to remark that, if n = n = 0 then it is easy to see from (5.45) that a = a = 0, sowe do not lose any generality with this representation. We have, a = a (1) n + a (2) n ⊥ + a (3) n × n ⊥ . As a first thing, recalling that det( + a ⊗ n ) = D , we deduce that a (1) = a · n = D −
1. On the otherhand, taking cross product of a with n follows that n × a = a (2) n × n ⊥ − a (3) n ⊥ . (5.53)A comparison of (5.53) with (5.50) and (5.51) thus leads to a (3) = n (1 − µ ), and a (2) = 0, whichimplies a = ( D − n + 1 n (1 − µ ) n × n ⊥ = ( D − µ ) n − n (1 − µ ) e , (5.54)27ith e = (0 , , T . As a first thing now we want to check whether it is possible to have | a | = µ ( D − µ ) . Since the orthogonal vectors n and n × n ⊥ satisfy | n | = 1 and | n × n ⊥ | = 1 − n , byrearranging the terms it is possible to obtain n = (1 − µ ) (1 − µ ) + µ ( µ − D ) − ( D − = (1 − µ ) µ ( D − µ ) . (5.55)It is easy to check that a pair of vectors ( n , a ) with a defined in terms of n as in (5.54), and where n is given by (5.55), satisfies the equations of (5.45). Thus, it turns out that (5.42) in necessary inorder not to contradict | n | = 1, 1 − n = n + n ≥ n > y ∈ W , ∞ (Ω; R ) such that ∇ y ( x ) satisfies (5.43)–(5.44) and ∇ y ( x ) = + a ( x ) ⊗ n ( x ) for a.e. x in Ω. We have to verify that conditions expressed in (5.32) hold, thatis, we have to check when the constructed deformations are actually gradients, by verifying that ∇ × ( a i n ) = 0 for i = 1 , ,
3, in the distributional sense. Since D and µ are constants, so is | n | = p ( µ − µ )( D − µ ) − . Hence, by (5.54), | a | is constant and non zero as long as µ = D and µ = 1,which is our case. Furthermore, n = sgn( n ) | n | and a = ±| a | sgn ( n ), where the sign dependson µ, D only and is hence fixed. We can hence suppose without loss of generality a = | a | sgn ( n ).Therefore, ∇ × ( a n ) = | a |∇ × (sgn( n ) n ) = 0 , implies ∇ × (sgn( n ) n ) = 0 , in the sense of distributions. This implies the existence of ψ ∈ W , (Ω) such that sgn( n ) n = ∇ ψ (seee.g., [23]). On the other hand, by Proposition 5.2 ∇ · ( a n ) = | n |∇ · (sgn( n ) a ) = 0 , which implies ∇ · (sgn( n ) a ) = 0 , in the sense of distributions. Furthermore, we have from (5.54) that ∇ · (sgn( n ) a ) = ( D − µ ) ∇ · (sgn( n ) n )which thus implies that ψ is harmonic in the sense of distributions. By using standard elliptic theory(see e.g., [19]) we can thus deduce that ψ ∈ C ∞ (Ω). On the other hand, as |∇ ψ | = 1, we have ∇ ψ T ∇ ψ = 0 , for all x ∈ Ω , which implies that one eigenvalue of the Hessian matrix is null. Another eigenvalue is 0 as e is aleft eigenvector related to 0 for ∇ ψ , and is not parallel to n unless n = e , in which case (5.55)forces sgn( n ) n = e everywhere. The fact that ψ is harmonic, i.e., tr ∇ ψ = 0, thus means that ∇ ψ is constant. Therefore, sgn( n ) n is constant, and as a consequence of (5.54), so is sgn( n ) a and a ⊗ n . 28 emark 5.3. In case µ = 1, it is not possible to deduce the same rigidity as in Lemma 5.2. Indeed,it can be deduced from equations (5.45) that µ = 1 implies either n = a = 0, or a = ( D − n ,but in the latter case the last equation in (5.45) implies a ⊗ n = . The problem becomes thustwo-dimensional, and we can rewrite n = (cid:0) n , n , (cid:1) T , n ⊥ = (cid:0) n , − n , (cid:1) T , a ( s ) = ( D − n + s n ⊥ , for some s ∈ R , n , n ∈ [ − ,
1] with n + n = 1. Now, take two martensite variants, for example, U = 12 λ m + λ M λ m − λ M λ m − λ M λ m + λ M
00 0 2 , U = 12 λ m + λ M λ M − λ m λ M − λ m λ m + λ M
00 0 2 , generating a compound twin, and such that their biggest and smallest eigenvalue, namely λ M and λ m , satisfy λ m < < λ M . Assume further, for simplicity, λ m + λ M > ≤ λ m λ M <
1. We cantake for example λ m = 0 . λ M = 1 .
1. It can be computed that ( SO (3) U ∪ SO (3) U ) qc coincideswith the set of matrices F satisfying (5.43)–(5.44), that is such that 0 < α, β ≤ ( λ m + λ M ) and αβ ≥ λ m λ M . Consider now F = + a ( s ) ⊗ n , then α = 1 + n ( D − s ) + sn n , β = 1 + n ( D − s ) − sn n . Choosing for simplicity n = 0 and s small enough, we get0 < α, β <
12 ( λ m + λ M ) , αβ ≥ λ m λ M = D . Thus, there exists ε > , and an open interval [ − ε, ε ] such that for every smooth function f : R → [ − ε, ε ]we have that + a ( f ( x · n ⊥ )) ⊗ n , is the gradient of a smooth map y , which is not constant, and which satisfies ∇ y ( x ) ∈ ( SO (3) U ∪ SO (3) U ) qc and (H1)–(H2) for each x . Therefore, a rigidity result as the one in Lemma 5.2 does nothold in general when µ = 1. Remark 5.4.
In [16] the author proved that for cubic to monoclinic II phase transitions (and hencealso for its special cases of cubic to orthorhombic and cubic to tetragonal phase transitions) necessaryand sufficient condition to satisfy (CC1)–(CC2) with a compound twin is to have µ = 1. Is thereforenot surprising that the case µ = 1 is a special case for Lemma 5.2. This is coherent also withProposition (5.3) below.By adding the further hypotheses that y | ∂ Ω is the restriction on ∂ Ω of a 1 − Proposition 5.3.
Let U , V ∈ R × Sym + and R I , R II ∈ SO (3) , b I , b II , m I , m II ∈ R \ { } satisfy R i V = U + b i ⊗ m i , i = I, II, where ( U , b I , m I ) , ( U , b II , m II ) do not fulfil (CC2). Assume y ∈ W , ∞ (Ω; R ) is such that y | ∂ Ω = y | ∂ Ω for some y ∈ C (Ω; R ) which is − in Ω , and ∇ y ( x ) ∈ (cid:0) SO (3) U ∪ SO (3) V (cid:1) qc , ∇ y ( x ) = + a ( x ) ⊗ n ( x ) , (5.56) a.e. in Ω , for some a ∈ L ∞ (Ω; R ) , n ∈ L ∞ (Ω; S ) . Then, ∇ y = constant . roof. Following the approach devised in [8], we introduce the orthonormal system of coordinates u i := U − m i | U − m i | , u i := b i | b i | , u i := u i × u i , with i = I, II to be chosen later, and let L i := U − (cid:0) − δ i u i ⊗ u i (cid:1) , δ i = 12 | U − m i || b i | . We set z i ( x ) := y ( L i x ) and the problem becomes equivalent to find a map z i ∈ W , ∞ (Ω; R ) suchthat ∇ z i ( x ) ∈ (cid:0) SO (3) S − i ∪ SO (3) S + i (cid:1) qc , a.e. x ∈ Ω L i := (cid:8) x : L i x ∈ Ω (cid:9) , (5.57)with S ± i = ± δ i u i ⊗ u i , and ∇ z i ( x ) = L i + a ( L i x ) ⊗ L Ti n ( L i x ) , a.e. in Ω L i , (5.58)where a ∈ L ∞ (Ω; R ), n ∈ L ∞ (Ω; S ) are as in (5.56). By [7], z i is a plane strain and satisfies z i ( x ) = Q (cid:0) z i ( s i , s i ) u i + s u i + z i ( s i , s i ) u i (cid:1) , (5.59)for some Q ∈ SO (3) , some Lipschitz functions z i , z i , and where s ij = x · u ij , j = 1 , , . As a consequence,from (5.58)–(5.59) we deduce u i = Q T ∇ z i ( x ) u i = Q T L i u i + Q T a ( L i x ) (cid:0) n ( L i x ) · L i u i (cid:1) , u i = ( ∇ z i ( x )) T Q u i = L Ti Q u i + L Ti n ( L i x ) (cid:0) a ( L i x ) · Q u i (cid:1) , a.e. in Ω L i . That is, a ( x ) (cid:0) n ( x ) · L i u i (cid:1) = ( Q − L i ) u i , (5.60) n ( x ) (cid:0) a ( x ) · Q u i (cid:1) = ( L − Ti − Q ) u i , (5.61)a.e. in Ω . Let us now consider the function f i ( µ ) = det (cid:0) ( U + µ b i ⊗ m i ) T ( U + µ b i ⊗ m i ) − (cid:1) , µ ∈ [0 , , i = I, II.
Thanks to [6, Prop. 5] we know that f i is a quadratic polynomial and f i ( µ ) = f i (1 − µ ). We firstnotice that f i ( µ ) = (cid:0) det U (cid:1) det (cid:0) ( U + µ b i ⊗ m i ) − ( U + µ b i ⊗ m i ) − T (cid:1) = (cid:0) det U (cid:1) det (cid:0) ( − U − ) + µ δ ( u i ⊗ u i + u i ⊗ U − u i ) (cid:1) = det (cid:0) ( U − ) + µ δ ( u i ⊗ U u i + u i ⊗ u i ) (cid:1) . A derivation of f i with respect to µ leads f ′ i ( µ )= 2 δ (cid:0) det U (cid:1) cof (cid:0) ( − U − ) + µ δ ( u i ⊗ u i + u i ⊗ U − u i ) (cid:1) : ( u i ⊗ u i + u i ⊗ U − u i )= 2 δ (cid:0) det U (cid:1)(cid:0) cof ( − U − ) u i · ( u i + U − u i ) + µ δ ( − U − ) u i · ( u i × U − u i ) (cid:1) = 2 δ (cid:0) cof ( U − ) u i · ( u i + U u i ) + µ δ ( U − ) u i · ( U u i × u i ) (cid:1) . cof (cid:0)P i v i ⊗ w i (cid:1) = P i It is clear from the proof of Proposition 5.3 that, if the type I solution ( U , b I , m I ) of the twinning equation (2.6) does not satisfy (CC2), but the type II solution ( U , b II , m II ) of (2.6) does, then we can guarantee that n in (5.56) is constant up to a change of sign. Similarly, if thetype I solution ( U , b I , m I ) of the twinning equation (2.6) does satisfy (CC2), but the type II solution ( U , b II , m II ) of (2.6) does not, then the direction of a in (5.56) is constant. That is, there exists v ∈ R such that a × v = 0 a.e. in Ω . We refer the reader to Proposition 6.1 and Proposition 6.2 forexamples of non-affine maps when (CC2) is not satisfied. In this section, we use the theory of the previous sections to prove some results about movinginterfaces in martensitic transformations. The results are different for different type of twins. We start31ith compound twins and we recall that, by Proposition 2.3, two martensite variants U , U ∈ R × Sym + form a compound twin if and only if there exist µ > , v ∈ S such that U v = U v = µ v . Thanksto Lemma 5.2 we can prove that in this case moving interfaces need to be planar and the relatedmacroscopic gradient constant. Theorem 6.1. Let U , U ∈ R × Sym + be a compound twin and such that U w = U w = µ w for some w ∈ R , µ = 1 . Then, every y satisfying the regular moving mask approximation and suchthat ∇ y ∈ ( SO (3) U ∪ SO (3) U ) qc , a.e. in Ω is constant, and the related moving interfaces planar.Proof. As y satisfies a regular moving mask approximation, by Theorem 4.1 we know that ∇ y = + a ⊗ n for some a ∈ L ∞ (Ω; R ) , n ∈ L ∞ (Ω; S ). Since µ = 1, we can apply Lemma 5.2 and thusdeduce that ∇ y is constant in Ω . The function z ( x ) := y ( x ) − x is such that z ∈ W , ∞ (Ω; R ) andis constant in every connected component of Ω A ( t ), for each t ≥ 0. Thus, Γ( t ) must be a (or at leastthe union of disconnected subsets of a) level-set for z , and hence a plane (or union of disconnectedplanes) as ∇ z is constant and rank-one in Ω.By arguing in the same way, Theorem 6.1 can be generalised to a wider range of situations asstated in Theorem 6.2 below. This is relevant, for example, in the cubic to monoclinic transforma-tion occurring in Zn Au Cu , where there are 3 sets of four deformation gradients satisfying thehypotheses of Theorem 6.2. Theorem 6.2. Let U , . . . , U N ∈ R × Sym + be such that U i w = µ w and det U i = D for some w ∈ R , µ = 1 and every i = 1 , . . . , n . Then, every y satisfying the regular moving mask approximation andsuch that ∇ y ∈ ( ∪ Ni =1 SO (3) U i ) qc , a.e. in Ω is constant, and the related moving interfaces planar. An equivalent result can be proved, in the same way, for the general two well problem under theadditional assumption that y coincides on ∂ Ω with a 1 − Theorem 6.3. Let U , V ∈ R × Sym + and R I , R II ∈ SO (3) , b I , b II , m I , m II ∈ R \ { } satisfy R i V = U + b i ⊗ m i , i = I, II, where ( U , b I , m I ) , ( U , b II , m II ) do not fulfil (CC2). Then, every y satisfying the regular moving maskapproximation, such that y | ∂ Ω = y | ∂ Ω for some y ∈ C (Ω; R ) which is − in Ω , and such that ∇ y ( x ) ∈ (cid:0) SO (3) U ∪ SO (3) V (cid:1) qc , is constant, and the related moving interfaces planar. Proposition 6.1. Let U and U be two martensitic variants and ˆ R ∈ SO (3) , b I , m I ∈ R be a typeI solution to (2.6) and satisfying the cofactor conditions. Then, there exist R , R ∈ SO (3) , ξ ∈ R , a ∈ R , n , n ∈ S such that for every λ ∈ L ∞ (Ω; [0 , satisfying ∇ λ × ( ξ n − n ) = in the senseof distributions, there exists y ∈ W , ∞ (Ω; R ) with ∇ y = R [(1 − λ ) U + λ ˆ RU ] = + a ⊗ (cid:0) λξ n + (1 − λ ) n (cid:1) , a.e. in Ω .Furthermore, y satisfies a regular moving mask approximation, the related moving interfaces arecurved, and ∇ · (cid:0) | λξ n + (1 − λ ) n | a (cid:1) = 0 in the sense of distributions.Proof. From Theorem 2.2 and in particular from (2.12) we know the existence of R , R ∈ SO (3), ξ ∈ R , a ∈ R , n , n ∈ S such that R [(1 − λ ) U + λ ˆ RU ] = + a ⊗ (cid:0) λξ n + (1 − λ ) n (cid:1) , for all λ ∈ [0 , λ ∈ L ∞ (Ω) to be a function such that λ ∈ [0 , 1] a.e., and define a ( x ) := a | n + λ ( x )( ξ n − n ) | , n ( x ) := n + λ ( x )( ξ n − n ) | n + λ ( ξ n − n ) | , so that n has unitary norm. In the notation of Theorem 2.2, we havedet( R [(1 − λ ) U + λ ˆ RU ]) = det( R ) det( U + λ b I ⊗ m I ) = det U where the Sherman-Morrison inversion formula and the fact that U − n · a = 0 have been used. Thatis, a · ( n + λ ( ξ n − n ))is constant independently of λ , or, in an equivalent way, a · ( ξ n − n ) = 0 . (6.65)We just need to check if it is possible to have ∇ × ( a i n ) = 0 . Exploiting the definition of a and n get that this is satisfied if and only if ∇ λ × ( ξ n − n ) = , (6.66)in a weak sense. Therefore, if Ω is convex λ must satisfy λ ( x ) = f ( x · ( ξ n − n )), for some f ∈ L ∞ ( R ; [0 , y = x + a ( n · x + F ( x · ( ξ n − n )) + c , for some constant c ∈ R , and where F ( s ) = R s f ( s ) d s . Therefore, after choosing I T := (cid:0) inf x ∈ Ω G ( x ) , sup x ∈ Ω G ( x ) (cid:1) , G ( x ) = n · x + F ( x · ( ξ n − n )), we deduce that the image of z ( x ) = y ( x ) − x is c + t a , for t ∈ I T . If Ω is connected but not convex, then λ might not be of the form f ( x · ( ξ n − n )), but theimage of z ( x ) = y ( x ) − x is still contained in the one-dimensional line c + t a , for some c ∈ R and t in some bounded interval ˆ I T . We can thus use Corollary 4.1 and deduce the existence of a family ofmoving interfaces, which are also level sets for z ( x ).In order to prove that ∇ · a , we first mollify λ and, defined m as m := ξ n − n , notice that thanksto Fubini’s theorem for distributions we can write (cid:10) ∇ λ ε × m , ψ (cid:11) D ′ , D = (cid:10) ∇ λ, ( m × ψ ε ) (cid:11) D ′ , D = (cid:10) ∇ λ × m , ψ ε (cid:11) D ′ , D = 0 , thanks to (6.66), for all ψ ∈ C ∞ c (Ω , R ). Therefore, ∇ λ ε k m , and, by (6.65), ∇ λ ε · a = 0 . (6.67)On the other hand, exploiting the smooth dependence on λ of a and the fact that λ ε → λ in L p (Ω)for every p ∈ [1 , ∞ ), we have Z Ω a ( λ ) · ∇ ψ d x = lim ε → Z Ω a ( λ ε ) · ∇ ψ d x = − lim ε → Z Ω g ′ ( λ ε ) a · ∇ λ ε ψ d x for every ψ ∈ C ∞ c (Ω) and where g ( s ) = | n + s ( ξ n − n ) | . The last term in the chain of identitiesabove is null due to (6.67), and therefore the proof is concluded.Finally, a result related to type II twins satisfying the cofactor conditions: Proposition 6.2. Let U and U be two martensitic variants and ˆ R ∈ SO (3) , b II , m II ∈ R be atype II solution to (2.6) and satisfying the cofactor conditions. Then, there exist R ∈ SO (3) , ξ ∈ R , a , a ∈ R , n ∈ S such that for every λ ∈ L ∞ (Ω; [0 , satisfying ∇ λ × n = in the sense ofdistributions, there exists y ∈ W , ∞ (Ω; R ) with ∇ y = R [(1 − λ ) U + λ ˆ RU ] = + (cid:0) λξ a + (1 − λ ) a (cid:1) ⊗ n , a.e. in Ω .Furthermore, y satisfies a regular moving mask approximation, and ∇ · (cid:0) λξ a + (1 − λ ) a (cid:1) = 0 in the sense of distributions.Proof. From Theorem (2.3) and in particular from (2.14) we know the existence of R , R ∈ SO (3), ξ ∈ R , a , a ∈ R , n ∈ S satisfying R [(1 − λ ) U + λ ˆ RU ] = + (cid:0) λξ a + (1 − λ ) a (cid:1) ⊗ n , for all λ ∈ [0 , λ ∈ L ∞ (Ω) to be a function such that λ ∈ [0 , 1] a.e., and define a ( x ) := ( a + λ ( x )( ξ a − a ))It is trivial to check, by arguing as in the proof of Proposition 6.1, that also (H2) holds.Now, by taking the curl of ∇ y i we deduce that ∇ × ( λ n ) = ∇ λ × n = 0 , in the sense of distributions. Therefore taking λ such that ∇ λ k n in a weak sense, by Proposition 4.1and Remark 4.1 follows the existence of a family of moving planar interfaces. The fact that ∇ · a = 0in the sense of distributions trivially follows from Proposition 5.2.34 Experimental evidence The physical assumptions which were made in this work and some of the properties that have beendeduced here are currently being investigated from an experimental perspective. Indeed, the authorsof [13] have used X-ray Laue microdiffraction to measure the orientations and structural parametersof variants and phases in Zn Au Cu . With this modern technique, the scanned area is meshedwith small rectangles (e.g., in [13] authors use 2 µ m wide squares), and one can identify the phase andvariant in each cuboid which has as a basis a rectangle of the mesh and depth of approximate 2 µ mfrom the sample surface. In cubes where a single phase or variant has been recognised, one can alsomeasure the lattice parameters necessary to compute the average deformation gradient. In this wayone can investigate what is happening at the phase interface by studying the lattice parameters inmesh rectangles where a martensite variant has been recognised and which have at least one neigh-bouring rectangle where the Laue microdiffraction was able to identify austenite.In this way, the authors of [13] compute in some of the mesh cubes lying on the interface the num-ber k cof ( ∇ y − ) k , which, as it is easy to verify, is zero if and only if ∇ y − is rank-one. Experimentalresults give k cof ( ∇ y − ) k to be of the order of 10 − , which seems small enough to be considered zero,and hence to justify (H1).Further investigations are ongoing to verify that ∇ y ( x ) remains constant in time when x is noton the interface. This seems a reasonable assumption, as long as no external force acts on the sampleand as long as one neglects other internal stresses giving rise to elastic deformations which, anyway,seem to be small compared to the deformations induced by the phase transition.In conclusion, the data collected up till now seem to confirm the validity of the assumptions thatwe made in the present work. However, in the images in [13] there are many mesh cubes close tosome of the phase interfaces where the X-ray Laue microdiffraction is not able to recognize any singlevariant or phase, and hence where the validity of the assumptions to get (H1) could be questioned orshould be verified in some other way. References [1] G. Alberti, S. Bianchini, and G. Crippa. Structure of level sets and Sard-type properties ofLipschitz maps. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) , 12(4):863–902, 2013.[2] L. Ambrosio, N. Fusco, and D. Pallara. Functions of bounded variation and free discontinuityproblems . 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