On non stress-free junctions between martensitic plates
Francesco Della Porta
TL;DR
This paper investigates non stress-free junctions between martensitic plates, specifically $V_{II}$ junctions in Ti74Nb23Al3, providing a mathematical characterization and proving their stability as local energy minimizers.
Contribution
It introduces a mathematical framework for $V_{II}$ junctions in martensitic microstructures and demonstrates their stability within elasto-plasticity theory, extending understanding beyond stress-free interfaces.
Findings
$V_{II}$ junctions are strict weak local minimisers of the energy functional.
The mathematical characterization aligns with experimental observations.
The study extends compatibility theory to non stress-free interfaces.
Abstract
The analytical understanding of microstructures arising in martensitic phase transitions relies usually on the study of stress-free interfaces between different variants of martensite. However, in the literature there are experimental observations of non stress-free junctions between martensitic plates, where the compatibility theory fails to be predictive. In this work, we focus on junctions, which are non stress-free interfaces between different martensitic variants experimentally observed in Ti74Nb23Al3. We first motivate the formation of some non stress-free junctions by studying the two well problem under suitable boundary conditions. We then give a mathematical characterisation of junctions within the theory of elasto-plasticity, and show that for deformation gradients as in Ti74Nb23Al3 our characterisation agrees with experimental results. Furthermore, we are…
| (approx. in dgs.) | Observed junction | (values) | |
|---|---|---|---|
| none | none | ||
| none | none | ||
| none | none | ||
| none | none | ||
| none | none | ||
| none | none | ||
| none | none | ||
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
00footnotetext: Acknowledgements: The author would like to thank the Max Planck Institute for Mathematics in the Sciences where part of this work was carried out. This work was partially supported by the Engineering and Physical Sciences Research Council [EP/L015811/1]. The author would like to thank John Ball, Tomonari Inamura and Angkana Rüland for the useful discussions. The author would like to acknowledge the two anonymous reviewers for improving this paper with their comments.
On non stress-free junctions between martensitic plates
Francesco Della Porta
Abstract
The analytical understanding of microstructures arising in martensitic phase transitions relies usually on the study of stress-free interfaces between different variants of martensite. However, in the literature there are experimental observations of non stress-free junctions between martensitic plates, where the compatibility theory fails to be predictive. In this work, we focus on junctions, which are non stress-free interfaces between different martensitic variants experimentally observed in Ti74Nb23Al3. We first motivate the formation of some non stress-free junctions by studying the two well problem under suitable boundary conditions. We then give a mathematical characterisation of junctions within the theory of elasto-plasticity, and show that for deformation gradients as in Ti74Nb23Al3 our characterisation agrees with experimental results. Furthermore, we are able to prove that, under suitable hypotheses that are verified in the study of Ti74Nb23Al3, junctions are strict weak local minimisers of a simplified energy functional for martensitic transformations in the context of elasto-plasticity.
1 Introduction
Martensitic phase transitions are abrupt changes occurring in the crystalline structure of certain alloys or ceramics when the temperature is moved across a critical threshold. The high temperature phase is called austenite or parent phase, and usually enjoys cubic symmetry, while the low temperature phase is called martensite, and has lower symmetry (e.g., tetragonal, orthorhombic, monoclinic [12]). For this reason, martensite has usually more variants, which are symmetry related, and which in experiments often appear finely mixed. Martensitic phase transitions are important because they are the physical motivation of shape memory, the ability of certain materials to recover on heat deformations which are apparently plastic.
After the seminal work of Ball and James [4] modelling martensitic phase transitions in the context of nonlinear elasticity (see Section 2), a vast literature has been developed to study energy minimisers, and energy minimising sequences for energy functionals describing this physical phenomenon at a continuum scale. Indeed, energy minimising sequences can be interpreted as microstructures, that is finely mixed martensitic variants, with no elastic energy at the macroscopic scale (see e.g., [6, 12, 27] and references therein). A key tool to understand and predict martensitic microstructures is the Hadamard jump condition (see e.g., [4, Prop. 1]) stating that if a continuous function is such that
[TABLE]
for some unit vector and two matrices , then
[TABLE]
This condition imposes some necessary compatibility between two martensitic variants, or between two average martensitic deformation gradients representing different homogeneous microstructures, in order to have stress-free junctions. If (1.1) holds, then we say that are compatible across the plane . Compatibility is a key ingredient not only to understand microstructures (see e.g., [4, 12]) but also to understand hysteresis of the phase transformation [37] and recently to construct materials undergoing ultra-reversible phase transformations [36, 16]. Nonetheless, in the literature experiments are reported where the above compatibility is not observed right off the phase interface, and where the phase junctions are not stress free. More precisely, martensite is elastically or plastically deformed to achieve compatibility between variants/phases. For example, in Figure 1(a) we show the situation of junctions observed in the cubic to orthorhombic transformation in Ti74Nb23Al3[25]. We have two different deformation gradients corresponding to two different martensitic variants, and the identity matrix , deformation gradient in the austenite region. In the case of junctions we have
[TABLE]
and therefore the interfaces between austenite and martensite are not stress-free close to the junction between with . Similarly, in the case of junctions (see Figure 1(b)), also observed in Ti74Nb23Al3[25], we have
[TABLE]
and therefore and are not compatible. In Figure 1(c) we show an incompatible junction between the two average deformation gradients representing the average of the martensitic microstructures on the left and on the right of the red line [9, 13]. In this case, as for the junctions, (1.2) holds. Non stress-free phase interfaces have also been observed in the X–interface configuration (Figure 1(d)) for which we refer the reader to [10, 34].
The following approach to measure the incompatibility between non-stress free junctions has been proposed in [9]. Assuming that are such that , and that has middle eigenvalue one, [4, Prop. 4] guarantees the existence of two rotations such that for . The incompatibility of can hence be measured by taking the minimum between the rotation angle of , and the rotation angle of . This is in agreement with the experimental results in [9, 25] where the observed non stress-free junctions are the ones where is small. Another way to measure how far three deformations gradients, say are to form a triple junction, that is to be all pairwise rank one connected, can be found in [21]. However, in the case for example of Ti74Nb23Al3[25] these approaches do not allow to predict when two martensitic variants will form a or a junction. Indeed, experiments show that some martensitic variants tend to meet only in junctions, while others form just junctions (see e.g., [25, Table 4]).
The aim of this work is to study junctions and their stability in the context of elasto-plasticity. The paper is organised as follows: in Section 2 we recall the nonlinear elasticity theory for martensitic phase transitions, and we introduce a simplified energy functional to describe the physical phenomenon when plastic shears occur. This energy functional is very general as it describes the transformation to all possible martensitic variants and all possible slip systems for body centred cubic austenite (as in Ti74Nb23Al3). In Section 3 we give a partial explanation of why we observe non stress-free junctions of type or like the ones in Figure 1(c). Our explanation is the following: these type of junctions usually form when two different plates of martensite, with deformation gradients , nucleate at different points in the domain, and expand until they meet (see Figure 2(a) and Figure 2(b)). We hence consider a bounded domain as in Figure 3 and two martensitic variants represented by their stretch tensors . We prove that, under some further geometric hypotheses which are verified by the non stress-free junctions in Ti74Nb23Al3[25] and in Ni65Al35 [9], there exists a one-to-one map satisfying
[TABLE]
with \mathsf{F}_{1},\mathsf{F}_{2}\in\bigl{(}SO(3)\mathsf{U}_{1}\cup SO(3)\mathsf{U}_{2}\bigr{)}^{qc} if and only if Therefore, no stress-free microstructure built with the two martensitic variants can fill the domain and match the previously nucleated plates .
In Section 4 we study when two simple shears are such that
[TABLE]
given with .
In Section 5 we give a mathematical characterisation of junctions as junctions reflecting (1.2), where the compatibility between is achieved thanks to single slips (and hence thanks to plastic effects). We also give sufficient conditions for junctions to be strict weak local minimisers for the simplified energy introduced in Section 2.
In Section 6 we study the possibility to form junctions in a one parameter family of deformation gradients, which approximates well the phase transformation in Ti74Nb23Al3. The obtained results are discussed at the end of the section, and seem to be in good agreement with experimental observations. Finally, in Section 7 we give some concluding remarks and possible directions to extend the present work.
2 A model for martensitic transformations with plastic shears
The most successful mathematical theory to describe martensitic phase transitions at a continuum level is based on the theory of nonlinear elasticity and was first introduced in [4]. This model has been successfully used to understand laminates and other microstructures (see [4, 12]), as much as the shape-memory effect (see [11]), and, more recently, hysteresis (see [37]).
In the nonlinear elasticity model, changes in the crystal lattice are interpreted as elastic deformations in the continuum mechanics framework, and legitimised by the Cauchy-Born hypothesis. The deformations minimize hence a free energy
[TABLE]
Here, denotes the temperature of the crystal, the domain (open and connected) stands for the region occupied by a single crystal in the undistorted defect-free austenite phase at the transition temperature , while denotes the position of the particle after the deformation of the lattice has occurred. By we denote the free-energy density, depending on the temperature and the deformation gradient . The behaviour of on must reflect the phase transition, that is when and , the energy is respectively minimised by martensite and austenite. At all phases are energetically equivalent.
Below, we assume to be fixed, and we consider to be defined by (omitting for ease of notation the dependence on )
[TABLE]
where are the positive definite symmetric matrices corresponding to the transformation from austenite to the variants of martensite at temperature . Here and below represents the set of symmetric and positive definite matrices. We remark that, defined as the point groups of austenite and martensite respectively (i.e., the sets of rotations that map the austenite and martensite lattices back to themselves), and denoting by their cardinality, we have . Also, for each there exists such that , so that share the same eigenvalues. We point out that this energy satisfies frame indifference. That is, for all and all rotations , , reflecting the invariance of the free-energy density under rotations. Furthermore, respects lattice symmetries, i.e., for all and all rotations . Such a has been already considered for example in [4, 3, 7, 20] and corresponds to the physical situation where the elastic constants are infinity, which, as remarked in [3], is usually a reasonable approximation when studying martensitic phase transitions with no external (or at least small) load. Considering to be out of the energy wells is also known as the elastically rigid approximation, and is often used in the context of elasto-plasticity since elastic effects in metals are usually much smaller than plastic ones (see e.g., [30]).
We now want to take in account the presence of plastic effects in the nonlinear elasticity model. Following [31, 32] and references therein, we use the multiplicative decomposition of the deformation gradient
[TABLE]
where respectively represent the elastic and the plastic component of the deformation gradient. The former describes the part of the deformation gradient which is reversible, while the latter captures the irreversible deformations given by the slip of atoms along planes. In solid crystals, atoms can slip just in particular directions on particular planes. For this reason, must be of the form
[TABLE]
where , , , , and Here, is called slip direction and is called the slip plane, while is the amount of shear. The set is the set of all possible slip systems. For body centred cubic austenite, which is the case of Ti74Nb23Al3, there are six planes of type each with two orthogonal directions, twenty-four planes and twelve planes each with one orthogonal direction.
Following the approach of [18, 2, 22] and references therein, we adopt the time discrete variational approach to elasto-plasticity [30], restricting ourselves to the first time step where most of the plastic events take place. We further assume cross hardening [2], which means that activity in one slip system suppresses the activity in all other slip systems at the same point. For this reason, we choose a plastic energy density of the type
[TABLE]
where is assumed to be continuous, strictly monotone and to satisfy Here, as for , could be finite and continuous. This approximation however simplifies the analytical study of the energy and allows to neglect any dependence of the results on the shape of the energy density out of its minima. We are now ready to introduce an elasto-plastic energy density defined as
[TABLE]
and an energy functional for the system
[TABLE]
We remark that the energy is not weakly lower semicontinuous and in general minimisers do not exist.
3 A rigidity result for the two well problem
In this section, we study the existence of solutions to Problem (1.3). As explained in the introduction, this gives a way to justify the formation of non stress-free junctions between martensitic plates.
Let , and let us set . For , we define (see Figure 3)
[TABLE]
Theorem 3.1 below states that, under suitable boundary conditions, the differential inclusion (1.3) has no solution. More precisely, under our assumptions, the boundary conditions on need to satisfy a compatibility condition, which is unexpected and strongly dictated by the structure of the two well problem. Also, in order to have no solution to the two well problem, we do not need to impose boundary conditions on the whole boundary of the domain, but just on a corner of it (namely, on ). By the work in [28] we know that, under suitable boundary conditions, there are infinitely many solutions to the differential inclusion \nabla\mathbf{y}(\mathbf{x})\in\bigl{(}SO(3)\mathsf{U}_{1}\cup SO(3)\mathsf{U}_{2}\bigr{)}, a.e. . Our result provides an example of boundary conditions where the convex-integration techniques used in [28] cannot be applied. Further, our result holds also for the relaxed differential inclusion \nabla\mathbf{y}(\mathbf{x})\in\bigl{(}SO(3)\mathsf{U}_{1}\cup SO(3)\mathsf{U}_{2}\bigr{)}^{qc}, a.e. . The proof relies on a result by Ball and James [5] which states that, after a suitable change of coordinates, in the two well problem there exists one direction (in the proof below ) where the martensitic deformation coincides with a constant elongation/contraction composed with a constant rotation. The proof exploits the fact that this direction and this rotation must be coherent across the whole domain and compatible with the boundary conditions. The result reads as follows:
Theorem 3.1**.**
Let such that there exists satisfying
[TABLE]
Suppose further that is such that . Then, there exists such that is in ,
[TABLE]
and
[TABLE]
for some , if and only if there exists such that
[TABLE]
Proof.
Necessity. We first notice that is Lipschitz, and therefore by Morrey’s imbeddings (see e.g., [1]). Therefore, is continuous on the line , that is
[TABLE]
Now, given (3.8), [16, Prop. 12] guarantees the existence of , , such that
[TABLE]
Without loss of generality, we can take from standard twinning theory (see e.g., [12]) \mathbf{b}=2\Bigl{(}\frac{\mathsf{U}_{1}^{-1}\hat{\mathbf{e}}}{|\mathsf{U}_{1}^{-1}\hat{\mathbf{e}}|^{2}}-\mathsf{U}_{1}\hat{\mathbf{e}}\Bigr{)}. The same results can be achieved by taking the only other solution of (3.12), that is \mathbf{m}=2\Bigl{(}\hat{\mathbf{e}}-\frac{\mathsf{U}_{1}^{2}\hat{\mathbf{e}}}{|\mathsf{U}_{1}\hat{\mathbf{e}}|^{2}}\Bigr{)}. We remark that by (3.12) we have that and hence, as , Following the strategy of [6], let us define the orthonormal system of coordinates
[TABLE]
and let
[TABLE]
Therefore, setting the problem becomes equivalent to finding a map such that
[TABLE]
with and
[TABLE]
Here,
[TABLE]
Following [6], we can characterise the set K_{L}:=\bigl{(}SO(3)\mathsf{S}^{-}\cup SO(3)\mathsf{S}^{+}\bigr{)}^{qc} as
[TABLE]
and where we denoted . Let us now define
[TABLE]
and remark that [5] together with the definition of yield
[TABLE]
for some Lipschitz scalar functions and some . Assume now that , the other cases can be treated similarly to deduce (3.17) below. In this case, the fact that on (cf. (3.14)) together with imply that
[TABLE]
where are coordinates on , that is
[TABLE]
Therefore, varying and in an open interval we deduce that
[TABLE]
There exists hence such that
[TABLE]
that is
[TABLE]
Taking the norm on both sides, we deduce that must satisfy
[TABLE]
We notice that implies that and hence . This yields
[TABLE]
Therefore, and (3.18) simplifies to
[TABLE]
that is or . In the same way, we can show that
[TABLE]
with or . We now claim that, even if , the only possible solution is . Indeed, let (the case can be treated similarly), and let us notice that
[TABLE]
for every as in (3.16). As a consequence, are linear on the boundary, and hence are linear on the set
[TABLE]
This is the subset of where the boundary condition is propagated along the characteristic lines in direction . Therefore, given (3.13), we deduce the existence of such that in . A version of the Hadamard jump condition (see e.g., [4, Prop. 1]) yields
[TABLE]
for some . The fact that together with (3.15) imply
[TABLE]
Exploiting (3.17) and (3.20) we deduce
[TABLE]
Now, polar decomposition implies , for some , As we also have and , as well as Thus, (3.21) becomes
[TABLE]
At the same time, the fact that implies that . But (3.20) entails,
[TABLE]
which implies that . The same argument can be applied to prove Therefore, (3.17) and (3.19) simplify to
[TABLE]
from which we deduce
[TABLE]
Here is given by the polar decomposition of , and is such that for some Now, as the hypothesis that implies that and are linearly independent. As a consequence, (3.11) and (3.23) imply
[TABLE]
and (3.10).
Sufficiency. Let us define
[TABLE]
It is easy to check that satisfies (3.13)–(3.14), proving the statement. ∎
Remark 3.1**.**
Let be the deformation gradients measured experimentally in Ti74Nb23Al3 (see [25] or Section 6 below) or in Ni65Al35 [9, 13]. By (1.2) we have , for some and such that . Taking and we have that is verified, and therefore Theorem 3.1 implies that no stress-free junction involving just two martensitic variants can be observed in Ti74Nb23Al3, nor in Ni65Al35 between the nucleated plates .
Remark 3.2**.**
The result is independent of the shape of
Remark 3.3**.**
By [16, Prop. 12], (3.8) is equivalent to the existence of , satisfying (3.12). If (3.8) fails, then, under some further physically relevant restrictions on the parameters of , [24] implies that , and that is affine.
Remark 3.4**.**
A similar result holds if we replace with
[TABLE]
for which we refer to Figure 4. In this case, however, necessary and sufficient conditions are (3.10) and, if
[TABLE]
This latter condition is to guarantee that the information carried by the characteristic lines in direction from the boundary conditions do not overlap.
Remark 3.5**.**
In general, the statement of Theorem 3.1 does not hold when . Consider for example
[TABLE]
for some These deformation gradients describe in a suitable basis an orthorhombic to monoclinic transformation. Let further ,
[TABLE]
and
[TABLE]
We choose such that
[TABLE]
so that the situation becomes fully two-dimensional (cf. Figure 5). Indeed, Then, we can construct as
[TABLE]
where continuity is guaranteed by the fact that In this case, following [23], if and only if satisfies
[TABLE]
It can be checked that both the first and the second property are satisfied for every Therefore, if (3.10) can fail. We remark that, in this situation, the key ingredient is not the type of transformation (represented here by its stretch tensors ), but the two-dimensional structure of the problem. Indeed, in this case, both the boundary conditions imposed on (which in direction are both a constant elongation/contraction of magnitude ) and the domain (whose shape does not depend on the axis) make the problem essentially two-dimensional.
4 Plastic junctions
In this section we want to investigate when, given two matrices with , there exist two simple shears , , , such that These results are useful for the mathematical characterisation of junctions given in the next section. Here and below, we denote by the set of admissible slip systems (or a suitable subset of it), and by the set of martensitic variants (or a suitable subset of it).
Under our hypotheses on there exist and such that
[TABLE]
Therefore, our problem becomes equivalent to finding and such that
[TABLE]
Lemma 4.1 below gives necessary conditions for the existence of solutions to (4.24). There and throughout this section, can be interpreted as .
Lemma 4.1**.**
Let and \operatorname{rank}\bigl{(}\mathbf{a}_{1}\otimes\mathbf{n}_{1}-\mathbf{a}_{2}\otimes\mathbf{n}_{2}\bigr{)}=2. Then, a necessary condition for the existence of such that
[TABLE]
is that at least one of the following four conditions hold:
[TABLE]
Proof.
Since if and only if , (4.25) is equivalent to
[TABLE]
Taking now the scalar product of (4.26) with and we respectively obtain
[TABLE]
Recalling that \operatorname{rank}\bigl{(}\mathbf{a}_{1}\otimes\mathbf{n}_{1}-\mathbf{a}_{2}\otimes\mathbf{n}_{2}\bigr{)}=2 implies that and , from (4.27) we deduce the claim. ∎
In general, the necessary conditions provided by Lemma 4.1 are not sufficient. In other cases, infinitely many solutions may exist given two slip systems In Proposition 4.1 we prove that, under certain hypotheses on the shear systems which are relevant in the following section, there exists a unique couple such that (4.25) is satisfied.
Proposition 4.1**.**
Let . Suppose further that \operatorname{rank}\bigl{(}\mathbf{a}_{1}\otimes\mathbf{n}_{1}-\mathbf{a}_{2}\otimes\mathbf{n}_{2}\bigr{)}=2. Then,
- •
if , for some , and if one out of holds, then are such that (4.25) is satisfied if and only if they satisfy
[TABLE]
- •
if , for some , and if one out of holds, then are such that (4.25) is satisfied if and only if they satisfy
[TABLE]
- •
if , and , for some , then are such that (4.25) is satisfied if and only if they satisfy
[TABLE]
In particular, there may be a one parameter family of solutions.
Proof.
We just prove the first case, as the second case can be proved in a similar way, and the third is a direct consequence of (4.31) below. Assuming and , solving (4.26) is equivalent to solving
[TABLE]
By testing this equation by and we obtain the necessity of (4.28). Now, let us show that, under our assumptions, (4.28) are also sufficient conditions. In order to do this, it is sufficient to show that, for as in (4.28) the equality in (4.31) tested with , for some such that , holds. Under our assumptions, and assuming (4.28), at least one out of and holds. Suppose without loss of generality the first one, as the other case can be deduced similarly. We can thus multiply
[TABLE]
by and deduce that the resulting number is zero, which concludes the proof of the first statement. ∎
The results above motivate Definition 4.1 below.
Definition 4.1**.**
Let and such that Let also and be such that satisfies
[TABLE]
for some . Then, we say that and form a plastic junction at for . In this case, we call the plane the plastic junction plane.
We say that the plastic junction formed by and at is locally rigid if there exists such that, for every with , and every satisfying , there exists no such that
[TABLE]
The following result gives sufficient conditions for a plastic junction to be locally rigid. The notation below refers to the notation of Definition 4.1.
Proposition 4.2**.**
Let and form a plastic junction at as defined in Definition 4.1. Let further , for some , such that , and
[TABLE]
Then the plastic junction formed by and at is locally rigid.
Proof.
Let us first notice that (4.32) can be written as
[TABLE]
for some , such that . Testing (4.34) by , we deduce that a necessary condition for to satisfy (4.32), is that the rotation axis of is . Furthermore, letting , a necessary condition for the existence of such that (4.32) holds is that
[TABLE]
which is (4.34) tested by . Let hence be the rotation of axis and angle . Let us also define the smooth function
[TABLE]
Necessary and sufficient condition to have local rigidity is that in a neighbourhood of . But
[TABLE]
Therefore, if condition (4.33) is satisfied, , and hence there exists a neighbourhood of radius of such that for every with
[TABLE]
which is the claim. ∎
5 Stability of plastic junctions
In this section we give sufficient conditions for plastic junctions to be weak local minimisers of the energy functional . We recall that any Lipschitz continuous map is a weak local minimiser if there exists such that for any Lipschitz continuous map satisfying . We start the Section by giving a mathematical definition of junctions. Then we state and prove our local stability result in Theorem 5.1 which gives sufficient conditions for junctions to be strict weak local minimisers. At the end of the section we state a stability result for plastic junctions, which relies on the same proof as Theorem 5.1.
The definition of junction reads as follows:
Definition 5.1**.**
Let and be such that Let also form a plastic junction at for which is locally rigid. Assume further:
- (1)
* and for some , ;* 2. (2)
(Domain) The domain (cf. Figure 6) is defined as for some . We also define for 3. (3)
(Geometry) Also, (cf. Figure 6) there exist (or in ) such that and
[TABLE]
where is the rotation of angle and axis of the half-plane . Furthermore, for any (resp. ). 4. (4)
(Structure) is defined by
[TABLE]
Then, we say that is a junction between and .
Remark 5.1**.**
The Hadamard jump condition implies that a necessary condition in order to form a junction between and is that
[TABLE]
Remark 5.2**.**
The hypothesis 3 requiring that guarantees the continuity of along the line for and justifies the bi-dimensional representation of stable plastic junctions given in Figure 6.
Before stating our stability result let us introduce the following definition:
Definition 5.2**.**
Let , , and . We say that enjoys the separation property if there exists such that for every , with , , , and where at least one out of and holds.
Remark 5.3**.**
If enjoys the separation property, then in a neighbourhood of there exists a unique decomposition of finite energy.**
We also introduce the definition of a locally stable junction:
Definition 5.3**.**
We say that a junction is locally stable if there exists such that, given any satisfying
- (A)
* for any open ball centred at and of arbitrary radius ,* 2. (B)
, 3. (C)
* is ,*
it holds:
- (T1)
for any measurable bounded
[TABLE]
- (T2)
the equality
[TABLE]
holds for any open ball centred at and of arbitrary radius if and only if for some .
Remark 5.4**.**
As pointed out in Section 2, the energy density is invariant under rigid motions. That is, given any , any and any , we have that and have the same energy. As we are not imposing any boundary condition on the variations in Definition 5.3, any , that is a rigid motion of a junction , has the same energy as . According to Definition 5.3 a locally stable junction is a strict weak local minimiser modulo rigid motions.
We are now ready to state and prove our stability theorem for junctions. The result relies on three main ingredients: first, we assume that any possible small variation of our junction (described by the map ) has locally finite energy. This, together with the structure of the energy density and the separation property (introduced in Definition 5.2) give a structure to the gradient of (cf. Remark 5.3). Second, we exploit the result of [5] characterising plane strains. Indeed, by using this result, we are able to prove that an Hadamard jump condition must hold for at the plastic junction plane of our plastic junction. Third, we use the local rigidity of the plastic junction to prove that our variation coincides, up to a rotation, with in a wedge of (namely ). Finally we prove (T1)-(T2). The theorem reads as follows:
Theorem 5.1**.**
*Let be a junction as in Definition 5.1. Let also enjoy the separation property. Then, if for , the junction is locally stable in the sense of Definition 5.3. *
Remark 5.5**.**
In Definition 5.1, Definition 5.3 and hence in the statement of Theorem 5.1 we consider an unbounded domain. This domain can be interpreted as a blow-up close to the line given by , where the incompatibility occurs. Mathematically, this choice is motivated by the argument in the proof, which relies on rigidity for plain strains. More precisely, this leads to the fact that the deformation gradient on the plane of compatibility is propagated in along the characteristic lines in direction , and in along the lines in direction . A similar theorem could be proved on any connected Lipschitz domain such that for every , , for every , where is such that This last condition guarantees that the information is transported by the characteristic lines from the plane of compatibility to every point in the domain.
Proof.
Let be as in Definition 5.2 such that respectively enjoy the separation property. Let also \delta_{3}:=\frac{1}{2}\min\bigl{\{}\|\mathsf{R}\mathsf{U}-\mathsf{V}\|:{\mathsf{U}\neq\mathsf{V}\in\mathcal{M}\cup\{\mathsf{1}\},\mathsf{R}\in SO(3)}\bigr{\}}, and let us take \varepsilon_{0}=\min\bigl{\{}\delta_{1},\delta_{2},\delta_{3}\bigr{\}}. Consider now any satisfying (A)–(C) in Definition 5.3. Then, since the energy is locally finite, by the separation property we have,
[TABLE]
for some locally Lipschitz continuous such that
[TABLE]
for some measurable , and , Define now . We notice that,
[TABLE]
where for any . It follows then by [5, Thm. 3.1] that is a plain strain, and we can hence deduce the existence of such that
[TABLE]
for some Lipschitz functions , and where
[TABLE]
Now, given the fact that the are Lipschitz continuous and that (and hence ) the value of is well defined on the plane . Indeed,
[TABLE]
for almost every and almost every such that . As a consequence, the value of on is well defined, and is respectively in , . By the continuity of and a weak version of the Hadamard jump condition (see [19, Remark 10]) we deduce that
[TABLE]
for some measurable .
We now claim that this implies the existence of such that a.e. in , . Indeed, let us consider the smooth functions
[TABLE]
and let be as in Definition 4.1. Since the ’s are continuous, as and if and only if and , there exists such that implies for . Let us hence fix . Therefore, if by (B) a.e. in with , then by (5.38) a.e. in . As a consequence, since with are plain strains and , (5.39) implies that for a.e. . By the fact that form a plastic junction which is locally rigid together with (5.38) and (5.40), it must hold . Therefore we deduce that there exists a measurable function such that a.e. on and for . By exploiting once more (5.38) and (5.39), we deduce that a.e. in But a result by Reshetnyak (see e.g., [33, 4]) implies that must be constant, concluding the proof of the claim.
As a consequence, since is a plain strain and linear, must be linear in , with , and of the form (5.38) with , for some . We remark that the energy of in is independent of . This, together with the fact that the energy density is non-negative imply (5.36). We remark that, every time we exploit (5.39) we implicitly rely on the fact that, for any , there exists and such that and that is convex.
Assume now that (5.37) holds. This, together with the fact that , the shape of and the rigidity result by Reshetnyak imply
[TABLE]
for some Again, by the Hadamard jump condition applied to on the planes and by [4, Prop. 4] we have , which leads to the claim of the theorem. ∎
An interesting consequence of the proof of Theorem 5.1 is the following rigidity result for plastic junctions:
Theorem 5.2**.**
Let be as in Definition 4.1, and form a plastic junction at for which is locally rigid. Assume further (1)–(3) in Definition 5.3 and:
- (4’)
(Local minimiser) is defined by
[TABLE] 2. (5)
* enjoy the separation property.*
Then, if for , there exists such that every satisfying
- a)
* for any open ball centred at and of arbitrary radius ,* 2. b)
, 3. c)
* is ,*
is of the form for some .
6 junctions in Ti74Nb23Al3
In this section we study the presence of junctions in cubic to orthorhombic transformations when the stretch tensors have both the middle eigenvalue and the determinant equal to one. This is done under the additional hypothesis that a parameter of the stretch tensors representing the lattice deformations lies in the physically relevant interval . A similar argument could be applied to study the case when . As explained below, this situation is a good approximation of the martensitic transformation in Ti74Nb23Al3 and similar materials. We prove that the existing junctions are locally stable in the case where the energy has all the wells, that is where the elastic energy is null on , where are the six matrices transforming a cubic lattice into an orthorhombic one, and where we consider all possible slip systems for body centred cubic austenite. However, the generality of the results leads to many long computations and, for this reason, in this section some of the hypotheses of Theorem 5.1 are verified numerically or with the aid of a plot. At the end of the section we compare the results obtained with experimental results.
The transformation in Ti74Nb23Al3 is from a cubic to an orthorhombic lattice, and therefore the stretch tensors describing the change of lattice vectors are given by
[TABLE]
Since in Ti74Nb23Al3 the middle eigenvalue of the transformation matrices is such that (see [25]) we implicitly assumed it to be equal to one in (6.42). Therefore, the eigenvalues of the ’s are , and, coherently with the lattice deformation in Ti74Nb23Al3, we assume also that A similar analysis could be worked out in the case where Under these assumptions, [4, Prop. 4] guarantees for every the existence of two couples of vectors and such that
[TABLE]
for some The different depending on are given by:
[TABLE]
[TABLE]
where
[TABLE]
As explained in the introduction, in experiments for Ti74Nb23Al3 [25] one observes the nucleation of different plates of martensite with , where and , which expand until they encounter another plate of martensite with similar properties. The nucleation is occurring at the interior of the domain, that is, an island of martensite with deformation gradient grows in the middle of an austenite domain with deformation gradient . Therefore, the deformation in the martensite region must be close to volume preserving, i.e., in a first aproximation , and hence, In reality, for Ti74Nb23Al3 some elasto-plastic effects take place to accommodate the nucleation at the interior. However, in order to simplify the analysis below, motivated by the experimental value of which is very close to one (the experimental values yield [25]), we assume We remark that, the analysis below holds also in the case (the lattice parameters for Ti74Nb23Al3) and for every , where is the image of , and are a finite number of polynomial curves . Furthermore we restrict ourselves to the physically relevant range . It is worth noticing that when the cofactor conditions are satisfied, and hence stress free triple junctions are possible (see e.g., [16, 21]). We now want to find plastic junctions as in Definition 4.1 and where and is of the form (cf. Remark 5.1)
[TABLE]
for and some . The case where has the form (6.43) but can be treated similarly, or simply deduced from our case by symmetry. We remark that, under our assumptions,
[TABLE]
for any . As a consequence \operatorname{rank}\bigl{(}\mathsf{R}_{1}\mathsf{V}_{1}-\mathsf{R}_{2}\mathsf{V}_{2}\bigr{)}=2.
We are now ready to state Theorem 6.1 which investigates the possibility to form plastic junctions and junctions in a one-parameter family of deformation gradients, and in particular in Ti74Nb23Al3. The stability of the existing junctions is also proved by verifying the hypotheses of Theorem 5.1. The results are compared with experimental results in Section 6.2. The theorem reads as follows:
Theorem 6.1**.**
Let . Let and be the set of all possible simple slips for body centred cubic lattices. Let us also define
[TABLE]
and
[TABLE]
Then, there exist a plastic junction (in the sense of Definition 4.1) for and with , if and only if
- (a)
, and
[TABLE] 2. (b)
, and
[TABLE] 3. (c)
, and
[TABLE] 4. (d)
, and
[TABLE]
All these plastic junctions can form locally stable junctions in the sense of Definition 5.3. There exists no junction (in the sense of Definition 5.1) between and .
Figure 7 shows the dependence of and on .
The results in Theorem 6.1 are compared with experimental observations in Section 6.2.
6.1 Verification of Theorem 6.1
The proof investigates first the existence of plastic junctions when . We then check that this plastic junctions can form a locally stable junction. To this aim, we need the verification of the assumptions of Theorem 5.1. These are technical and require long and uninteresting computations. Therefore, the verification of some of the assumptions of Theorem 5.1 is checked numerically or by means of a plot. Finally, we show that no junction (according to Definition 5.1) exists when We divide the argument into steps to simplify the presentation.
Existence of plastic junctions.
By Lemma 4.1, and taking in consideration all the slip systems for body centred cubic lattices (see Section 2), we can see that the necessary conditions to have plastic junctions for and with , are satisfied by each of the points (i)–(iv) below:
- (i)
and ; 2. (ii)
and ; 3. (iii)
and ; 4. (iv)
and .
In all the above cases and we therefore simplified notation by writing We now show that these conditions are sufficient to have plastic junctions. Thanks to Proposition 4.1 we can find such that
[TABLE]
Here, again, are the two different Burger’s vectors in the plane orthogonal to , among the slip systems for body centred cubic lattices. We recall that, in these cases, for every there are exactly two (up to sign change) such that is a slip system for body centred cubic lattices. By post-multiplying the above equation by we get
[TABLE]
Therefore, if the solution of (6.45) is unique, it can be identified with the unique solution of (6.44). Some computations conclude the proof of (a)–(d).
Local rigidity of plastic junctions.
In order to verify that the constructed plastic junctions are locally rigid (in the sense of Definition 4.1) we make use of Proposition 4.2. Under our hypotheses, , and, in the notation of Proposition 4.2, and . Furthermore, defining
[TABLE]
we have that for the first option in the cases (a)–(d) is respectively parallel to
[TABLE]
For the second option in the cases (a)–(d), can be deduced by pre-multiplying the vectors in (6.46) by . We now have all the ingredients to show (see (4.33))
[TABLE]
The easiest way to show this is graphically, by plotting the function for the cases (a)–(d) in Figure 8.
Separation property.
Let and , where and are as in (a)–(d). We first claim that for each there exists such that
[TABLE]
for any , whenever at least one out of
[TABLE]
holds. The amount of cases to be checked is huge. Indeed, there are four different junctions to be checked, that is case (a)–(d), each with two subcases. For each of these cases we have to verify two inequalities, namely (6.48)–(6.49), which must hold for six possible different ’s, and for forty-eight possible slip-systems. The total amount of cases to be checked is hence . Since we were not able to identify a unique simple algorithm to verify (6.48)–(6.49) in all these cases, we verified it numerically. Indeed, for any , any , and the functions are fourth order polynomials in which can be minimised numerically. The smooth dependence of on make the numerical problem well posed. Numerically one observes that the claim is true for any (cf. Figure 9).
Now, given as in the claim, we know that there exists such that if satisfies then Furthermore, the function defined by is Lipschitz on its domain, and hence there exists such that
[TABLE]
Therefore, combining this inequality with the claim we obtain that enjoys the separation property with .
junctions and local stability.
First, we have to construct such that (2)–(3) in Definition 5.1 are satisfied. But for as in (a)–(d), fixed we can choose such that (2)–(3) are satisfied. Let us now define as in (5.41). This is well defined because of the Hadamard jump condition, and leads to a junction for each of the cases (a)–(d). Given the steps above, in order to show that the junctions are stable, we just need to verify the assumption in Theorem 5.1 that , with , where in the notation of Theorem 5.1 and and is given by (a)–(d). This is done by using (6.46). We plot against in Figure 10, and we deduce that it is satisfied for all the cases (a)–(d) and . The junctions given by (a)–(d) are hence locally stable.
junctions between and .
In this case there are many slip systems which make plastic junctions possible. However, the only ones which satisfy the necessary conditions of Lemma 4.1, and such that (where is parallel to ) as required by hypothesis 3 in Definition 5.1, are couples of slip systems among
- (I)
and ; 2. (II)
and ; 3. (III)
and ; 4. (IV)
and .
Below we denote by case the case where are respectively given by and among (I)–(IV) above. Let us study the situation in the different cases:
Case and case . In these cases Proposition 4.1 guarantees that there are no plastic junctions as , but for , in (4.28).
Cases . By Proposition 4.1 there exists a unique plastic junction, and for the slip on the plane . Therefore, this cases can be studied within the context of cases and below.
Case and case . In these cases, Proposition 4.1 guarantees the existence of a one parameter family of plastic junctions. However, no local rigidity (in the sense of Definition 4.1) holds. Indeed, let , and be such that
[TABLE]
Let be a rotation of angle and axis We notice that , and hence
[TABLE]
for any Therefore, if for any small we can show that there exists such that
[TABLE]
we have for any small ,
[TABLE]
for some , and hence the plastic junction is not rigid. But (6.50) simplifies to
[TABLE]
If or , that is if or if , then by hypothesis 3 in Theorem 5.1 the case reduces to case or case below. Otherwise, since and are linearly independent, there exists an open neighbourhood of [math] such that and are linearly independent for any . Taking in account that all the terms in (6.51) are orthogonal to , (6.51) is solvable for some . As a consequence the junctions are not locally rigid.
Case and case . In these cases Proposition 4.1 guarantees the existence of a one parameter family of solutions respectively given by
[TABLE]
In the cases (I,I) and (II,II), we respectively have
[TABLE]
By arguing as in the case and the case we can deduce that, as long as and then the plastic junctions constructed in the case and in the case are not locally rigid. But we notice that, given and as in (6.52) this never occurs, concluding that no local rigidity holds for these junctions.
Case and case . In these cases there exists plastic junctions if and only if , and . Let now be a rotation of angle and axis In this case we can solve explicitly
[TABLE]
in terms of , and deduce that the unique solution is given by
[TABLE]
In this case, however,
[TABLE]
for some depending on . Therefore, also in this case no local rigidity holds.
The verification of the Theorem is thus completed.
6.2 Comparison with experimental results
We now compare the results obtained in Theorem 6.1 to the experimental observations in [25] for Ti74Nb23Al3. We recall that for Ti74Nb23Al3, junctions with are observed only for , with equal to and . This is coherent with the result in Theorem 6.1. Indeed, although Theorem 6.1 predicts the existence of junctions also for the cases equal to and , Figure 7 shows that the energy required for a single slip in these cases is consistently bigger than the energy required in the case equal to and .
If we approximate the transformation matrices for the phase transition in Ti74Nb23Al3 with the matrices in (6.42) with , we get that, in some regions of the domain, the shear amount required to form junctions in the cases equal to and , is about ten times bigger than in the case equal to and . Therefore, one can explain the lack of junctions between and , with equal to and with the fact that they are energetically expensive. We report the above discussed results in Table 1.
Another factor influencing the presence of junctions may be the norm of the dislocation density tensor (see e.g., [31]). For junctions as in Definition 5.3 we have that is a Radon measure and \nabla\times\mathsf{F}^{p}=\bigl{(}\bar{t}_{1}\boldsymbol{\phi}_{1}\otimes\boldsymbol{\psi}_{1}-\bar{t}_{2}\boldsymbol{\phi}_{2}\otimes\boldsymbol{\psi}_{2}\bigr{)}\times\mathbf{m}\,\mathscr{H}^{2}\,\mathbin{\vrule height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule height=0.55974pt,depth=0.0pt,width=5.59721pt}\{\mathbf{x}\cdot\mathbf{m}=0\}. Here is the two-dimensional Hausdorff measure restricted to the plane , and the cross product is taken row-wise. We report in Figure 11 the values of |\bigl{(}\bar{t}_{1}\boldsymbol{\phi}_{1}\otimes\boldsymbol{\psi}_{1}-\bar{t}_{2}\boldsymbol{\phi}_{2}\otimes\boldsymbol{\psi}_{2}\bigr{)}\times\mathbf{m}| for the the constructed junctions. Again, the results confirm that the cases equal to and are more preferable than the cases equal to and .
7 Concluding remarks
In Section 5 we provided a mathematical characterisation of junctions in martensitic transformations. Our junctions are weak local minimisers of a physically relevant energy introduced in Section 2. In Section 6 we have showed that our model is successful in capturing the junctions observed in Ti74Nb23Al3. There are nonetheless a few directions in which the present work can be extended or improved.
Despite junctions look very similar to the inexact junctions observed in Ni65Al35 [9, 13], the theory developed in this paper cannot be applied to that case. This is mainly for three reasons: first, as reported in [8] elastic distortions are experimentally observed and seem to play an important role for the formation of incompatible junctions in Ni65Al35. Second, when considering average deformation gradients like laminates (and hence a relaxed elastic energy), one should also consider average plastic shears (and thus a relaxed plastic energy). In that case, also the compatibility results of Section 4 should be re-proven. Third, it seems that a rigidity argument based on the separation of wells as the one in the proof of Theorem 5.1 does not work for a relaxed elastic energy.
The aim of this work is to study junctions; it would be interesting to understand also junctions within this framework. This would allow to better understand nucleation of martensite in Ti74Nb23Al3. Indeed, as reported in [25], nucleation in Ti74Nb23Al3 occurs mostly through the formation of new junctions. However we were not able to find a mathematical characterisation of junctions which is both simple and well-defined, as in this case one should consider plastic deformations both in austenite and in the martensite plates. This will hopefully be discussed in future work.
In our opinion, taking in account small elastic effects would improve the physical accuracy of the model discussed in Section 2, but would make the proof of local stability much harder. The context of linear elasto-plasticity and the geometrically linear theory of elasticity for martensitic transformations (see e.g., [12]) may provide a better framework to approach this problem analytically. Indeed, in geometrically linear elasticity the composition of subsequent deformations reduces to summing the respective deformation gradients, rather than multiplying them as in the context of nonlinear elasticity. Therefore, by giving up some accuracy in the model, this theory guarantees a more approachable framework for analytic results. Examples of recent studies of martensitic transformation within this context are [17, 14, 35]. However, we remark that in some particular cases the nonlinear elasticity theory and the geometrically linear theory may give different results (cf. the case of triple stars in [15, Sec. 2-3]).
Acknowledgments
This work was partially supported by the Engineering and Physical Sciences Research Council [EP/L015811/1]. The author would like to thank John Ball, Tomonari Inamura and Angkana Rüland for the useful discussions. The author would like to acknowledge the two anonymous reviewers for improving this paper with their comments.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] R.A. Adams and J.J.F. Fournier. Sobolev spaces , volume 140. Elsevier, 2003.
- 2[2] K. Anguige and P.W. Dondl. Energy estimates, relaxation, and existence for strain-gradient plasticity with cross-hardening. In Analysis and computation of microstructure in finite plasticity , pages 157–173. Springer, 2015.
- 3[3] J.M. Ball, C. Chu, and R.D. James. Hysteresis during stress-induced variant rearrangement. Le Journal de Physique IV , 5(C 8):C 8–245, 1995.
- 4[4] J.M. Ball and R.D. James. Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. , 100(1):13–52, 1987.
- 5[5] J.M. Ball and R.D. James. A characterization of plane strain. Proc. Roy. Soc. London Ser. A , 432(1884):93–99, 1991.
- 6[6] J.M. Ball and R.D. James. Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. R. Soc. Lond. A , 338(1650):389–450, 1992.
- 7[7] J.M. Ball and K. Koumatos. Quasiconvexity at the boundary and the nucleation of austenite. Arch. Ration. Mech. Anal. , 219(1):89–157, 2016.
- 8[8] J.M. Ball and D. Schryvers. The formation of macrotwins in Ni Al martensite. In IUTAM Symposium on Mechanics of Martensitic Phase Transformation in Solids , pages 27–36. Springer, 2002.