Nucleation barriers at corners for cubic-to-tetragonal phase transformation
aa r X i v : . [ m a t h . A P ] N ov Nucleation barriers at corners for cubic-to-tetragonal phasetransformation
Peter Bella Michael Goldman ∗ July 24, 2018
Abstract
We are interested in the energetic cost of a martensitic inclusion of volume V in austenitefor the cubic-to-tetragonal phase transformation. In contrast with the work of [Kn¨upfer,Kohn, Otto: Comm. Pure Appl. Math. 66 (2013), no. 6, 867–904], we consider domain witha corner and obtain a better scaling law for the minimal energy ( E min ∼ min( V / , V / )).Our predictions are in a good agreement with physical experiments where nucleation ofmartensite is usually observed near the corners of the specimen. In this short note we study the energetic cost of a nucleation for the cubic-to-tetragonal phasetransformation in domains with a corner. We prove that in the geometrically linear setting, itis energetically more favorable to start the nucleation at the corner of the domain comparedto the nucleation inside the specimen. This is in good agreement with experiments and withrelated results for cubic-to-orthorhombic phase transitions [1].We work in the framework of geometrically linear elasticity so that the elastic energy ofour material depends only on the symmetric part e ( u ) := ( ∇ t u + ∇ u ) of the gradient of thedisplacement u : Ω → R from a reference configuration (see [6]). We choose the referencelattice to be the austenite phase so that the stress-free strains are given by e (0) = 0 in theaustenite and by e (1) , e (2) , and e (3) in the three variants of the martensite (see Section 2 for thedefinition of e ( i ) ). We introduce the characteristic functions of the martensitic variants χ i ∈ BV (Ω , { , } ) , and χ + χ + χ ≤ . (1)The volume of the martensitic inclusion is then given by V := Z Ω χ + χ + χ d x. (2)The normalized elastic energy is given by: E elast [ χ ] = inf u ∈ H (Ω , R ) Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e ( u ) − X i =1 χ i e ( i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d x. ∗ Max Planck Institute for Mathematics in the Sciences, Leipzig (Germany), email: [email protected], [email protected]
1e also include an interfacial energy which penalizes the creation of interfaces between austeniteand martensites or between two variants of martensites (see [2]): E interf [ χ ] := Z Ω ( |∇ χ | + |∇ χ | + |∇ χ | ) d x, where for χ ∈ BV (Ω , { , } ), R Ω |∇ χ | denotes the total variation of χ . The total energy is thengiven by E [ χ ] := E interf [ χ ] + E elast [ χ ] . We are interested in the energetic cost of the formation of a nucleus of martensite of volume V inside the austenite. This energetic cost determines the energy barrier which needs to beovercome to start the nucleation of martensite in the austenite. In [6], it was proven that whenΩ = R , inf χ satisfies (1) , (2) E [ χ ] ∼ (cid:26) V / if V ≤ V / if V ≥ , which gives the cost of such a nucleus inside the domain. We are instead interested in the casewhen Ω is a generic corner . Our main result reads as follows: Theorem 1.
If the domain Ω is a generic corner (see (3) for a precise definition), then theminimum of the energy scales like inf χ satisfies (1) , (2) E [ χ ] ∼ (cid:26) V / if V ≤ V / if V ≥ . This results show that it is indeed energetically favored to start nucleation at corners. Weshould however warn the reader that we do not say anything about the nucleation on faces oredges of the specimen. We believe that in these cases, the scaling should be V / as for thenucleation inside but since they do not seem accessible using the presently-available techniques ,we do not investigate this question here. We would also like to point out that our result doesnot hold for the corner ( R ) + since in this case there is no habit plane which is cutting thecorner. By adapting the construction of [6], we see that in this case, nucleation both at thecorner and at the edge can be achieved with an energy of order at most min( V / , V / ). Sincethis case is very degenerate and since again, a proof of the corresponding lower bound wouldprobably require completely new ideas, we do not push further this question here.The main difference between nucleation at a corner and nucleation in the bulk is that in thelatter, the notion of self-accommodation plays a central role. In fact, in order to create a nucleusin the bulk, one needs to be able to construct a three-dimensional macroscopically stress-freeconfiguration. As shown in [6, 7], this is possible only if the three variants of the martensite arepresent (in similar volume fractions). Indeed, if only two variants of martensite are used, thenthe energy must be larger – of order V / ) [7]. Moreover, to avoid large energetic costs themartensitic inclusion prefers to have boundary parallel to one of the habit planes. This is notcompletely possible for the inclusion in the whole space, and so one constructs a lens-shapedinclusion [6] to (at least approximately) have the boundary parallel to the habit plane. Theenergy scaling law V / is partly due to the macroscopic bending of the lens. On the contrary,for a generic corner it is possible to construct a macroscopically stress-free configuration with aboundary between martensitic inclusion and austenite exactly parallel to the habit plane. The The proof of the lower bound for the whole space in [6] is based on the Fourier transform, which for obviousreasons can not be used here. V / is then obtained by the classical branching pattern usingonly two variants of martensite (see [8, 3, 5]).Our proof of the lower bound strongly relies on a result by Capella and Otto [4] which givesa local lower bound on the energy of austenite-martensite interfaces. More precisely, we use thefact that if in a ball the volume fraction of austenite is bounded away from 0 and 1 (i.e. bothaustenite and martensites are present), then in a larger ball the energy cannot be small.Let us emphasis that we are dealing only with the cubic-to-tetragonal phase transformationwhere the possible microstructures are very rigid (see [3, 4] and references therein). Much lessis known in other situations such as cubic-to-orthorhombic or cubic-to-monoclinic phase trans-formations (see [10, 9] for some available results). Notation
In the paper we will use the following notation. The symbols ∼ , & , . indicateestimates that hold up to a constant, which could depend on the domain Ω, but not on χ or u .For instance, f . g denotes the existence of a constant C = C (Ω) > f ≤ Cg . For u, v in R , the tensor product u ⊗ v is the 3 × u ⊗ v ) ij := u i v j .The symmetrized tensor product is defined by u ⊙ v := ( u ⊗ v + v ⊗ u ). Finally, for a matrix A , we denote e ( A ) := ( A t + A ) its symmetric part and its norm by | A | := qP ij A ij . We adopt the notation from [6]. We assume that the atoms in the specimen are aligned withthe coordinate axes, i.e. e (0) := 0 , e (1) := − , e (2) := − , e (3) := − . Two strains
A, B will be called compatible if there exists a plane and a continuous function u with e ( u ) = A on one side of the plane and e ( u ) = B on the other side. Two such strainsare called twins, the corresponding plane is called the twinning plane and its normal is calledthe twin direction. It can be proven that two strains A and B are compatible if and only if A − B = a ⊙ b for some vectors a, b in R . In the case of cubic-to-tetragonal phase transformationsit can be computed that the martensitic variants are pairwise compatible in the sense that forany permutation ( ijk ), letting ε ijk be the signature of the permutation, there holds e ( i ) − e ( j ) = 6 ε ijk ( b ij ⊙ b ji ) , where the six twinning planes have normals b := 1 √ , b := 1 √ , b := 1 √ ,b := 1 √ − , b := 1 √ − , b := 1 √ − . Moreover, though no single martensitic variant is compatible with the austenite, there is com-patibility of austenite with some convex combinations of the martensite variants in the sensethat for i = j , 13 e ( i ) + 23 e ( j ) = 2 ε ijk ( b jk ⊙ b kj ) . { αa + βb + γc : α ≥ , β ≥ , γ ≥ } , (3)where a, b, c is some basis of R with | a | = | b | = | c | = 1. Moreover, we assume that (at least)one of the habit planes can cut off a corner of our specimen, i.e. that at least one of the normalsto the habit planes (we denote that normal by n ) satisfies either a · n > , b · n > , c · n > , (4)or a · n < , b · n < , c · n < . Without loss of generality we can assume that n = b and that condition (4) holds. Moreover,we can also assume that the vectors a , b , and c are in positive order, i.e. that ( a × b ) · c > b × c ) · a >
0, and ( c × a ) · b >
0. Finally, we define µ := min (cid:26) ( a × b ) | a × b | · c, ( b × c ) | b × c | · a, ( c × a ) | c × a | · b (cid:27) ∈ (0 , . In this section we prove the lower bound, that isinf χ satisfies (1) , (2) E [ χ ] & (cid:26) V / if V ≤ ,V / if V ≥ . (5)If V ≤
1, the lower bound immediately follows from the isoperimetric inequality. Hence weneed to focus only on the case V ≥
1. In the proof we will use the following lemma, which canbe obtained by simple scaling from [4, Theorem 1 - part i)]:
Lemma 1.
There exists a small but universal κ ∈ (0 , with the following property: for every R ≥ /κ , every u ∈ H ( B R (0) , R ) and every χ with the property | B κR | ≤ Z B κR χ + χ + χ d x ≤ | B κR | , (6) we have Z B R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e ( u ) − X i =1 χ i e ( i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + X i =1 |∇ χ i | d x & R / . (7)It is easy to see that for R < /κ , (7) (with possibly different universal constant) followsfrom (6) by the isoperimetric inequality.Let us introduce the following notation for the portion of the space occupied by the marten-sites M := { x ∈ Ω : χ ( x ) + χ ( x ) + χ ( x ) = 1 } , (8)so that | M | = V . Let us now consider χ which satisfies (1), (2). To prove (5) it is enough toconstruct a (at most countable) set of balls B ( x i , r i ) with the following properties:4. the balls B ( x i , r i ) are disjoint and are subset of Ω;2. (cid:12)(cid:12) M \ S i B ( x i , µ − r i ) (cid:12)(cid:12) = 0;3. | B ( x i , κr i ) | ≤ R B ( x i ,κr i ) χ + χ + χ d x ≤ | B ( x i , κr i ) | . Indeed, let us assume that we have such a covering. Then by Lemma 1 we have for every i Z B ( x i ,r i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e ( u ) − X i =1 χ i e ( i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + X i =1 |∇ χ i | d x & r / i . Since B ( x i , r i ) are disjoint and B ( x i , r i ) ⊂ Ω, summing the above relation in i implies Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e ( u ) − X i =1 χ i e ( i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + X i =1 |∇ χ i | d x & X i r / i . Finally, because of the second property, we see that X i r i & X i | B ( x i , µ − r i ) | ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)[ i B ( x i , µ − r i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ | M | = V, and by Jensen’s inequality applied to the concave function t / , we see that Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e ( u ) − X i =1 χ i e ( i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + X i =1 |∇ χ i | d x & X i ( r i ) / & X i r i ! / & V / . Since this holds for any u and χ which satisfies (1) and (2), (5) immediately follows.It remains to show the existence of the balls B ( x i , r i ). Let ¯ x be any point of density of M . Then there exists a radius r > B (¯ x, r ) ⊂ Ω and | B (¯ x, κr ) ∩ M | ≥ | B (¯ x, κr ) | .Consider the function f defined by f ( t ) := | B (¯ x + t ( a + b + c ) , κ ( r + µt )) ∩ M || B (¯ x + t ( a + b + c ) , κ ( r + µt )) | , which measures the volume fraction of balls obtained from B (¯ x, r ) by dilation in such a waythat all these balls belong to Ω.We claim that for some t ¯ x ≥ ≤ f ( t ¯ x ) ≤ . Indeed, we know that f (0) ≥ andthat f is a continuous function on [0 , ∞ ). Since the volume of M is finite, we have lim t →∞ f ( t ) =0, and the claim follows. In this way, to each ¯ x we assign a ball B (¯ x + t ¯ x ( a + b + c ) , r + µt ¯ x ).Before proceeding we point out that B (¯ x + t ¯ x ( a + b + c ) , r + µt ¯ x ) ⊂ Ω.Now let B denotes set of the constructed balls enlarged by a factor 3 µ − , i.e. we define B := (cid:8) B (¯ x + t ¯ x ( a + b + c ) , µ − ( r + µt ¯ x )) : ¯ x point of density of M (cid:9) . We note that ¯ x ∈ B (¯ x + t ¯ x ( a + b + c ) , µ − ( r + µt ¯ x )), in particular M is covered (up to a negligible set) by the balls in B . Now we apply Vitali’s covering lemma to B . We observe that for every ¯ x and corresponding t ¯ x we have | M | ≥ | B (¯ x + t ¯ x ( a + b + c ) , κ ( r + µt ¯ x )) ∩ M | ≥ | B (¯ x + t ¯ x ( a + b + c ) , κ ( r + µt ¯ x )) | , andso r + µt ¯ x . | M | / . Therefore the supremum of radii of balls in B is bounded, and so by Vitali’scovering lemma there exists a set of balls B i = B ( x i , r i ) = B ( x i + t x i ( a + b + c ) , r + µt x i ) ⊂ Ωsuch that B ( x i , µ − r i ) ∈ B , B ( x i , µ − r i ) cover M (up to a negligible set), and B ( x i , µ − r i )(hence also B ( x i , r i )) are disjoint. This completes the proof of the lower bound. That is a Lebesgue point of χ M . Notice that by definition, ¯ x has to belong to the interior of Ω. Upper bound
In this section we prove the upper bound, i.e.inf χ satisfies (1) , (2) E [ χ ] . (cid:26) V / if V ≤ ,V / if V ≥ . (9)If V ≤
1, it is enough considering χ = χ B R ( x ) with | B R ( x ) | = V and B R ( x ) ⊂ Ω, χ = χ = 0and u = 0 to get E ( χ ) ≤ CV / + Z B R ( x ) | e (1) | . V / + V . V / . For V ≥
1, we use a classical branching construction which dates back to Kohn and M¨uller [8](see also [3, 5]). Our construction will be essentially two-dimensional and will use branchingclose to the martensite-austenite interface. We will use a simplified version of the constructionused in [6] and will follow their notations.Let n := b be the normal to the habit plane cutting off the corner and let us denote b := b × n | b × n | , b := n × b | n × b | , b := b × b | b × b | , which is a basis of R and let y i := x · b i be the associated coordinates. For R >
1, let C R := {− R ≤ y ≤ R, ≤ y ≤ R, ≤ y ≤ R } . We are going to construct a pair ( u, χ ) suchthat χ + χ + χ = 1 in C − R := C R ∩ { y ≤ } , χ + χ + χ = 0 in C + R := C R ∩ { y > } , u = 0in C + R and Z C R ( |∇ χ | + |∇ χ | + |∇ χ | ) d x + Z C R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e ( u ) − X i =1 χ i e ( i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d x . R / . (10)Let us first see how this would give the proof of (9). By the non-degeneracy condition (4),we see that for every V > R ∼ V / and a translation z R ∈ R such that | Ω ∩ ( z R + C − R ) | = V and { x ∈ Ω : x · n < } ⊂ z R + C − R . Then by extending the previouslyconstructed pair ( u, χ ) to the whole Ω by zero we find that E ( χ ) ≤ Z C R ( |∇ χ | + |∇ χ | + |∇ χ | ) d x + Z C R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e ( u ) − X i =1 χ i e ( i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d x . R / . V / and the upper bound is proven. Let us now turn to the construction of the pair ( u, χ ) satisfying(10). In our construction we are going to use only variants 1 and 2 of the martensite, i.e. χ ≡ C − R , we will have fine-scale oscillations of the martensite variants inthe direction b and branching in the direction b . The whole construction will be invariant inthe direction b . As in [6] first we need to choose the gradients that will be involved and whichallow for twinning between martensites and compatibility with the austenite. For this we let D (1) := − − − D (2) := − − − D M := − − , so that e ( D (1) ) = e (1) , e ( D (2) ) = e (2) , D (1) − D (2) = 6( b ⊗ b ), and D M = 13 D (1) + 23 D (2) = 2( b ⊗ n ) . C − R we are going to let u = u m + D M , where u m will be the microscopic displacementaccounting for the twinning of the martensites and for the branching process, and where D M is the macroscopic displacement, which ensures compatibility with the austenite phase.Since the construction consists of self-similar cells, let us now define the basic cell Z . For1 < h ∼ w / (which would then imply w . h ), let Z := { ≤ y ≤ h, ≤ y ≤ w, ≤ y ≤ w } . We let χ := 1 on the sets (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) y w − (cid:12)(cid:12)(cid:12)(cid:12) ≤ y h (cid:27) ∩ Z, (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) y w − (cid:12)(cid:12)(cid:12)(cid:12) ≤ y h (cid:27) ∩ Z, (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) y w − (cid:12)(cid:12)(cid:12)(cid:12) ≤ − y h (cid:27) ∩ Z, and χ := 0 on the rest of Z . We then let χ = 1 − χ . Notice that on each slice { y = c } , thereholds Z { y = c }∩ Z χ = w Z { y = c }∩ Z χ = 2 w . We define the microscopic displacement u m by imposing that u m = 0 on { y ∈ { , w } or y ∈ { , w }} ∩ Z (11)and so that the derivatives of u m in the b and b directions are given by ∂ b u m := [( D (1) − D M ) χ + ( D (2) − D M ) χ ] b ,∂ b u m := [( D (1) − D M ) χ + ( D (2) − D M ) χ ] b = 0 . This together with (11) implies that ∂ b u m is constant on each connected component of thesupport of χ and χ , and has a jump of order wh at the interfaces. Let us now estimate theenergy of such a configuration. Since ( D (1) − D M ) b = ( D (2) − D M ) b = 0, the definition of ∂ b u and ∂ b u imply Z Z | e ( u m ) − X i =1 χ i ( e ( i ) − e ( D M ) | ≤ Z Z |∇ u m − X i =1 χ i ( D ( i ) − D M | = Z Z | ∂ b u | . w h . Since w . h , the interfacial energy can be estimated by wh and thus Z Z X i =1 |∇ χ i | + Z Z | e ( u m ) − X i =1 χ i ( e ( i ) − e ( D M ) | . hw + w h ∼ w / . (12)We now decompose C − R into cells. Let us first denote by C blR := C − R ∩ {− ≤ y ≤ } theboundary layer of thickness 1 and by C intR := C − R ∩ {− R ≤ y ≤ − } the interior domain. Wethen decompose the square { ≤ y ≤ R, ≤ y ≤ R } into cubes Q k , k = 1 , .., K of sidelength w ∼ R / (so that K ∼ R / ) and consider the corresponding cylinderΣ k := { q + αb : q ∈ Q k , − R ≤ α ≤ − } ⊂ C − R . Next, we decompose any such cylinder in refining cells which are going to be rescaled versionsof Z . The first cell is of width w and height h := C w / (with C to be fixed later) and thei-th generation of cells is defined by w i := w i − h i := C w / i
7o that above each cell of the generation i − i . Theconstruction stops after M iterations when h M ≤ w M . (13)The constant C is finally chosen so that P Mi =0 h i = R −
1. Notice that since h i is geometricand since w ∼ R / , we have C ∼
1. The functions χ , χ and u m are then defined on theconstructed cells by rescaling of those defined in Z . In the boundary layer, we set χ = 1 andextend continuously u m so that u = 0 on { y = 0 } and k∇ u m k L ∞ ( C blR ) . k u m k L ∞ ( ∂C blR ) + k∇ u m k L ∞ ( ∂C blR ) . We can now compute the energy of such a configuration. The energy can be split into two parts,one coming from the contribution in C intR and the other coming from the contribution in C blR .Let us start by estimating the energy coming from C intR . For this, consider first one cylinderΣ k . Recalling (12) and the definitions of w i and h i , we find Z Σ k X i =1 |∇ χ i | + Z Σ k | e ( u m ) − X i =1 χ i ( e ( i ) − e ( D M )) | . w / ∞ X i =0 i (cid:18) (cid:19) i/ . R / which, summing for k = 1 , .., K (and recalling that K ∼ R / ) gives Z C intR X i =1 |∇ χ i | + Z C intR | e ( u m ) − X i =1 χ i ( e ( i ) − e ( D M )) | . R / . We finally estimate the contribution of the energy coming from the boundary layer. Noticethat since its thickness is one, | C blR | ∼ R and H ( ∂C blR ) ∼ R . Moreover, since in the lastgeneration of cells w M ∼ k u m k L ∞ ( ∂C blR ) + k∇ u m k L ∞ ( ∂C blR ) . Z C intR X i =1 |∇ χ i | + Z C intR | e ( u m ) − X i =1 χ i ( e ( i ) − e ( D M )) | . R . R / , from which (10) follows. Acknowledgment
The authors warmly thank F.Otto for suggesting the problem and for very valuable discussions.They also thank H. Seiner for many comments on the experimental and physical background.M. Goldman was funded by the Von Humboldt foundation.
References [1] J.M. Ball, K. Koumatos, and H. Seiner. Nucleation of austenite in mechanically stabilizedmartensite by localized heating.
Proceedings ICOMAT11, Journal of Alloys and Com-pounds , in press.[2] K. Bhattacharya.
Microstructure of martensite . Oxford Series on Materials Modelling.Oxford University Press, Oxford, 2003. Why it forms and how it gives rise to the shape-memory effect. 83] A. Capella and F. Otto. A rigidity result for a perturbation of the geometrically linearthree-well problem.
Comm. Pure Appl. Math. , 62(12):1632–1669, 2009.[4] A. Capella and F. Otto. A quantitative rigidity result for the cubic-to-tetragonal phasetransition in the geometrically linear theory with interfacial energy.
Proc. Roy. Soc. Edin-burgh Sect. A , 142(2):273–327, 2012.[5] R. Choksi, S. Conti, R.V. Kohn, and F. Otto. Ground state energy scaling laws during theonset and destruction of the intermediate state in a type I superconductor.
Comm PureAppl. Math. , 61(5):595–626, 2008.[6] H. Kn¨upfer, R. V. Kohn, and F. Otto. Nucleation barriers for the cubic-to-tetragonal phasetransformation.
Comm. Pure Appl. Math. , 66(6):867–904, 2013.[7] H. Kn¨upfer and F. Otto. In preparation.[8] R. V. Kohn and S. M¨uller. Surface energy and microstructure in coherent phase transitions.