Surface Energies Emerging in a Microscopic, Two-Dimensional Two-Well Problem
In this article we are interested in the microscopic modeling of a two-dimensional two-well problem which arises from the square-to-rectangular transformation in (two-dimensional) shape-memory materials. In this discrete set-up, we focus on the surface energy scaling regime and further analyze the Hamiltonian which was introduced in [KLR15]. It turns out that this class of Hamiltonians allows for a direct control of the discrete second order gradients and for a one-sided comparison with a two-dimensonal spin system. Using this and relying on the ideas of Conti and Schweizer [CS06c], [CS06b], [CS06a], which were developed for a continuous analogue of the model under consideration, we derive a (first order) continuum limit. This shows the emergence of surface energy in the form of a sharp-interface limiting model as well the explicit structure of the minimizers to the latter.
- 1 Introduction
- 2 Rigidity
- 3 Surface Energies
- A Mapping the Microscopic Two-Well Problem to a Spin System
- B Second Derivative Control
- C Sketch of Proof of the Discrete Coarea Formula
- D Proof of the Well-Definedness of the Algorithm for the Perturbed Grid Construction
In this article we are concerned with the modeling of a discrete, two-dimensional square-to-rectangular martensitic phase transition in the regime of surface energy scaling. Due to their interesting thermodynamical and mathematical behavior, martensitic phase transitions, being examples of diffusionless, solid-solid phase transitions, have attracted a large amount of attention (c.f [Bha03] and [Mü99] for overviews). Hence, a number of models for these phase transitions, describing them from both microscopic and macroscopic points of view, exist in the literature. In the present article we continue to analyze the microscopic discrete model for the square-to-rectangular martensitic transition which was introduced in [KLR15]. In particular, we compare it with its continuous analogues, which have been considered previously in the literature.
1.1 Macroscopic, continuum models
Before describing our microscopic, discrete model, we recall the most commonly used features of macroscopic, continuum models for martensitic phase transitions (see e.g [BJ87, KM92, KO12, Con00, CS06c] and the references therein). In this context, a classical modeling approach is the analysis of purely elastic multi-well energies of the form
Here is an invariant, in general non-quasiconvex (bulk) energy density which describes the energy cost of deforming a reference configuration into its image configuration by the deformation which is considered under appropriate boundary conditions. It is assumed that the deformation, , and the associated deformation gradient, , are in appropriate Sobolev spaces which are determined by the growth conditions imposed on the energy density .
In modeling our phase transitions, we focus on the regime, in which the martensitic phase is favored and has multiple energy wells, i.e. there exist such that if and only if . Deformations which (almost everywhere) satisfy are denoted as exactly stress-free states. If there are rank-one connections between the energy wells, i.e. if for , with , there exist a rotation and vectors such that
then examples of stress-free states are provided by so-called simple laminates. These are Lipschitz continuous deformations which only depend on the variable and whose gradient alternates between the two values , , e.g.
However, under general boundary conditions, due to the non-quasiconvex nature of the energy density, , exact minimizers of (1) do not exist. Instead, in many cases infimizing sequences display highly oscillatory behavior.
In order to remedy this non-existence issue and the “unphysical”, infinitely fine oscillations, higher order regularizations are added [KM92, KO12, Con00, CS06c], leading to energies like for instance
These additional higher order contributions are interpreted as surface energies since they penalize oscillations and transitions between different energy wells. Due to compactness, in general in the associated spaces minimizers to (2) exist and display characteristic length scales (c.f. [KM92], [KO12], [Con00]).
While the basic intention of higher order regularizations always consists of penalizing too high oscillations, their precise functional form for macroscopic models is in general not known (from experiments for instance). Hence a number of different possible regularizations exist, which range from various kinds of diffuse to sharp interface models. Strikingly, experiments show the presence of both diffuse and sharp interfaces around twin planes for different materials [BVTA87, BMC09]. Hence, in order to answer the question which of these energies is appropriate in which situation, a more “first principles” approach coming from microscopic considerations seems to be desirable.
1.2 The microscopic two-well problem
While the previously described models have had enormous success in predicting material patterns and microstructure, they are all continuum models. As such they are macroscopic and “phenomenological”. In order to develop a more rigorous foundation for these and other macroscopic models in mathematical physics, there has been a great activity in introducing microscopic discrete models describing different phenomena in mathematical physics and relating them to their continuum analogues, see e.g. [Bra] and the references therein.
In these microscopic models it is assumed that the elastic sample is given as a deformation of a ground state (atomic) lattice, e.g. or subsets thereof. This deformation is energetically described by a Hamiltonian, i.e. a sum of local energies originating from
interaction of the “atoms” involved in the microscopic sample.
The described discrete models have been thoroughly analyzed in one-dimension (e.g. for elastic chains) [BC07]. As in the continuous analogue, vectorial problems are less well understood. Here various approaches are pursued [LM10], [FT02], [BBL02], [AC04], [BS13], [Ros14]. Moreover, a direct comparison of the scaling behavior of a discrete and a continuous model of a two-dimensional two-well problem is given in [Lor06], [Lor09].
In the context of the vectorial set-up in martensitic phase transitions a key (mathematical) difficulty which distinguishes it from the one-dimensional case is the presence of a continuum of energy wells, i.e. .
In the sequel, we address a specific two-dimensional martensitic phase transition, the square-to-rectangular phase transition, from a microscopic point of view. In the following subsections, we introduce our precise set-up based on the discrete two-well Hamiltonian (with symmetry) and present our main results.
In this section we describe the basic set-up of our discrete two-well problem. We define the underlying domains and function spaces and explain our explicit model Hamiltonian. In the sequel, we seek to model the two-dimensional square-to-rectangular phase transition in the martensitic phase in which the variants of martensite constitute the energy wells. For this purpose we introduce the following energy wells
Although for the mathematical treatment of our problem this is not necessary, we restrict ourselves to the most relevant physical situation of volume preserving transformations. This corresponds to the assumption . In this study, for convenience of notation, we denote by any positive constant depending only (if not stated explicitly otherwise) on the lattice parameters and . Moreover, we often use the special constant
We recall that for every deformation there are exactly two rank-one connected matrices in , i.e. there exist (exactly) two rotations such that
Macroscopically, we expect that a shape memory alloy with these wells can form two variants of simple laminates (without bulk energy cost), where the normals to the jump planes are either given by or . Our first compactness result (c.f. Proposition 2.2) shows that indeed this macroscopic expectation can be justified by starting from a microscopic point of view and passing to the corresponding continuum limit. Moreover, we note that due to the rank-one connections between the wells, and are both rank-one connected with their convex combinations
where is the rotation from (5) and .
In order to prepare the definition of our model Hamiltonian which is formed by a sum of local energies having the set as the set of their local minimizers, we first define the precise set-up regarding the underlying domains and function spaces (c.f. Figure 1):
Definition 1.1 (Domains, deformations, admissibility).
In the sequel, we consider the following domains:
For , , we define the translated parallelograms
If , we also omit the point in the notation and simply write , .
Using the previous notation, we set and .
In the sequel, we work on the following triangles:
denotes the set of all edges involved in the triangles . With slight abuse of notation, we also refer to “the grid ” in the sequel, by which we mean the pair for .
Moreover, for any given set and any we define
In particular, this defines the sets and .
On the respective domains, we consider associated deformations and denote their values on lattice points as
Additionally we restrict ourselves to admissible lattice deformations satisfying a non-interpenetration condition, i.e. on any domain under consideration , where
Finally, we impose boundary conditions prescribed by the matrices from (6) on admissible deformations considered on the whole domain , i.e. for any admissible deformation defined on the whole domain , we demand that , where
Let us comment on these notions. We remark that we can easily switch from functions to grid functions using the definition given in (8) above. Conversely, starting from a discrete lattice function , we can pass to a function by
piecewise affine interpolation on the triangles . Hence most of the functions in our applications will be piecewise affinely interpolated lattice functions satisfying (10). Thus, with slight abuse of notation, in the sequel we will use the phrase “let be a sequence of piecewise affine functions on ” to denote a sequence of admissible functions that is affine on all of the grid triangles .
In the context of our admissible lattice functions, the non-interpenetration condition contained in (9) corresponds to requiring that the labeling of the lattice triangles is not reversed or interchanged under the deformation . Hence, it can be interpreted as a local invertibility constraint for piecewise affine deformations on each of the lattice triangles (c.f. the recent article of Braides and Gelli, [BG15], for a criticism of this in the context of discrete-to-continuum fracture mechanics).
By applying the conventions from Definition 1.1 and in particular by interpreting as the above described piecewise affine interpolation of the lattice deformation , the usual spatial gradient, of the deformation is well-defined and satisfies . Here denotes the space of piecewise constant matrix valued functions. In addition, we further agree on working with the following Lebesgue representative for the equivalence class of our deformation gradient : On the edges of the lattice triangles we define the gradient of to be equal to its value in the interior of the corresponding triangle which contains the considered edge. Also at the lattice point the gradient is identified with the one on . Using this convention, the abbreviations
which are used in the sequel, are properly defined.
In Propositions 2.3 and 3.1 we will also deal with functions of , which are compositions of Lipschitz functions and . Consequently, these are piecewise constant themselves. In this context, we will work with discrete gradients which we denote by . Here, for any lattice function , we set
Note that in order to avoid additional technicalities connected with boundary effects and in order to keep the leading order of the bulk elastic energy zero in all considerations below we define our reference domain as the parallelogram from Definition 1.1 (with sides orthogonal to one of the normals of the rank-one connections from (5)).
Keeping these conventions in mind, we proceed with the definition of our model Hamiltonian, which had already been previously introduced in [KLR15].
Definition 1.2 (Model Hamiltonian, I).
Let . Then the model Hamiltonian on the lattice is defined as
where and denotes the scalar product in . In the above definition and below, we use a summation agreement: The sign in a term indicates that the latter should be replaced by the sum of the respective terms having all possible sign combinations, e.g.
Remark 1.3 (Boundary conditions).
We stress that in our model we impose “hard” boundary conditions in the form of (10). Already at this stage we emphasize that they are “seen” by our Hamiltonian, as for points on the layers with , the Hamiltonian still takes the left and right neighbors of these into account. On these the boundary conditions have already been prescribed. This will give rise to boundary layer energies (c.f. Theorem 1).
The Hamiltonian in Definition 1.2 is constructed in such a way that the matrices from (3) indeed form its energy wells, i.e. for all admissible and if and only if or on the whole of . On the level of the local energies the deviation from the wells is measured by penalizing deformations which do not map horizontal and vertical unit line segments onto line segments of either the lengths or . Physically, this corresponds to two-body interactions between the five neighboring atoms , and . Moreover, the local energy favors deformations which enforce that orthogonal line segments are mapped to orthogonal line segments, i.e. deviations from orthogonality of the pairs and are penalized. The latter is physically achieved by three-body interactions measuring the angle between and , respectively. We emphasize that the condition on the angles is necessary in modeling the deformation of an elastic body, as otherwise no shear resistance would be present.
Using the notation from Definition 1.1, we can rewrite the brackets in the definition of the Hamiltonian (12) in terms of lengths and angles of the horizontal and vertical derivatives of the deformations. For instance, the first bracket in (12) turns into
Hence the local energy density
can also be regarded as a function of the lengths and angles formed by the corresponding partial derivatives of the deformation. Here
This also serves as a guiding intuition in defining a more general class of Hamiltonians for which our results are valid (see Definition 1.4 below).
In this sense the density in the Hamiltonian from Definition 1.2 (and later also the ones from Definitions 1.3 and 1.4) can be viewed as a Lipschitz continuous function. In order to construct the Hamiltonian it is composed with the discrete function (or, depending on which point of view is more suitable in the respective situation, with the piecewise constant function ).
For the individual brackets of the Hamiltonian from Definition 1.2 we introduce the notation
This notation is used in the Step 2 of the proof of the -convergence result in Section 3.3.
Apart from this geometric interpretation of the energy density, the definition of also implies the following pointwise control on the distance of to the energy wells .
Lemma 1.1 (Lower bound).
Let . Then for each the following bound holds:
The proof follows immediately from noticing that for
We further remark that the Hamiltonian defined in Definition 1.2 is of mixed growth: If on the gradient is close to the wells, the local energy is comparable to , while if is at a finite distance from the wells, the energy controls the norm of . Hence, in total
In order to avoid technical difficulties with the mixed growth behavior at infinity, we truncate the energy density for large gradient values.
Definition 1.3 (Hamiltonian, II).
Let . Let be a Lipschitz continuous function satisfying the bound
for any and for some universal constants . Using this function, we define a modified energy density as
with and a cut-off function which is chosen such that for all and for . Here the constant is the one from (4). Using this, we define the (final) model Hamiltonian as
We remark that the energy density from Definition 1.3 in particular satisfies bounds at infinity and for each the global estimate
In concluding this section we stress that the following results are not only valid for our model Hamiltonian from Definition 1.3 but hold for the following more general class of discrete Hamiltonians.
Definition 1.4 (General class of Hamiltonians).
for each and some positive constant . Here the abbreviation is used as above to denote the pointwise evaluation at (analogously as the evaluations defined in Remark 1.4). We set
We emphasize two important properties of this class of Hamiltonians: These are the lower bound from Lemma 1.1 (which follows from that for ) and the lower bound (16), which allows us to invoke the comparison arguments from Appendix A. The lower bound (16) contains the origin of the surface energies which are more closely analyzed in Section 3.
As there are only small modifications in the proofs of the following results, we always carry them out for our model Hamiltonian from Definition 1.3 and leave the corresponding modifications for the general class to the reader.
1.2.2 Main results
Our main objective in this article is an analysis of the discrete square-to-rectangular phase transition in the regime of surface energy scaling. In this context we will analyze the microscopic Hamiltonians from Definitions 1.3 and 1.4 on “low energy deformations”. More precisely, as in [KLR15] in the sequel we study sequences of admissible deformations for which
for some (in ) uniform constant .
As an immediate property of this scaling, we observe that, since the Hamiltonian controls the distance of to the wells, we in particular obtain in measure for deformations satisfying (17).
As in the case of atomic chains, which were investigated in [KLR15], we expect that this scaling in (in combination with the boundary conditions given by (6)) yields deformations which are locally simple laminates in the limit (see Proposition 2.2 below).
In this context our main interest is driven by the modeling side of the problem and by the question of whether the discrete problem can be regarded as an “equivalent” of the continuous regularization.
Mathematically, our analysis is strongly based on the fundamental ideas introduced in the treatment of the continuum version of the two-well problem by Conti and Schweizer [CS06c], [CS06b], [CS06a].
Let us also mention that these methods differ substantially from the ones introduced in our preceding paper [KLR15], where, after a special reduction of the Hamiltonian from Definition 1.2 onto one-dimensional chains, low energy deformations were treated both analytically and numerically. Due to the presence of “atomic chains”, additional structural conditions had been exploited in that context.
As the main result of this article, we prove that in the surface energy scaling regime and in the continuum limit, i.e. as , there exists a limiting surface energy which resembles the analogous limiting energy from the continuous set-up [CS06c]. This result is formulated precisely in the following theorem.
Let us comment on the result of Theorem 1 and its relation to [Lor09]: Similarly as in [Lor09] our analysis is motivated by understanding the relation between the discrete and continuous regularizations of the square-to-rectangular phase transition. In this context we are in particular interested in studying the origins of surface energies. The article [Lor09] compares the scaling of infimizers of a functional of the type (2) with a discretization of (1) and shows that the corresponding scaling behaviors coincide. In particular, (for infimizers) this allows to switch between the discrete and continuum functional (up to giving up constants) and to transfer bounds from the discrete to the analogous continuum model. In principle, this would permit us to establish compactness properties for sequences for the discrete (minimizing) sequences from the compactness properties of their continuous analoga. However, due to the loss of the constants, [Lor09] does not imply our -convergence result. Instead of exploiting the result of [Lor09], we give an independent proof of the compactness properties, since in our discrete setting this step is simplified by a comparison with a spin system (due to the presence of next-to-nearest neighbor interactions). Having established compactness, our proof then follows the ideas outlined by Conti and Schweizer [CS06c], [CS06b], [CS06a] adapted to our discrete set-up.
Let us further note that while Theorem 1 does not explain the different diffuse and sharp interface features observed in experiments, it does show that as in the one-dimensional case and as in [Lor06], [Lor09], discrete two-well energies such as in Definitions 1.3 and 1.4 naturally lead to higher order regularizations. If understood on the level of finite, but large sample sizes they might even give indications for the experimentally observed behavior. The analogy between the discrete and the continuous setting is highlighted in Table 1. It provides a natural correspondence between continuum objects considered by Conti and Schweizer [CS06c], [CS06b], [CS06a] and our discrete setting introduced in the previous paragraph.
|continuum, macroscopic||discrete, microscopic|
|regularity of||is piecewise|
|the deformation||affine on the underlying|
|lattice (after interpolation)|
1.3 Organization of the article
The remainder of the article is organized as follows. In Section 2 we derive compactness (Proposition 2.1) and rigidity properties (Propositions 2.2, 2.3) of the sequences of deformations obeying the surface energy scaling (17). In Section 3 we describe the limiting surface energies and prove Theorem 1. Important auxiliary results are proved in the Appendices: In Appendix A we provide a mapping of our discrete two-well problem to a spin system. This yields a one-sided estimate of the original Hamiltonian from below. These results are crucially used in our proof of the compactness results of Propositions 2.1, 2.2. In Appendix B we show that our discrete Hamiltonian provides upper bounds of the discrete second derivatives of admissible deformations . While the latter bounds are not actually necessary for our argument, they are included as an illustration of the comparability of our discrete and the continuous model from [CS06c]. Last but not least in Appendix C and Appendix D we give sketches of the proofs of the discrete coarea formula and the well-definedness of the algorithm yielding the perturbed grid in the proof of Proposition 3.1, Step 4a in Section 3.
In this section we prove various rigidity estimates. On the one hand they yield compactness properties (c.f. Propositions 2.1, 2.2) for admissible sequences which obey the energy bound (17). On the other hand, adapting to our discrete setting the ideas of Conti and Schweizer [CS06c], [CS06b], [CS06a], we show finer rigidity estimates for sequences, which, in addition to (17), also satisfy the smallness condition (21) for a local one-well energy (see Proposition 2.3 below). The latter estimates play a crucial role for the cutting mechanism which is used for the construction of the recovery sequence in the - inequality (c.f. Section 3.3).
Let be a sequence of piecewise affine functions on the grid satisfying (17). Then there exist (up to zero sets) disjoint Caccioppoli sets and such that
Let be a sequence of piecewise affine functions on the grid satisfying the energy bound (17). Then there exist a subsequence and a limiting deformation such that
An analogous statement holds in .
In order to show this, we follow an argument of Kinderlehrer, c.f. also [Mü99] Theorem 2.4, in which the function is replaced by a lower semi-continuous analogue.
Step 1: Set-up.
We set and remark that and with and . Moreover, we recall the truncation argument from [FJM02], which allows to replace by a sequence with the following properties:
Here are universal constants (independent of ). The last estimate follows from the energy bounds (17) and (15). As a consequence of the comparability of and , the two functions have the same weak limit . Therefore, it suffices to prove the strong convergence of to .
Step 2: Convergence of . We claim that
Step 3: Using the convergence from Step 2, leads to
Here the first equality follows from Step 2 above. The fourth estimate is a consequence of the lower semi-continuity of the norm together with the weak continuity of the determinant and the fact that in as . Hence, we have equalities everywhere in (20) and therefore,
along a subsequence. As a consequence, along a subsequence in and (we remark that due to the closeness of to which follows from Lemma A.3). ∎
With these results at hand, we can now invoke the two-well rigidity result of Dolzmann and Müller [DM95]. This yields a structure result for the sets , . More precisely, the associated liming deformation, , has to be a simple laminate, which (by virtue of the energy bound (17)) implies that consists of a union of finitely many stripes (and triangles), c.f. Figure 2.
Proposition 2.2 (Rigidity).
Let be a sequence of piecewise affine functions on the grid satisfying the energy bound (17). Then there exist a subsequence and a limiting deformation such that
in , where a.e. is a piecewise constant BV function,
the associated domains and which were defined in Lemma 2.1 consist of a union of finitely many polygonal domains which extend up to the boundary of . The interfaces of the polygonal domains are either given by lines with the normals or (which are determined by the rank-one connections from (5)) or by the boundary of .
there exists a constant with
for all .
By Proposition 2.1 there exists a subsequence such that and (and ) in (and , respectively). As and have finite perimeter, satisfies the conditions of Theorem 5.3 in [DM95] and the limiting deformation locally is a laminate. As , there can only be a finite number of phase transitions between and . The remaining parts of the statements (a), (b) follow from Theorem 5.3 in [DM95].
The boundary conditions which are stated in (c) are a consequence of the convergence of and the prescribed boundary data (10). The finiteness of