ON RIGIDITY FOR THE FOUR-WELL PROBLEM ARISING IN THE CUBIC-TO-TRIGONAL PHASE TRANSFORMATION ANGKANA R ULAND AND THERESA M. SIMON

2025-05-02 0 0 504.1KB 19 页 10玖币

侵权投诉

ON RIGIDITY FOR THE FOUR-WELL PROBLEM ARISING IN THE

CUBIC-TO-TRIGONAL PHASE TRANSFORMATION

ANGKANA R ¨

ULAND AND THERESA M. SIMON

Abstract. We classify all exactly stress-free solutions to the cubic-to-trigonal phase trans-

formation within the geometrically linearized theory of elasticity, showing that only simple

laminates and crossing-twin structures can occur. In particular, we prove that although this

transformation is closely related to the cubic-to-orthorhombic phase transformation, all its

solutions are rigid. The argument relies on a combination of the Saint-Venant compatibil-

ity conditions together with the underlying nonlinear relations and non-convexity conditions

satisﬁed by the strain components.

1. Introduction

Shape-memory alloys are materials with a thermodynamically very interesting behaviour:

They undergo a diﬀusionless, solid-solid phase transformation in which symmetry is reduced.

More precisely, a highly symmetric high temperature phase, the austenite, transforms into a

much less symmetric low temperature phase, the martensite, upon cooling below a certain critical

temperature [Bha03]. Mathematically, these materials have very successfully been described by

an energy minimization [BJ92] of the form

Ω

W(∇u, θ) dx→min .(1)

Here Ω ⊂R3denotes the reference conﬁguration, which often is chosen to be the austenite state

at the critical temperature θc>0, the deformation of the material is u: Ω →R3, the temperature

is denoted by θ: Ω →[0,∞) and W:R3×3

+×[0,∞)→[0,∞) corresponds to the stored energy

function. This function encodes the physical properties of the material and is assumed to be

(i) frame indiﬀerent, i.e. W(F, θ) = W(QF, θ) for all F∈R3×3

+and Q∈SO(3),

(ii) invariant with respect to the material symmetry, i.e. W(F, θ) = W(F H, θ) for H∈ Pa

where Padenotes the symmetry group of the austenite phase, which we assume to strictly

include the symmetry group of the martensite phase.

Here (i) can be viewed as a geometric nonlinearity, while (ii) encodes the main material nonlin-

earity which, for instance, reﬂects the transition from the highly symmetric austenite to the less

symmetric martensite phase. Both structure conditions imply that the energies in (1) are highly

non-quasiconvex and thus give rise to a rich energy landscape. As a result, minimizing sequences

can be rather intricate, which physically leads to various diﬀerent microstructures.

In this note, it is our objective to study a speciﬁc phase transformation for which experimen-

tally interesting microstructures are observed. Seeking to capture “crossing-twin structures” in a

fully three-dimensional model (see Figure 1, left), we focus on the cubic-to-trigonal phase trans-

formation in three dimensions. This deformation, for instance, arises in materials such as Zirconia

or in Cu-Cd-alloys but also in the cubic-to-monoclinic transformation in CuZnAl. We refer to

[Sim97,PS69] for experimental studies, to [HS00] for an investigation of special microstructures

in a geometrically nonlinear context and to [BD00,BD01] for mathematical relaxation results for

arXiv:2210.04304v1 [math.AP] 9 Oct 2022

2 ANGKANA R ¨

ULAND AND THERESA M. SIMON

strains possible normals (aij , nij )

e(1), e(2) [1,0,0],[0,4,4]

e(1), e(3) [0,0,1],[4,4,0]

e(1), e(4) [0,1,0],[4,0,4]

e(2), e(3) [0,1,0],[−4,0,4]

e(2), e(4) [0,0,1],[−4,4,0]

e(3), e(4) [1,0,0],[0,4,−4]

Table 1. The possible normals arising in simple laminate constructions. For

these the strain alternates between the two strain values given in the left column

of the table.

the associated geometrically nonlinear problems. Since the study of the minimization problem

(1) can be rather complex, in this note we make the following three simplifying assumptions

which are common in the mathematical analysis of martensitic phase transformations:

•We ﬁx temperature below the transition temperature,

•we consider only the material nonlinearity while linearizing the geometric nonlinearity,

•and we study only exactly stress-free structures.

Instead of investigating the full minimization problem (1), we thus study the diﬀerential inclusion

e(u)∈ {e(1), e(2), e(3), e(4)}in T3,(2)

where

e(1) =



d11 1

1d21

1 1 d3



, e(2) =



d1−1−1

−1d21

−1 1 d3



,

e(3) =



d11−1

1d2−1

−1−1d3



, e(4) =



d1−1 1

−1d2−1

1−1d3



,

(3)

and d1, d2, d3are material-speciﬁc constants. In order to avoid additional mathematical diﬃcul-

ties, we assume that the reference conﬁguration is given by the torus T3:= T1×T2×T3, where

Ti:= [0, λi) for some λi>0 and i∈ {1,2,3}.

We observe that all the matrices in (3) are symmetrized rank-one connected, i.e. for each

i, j ∈ {1,2,3,4}there exist aij ∈R3\ {0}, nij ∈S2such that

e(i)−e(j)=1

2(aij ⊗nij +nij ⊗aij ).

It is well-known that, as a consequence, the diﬀerential inclusion (2) thus allows for so-called twin

or simple laminate solutions, i.e. solutions u(x) = u(nij ·x) with nij ∈S2denoting the vectors

from above. These are rather rigid, one-dimensional structures, which are frequently observed

in experiments [Bha03], see also Figure 1, right. The possible pairs (aij , nij )∈R3×S2for the

cubic-to-trigonal phase transformation are collected in Table 1.

Contrary to other materials such as alloys undergoing a cubic-to-tetragonal phase transfor-

mation, simple laminates are not the only possible solutions to (2). As in the (more complex)

cubic-to-orthorhombic phase transformation, also in the cubic-to-trigonal phase transformation

“crossing-twin structures” can emerge. These are two-dimensional structures involving “lam-

inates within laminates” (see Figure 1, left). In particular, these patterns locally consist of

zero-homogeneous deformations which involve speciﬁc “corners” which are formed by four dif-

ferent variants of martensite.

RIGIDITY FOR THE FOUR-WELL PROBLEM 3

1.1. The main result. As our main result, we classify all solutions to the diﬀerential inclusion

(2) and prove that in addition to the simple laminate solutions only crossing twin structures

arise.

Theorem 1. Let e(u) := 1

2(∇u+ (∇u)t)with e(u) : T3→R3×3

sym be a periodic symmetrized

gradient. Assume that

e(u)∈ {e(1), e(2), e(3), e(4)}.

Then the following structure result holds:

(i) There exists j∈ {1,2,3}such that ∂je(u) = 0.

(ii) Assuming that j= 2, there exist functions f1:T1→Rand f3:T3→Rsuch that

f1, f3∈ {−1,1}and either e13(u)(x) = f1(x1)for a.e. x∈Ωor e13(u)(x) = f3(x3)for

a.e. x∈Ω.

(iii) Assuming that e13(u)(x) = f3(x3), consider Φ(s, t) := (t−F3(s), s), where F3(s)is such

that F0

3(s) = f3(s)a.e. and F3(0) = 0. Then, there exists g: Φ−1(T3)→R,(s, t)7→ g(t)

such that

(e12 ◦Φ)(s, t) = g(t),(e23 ◦Φ)(s, t) = f3(s)g(t).

Remark 2. All cases not listed result from the symmetries of the model under permutation of

the space directions, as these only permute the side lengths of the torus T3and the constants d1,

d2, and d3, the precise values of these constants do not enter the argument. The permutations

play the following roles: If an index i∈ {1,2,3}has been ﬁxed, the other two can be exchanged

via transposition. A ﬁxed index can be transformed into a diﬀerent ﬁxed index by a full cyclic

permutation.

Let us comment on this result: From a materials science point of view, it gives a complete

classiﬁcation of exactly stress-free solutions for the cubic-to-trigonal phase transformation in the

geometrically linear framework. Mathematically, Theorem 1provides a rigidity result for a phase

transformation which leads to more complex structures than simple laminates. While a similar

classiﬁcation and rigidity result had been obtained in [R¨ul16a] for the cubic-to-orthorhombic

phase transformation in three dimensions, this required strong geometric assumptions on the

smallest possible scales. These assumptions were also necessary as a result of the presence of

convex integration solutions for the corresponding diﬀerential inclusion in the case of the cubic-

to-orthorhombic phase transformation. In contrast, in our model, such conditions are not needed.

Due to the smaller degrees of freedom that are present (four instead of six possible strains), in

fact any exactly stress-free solution must satisfy the structural conditions and “wild” convex

integration solutions are ruled out.

1.2. Relation to the cubic-to-orthorhombic phase transformation. Due to the outlined

rather diﬀerent behaviour (in terms of rigidity and ﬂexibility) of stress-free solutions of the cubic-

to-orthorhombic and the cubic-to-trigonal phase transformation, we explain the algebraic relation

between these two transformations: To this end, we recall that for the cubic-to-orthorhombic

phase transformation, the exactly stress-free setting in the geometrically linearized situation

corresponds to the diﬀerential inclusion

e(u)∈ {e(1), . . . , e(6)},

4 ANGKANA R ¨

ULAND AND THERESA M. SIMON

e(4) e(4)

e(3) e(3

e(3) e(3)

e(1)

e(2)

[0,1,0]

[1,−1,0]

e(1)

e(2)

Figure 1. Schematic illustration of a crossing twin structure (left) and a simple

laminate (right). Only very speciﬁc twins can be used to form crossing twin

structures. These are obtained as a consequence of the compatibility conditions

satisﬁed by the strain equations.

with

e(1) := 



1δ0

δ1 0

0 0 −2



, e(2) := 



1−δ0

−δ1 0

0 0 −2



, e(3) := 



1 0 δ

0−2 0

δ0 1



,

e(4) := 



1 0 −δ

0−2 0

−δ0 1



, e(5) := 

−200

0 1 δ

0δ1



, e(6) := 

−2 0 0

0 1 −δ

0−δ1



.

Here δ > 0 is a material dependent parameter. If now one assumes that a microstructure only

involves the inﬁnitesimal strains {e(1), . . . , e(4)}and if one carries out the change of coordinates

x7→ ˆx:= C−tx,u7→ ˆu:= Cu with

C:= 





√30 0

0√3

√2δ0

0 0 1

√3





·



0 1 1

√2 0 0

0 1 −1



,

using that the (inﬁnitesimal) strain transforms according to e(ˆu) = Ce(u)Ct, one exactly arrives

at the diﬀerential inclusion (2) for the cubic-to-trigonal phase transformation with the parameters

d1=−1

3,d2=3

δ2,d3=−1

3. This shows that the diﬀerential inclusion for the cubic-to-trigonal

phase transformation indeed corresponds to a subset of the diﬀerential inclusion for the cubic-

to-orthorhombic phase transformation. Due to the fewer degrees of freedom, in contrast to the

full cubic-to-orthorhombic phase transformation, it however displays strong rigidity properties.

1.3. Main ideas. The arguments for the proof of Theorem 1rely on a combination of the

linear compatibility conditions for strains in the form of the Saint-Venant equations and the

nonlinear constraints in our model. More precisely, the Saint-Venant conditions imply structural

conditions on the possible space dependences of the strains. Furthermore, the full classiﬁcation

result requires a breaking of symmetries that can only be deduced in combination with the non-

convexity of the problem, i.e., the fact that for all i, j ∈ {1,2,3}we have that eij (u) attains

at most three possible values and the nonlinear relation e23 −e12e13 = 0. For a simpliﬁed

model with only two-dimensional dependences similar arguments had earlier been considered in

[R¨ul16a]. However, in contrast to [R¨ul16a], in the present setting we do not need to make use of

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

ONRIGIDITYFORTHEFOUR-WELLPROBLEMARISINGINTHECUBIC-TO-TRIGONALPHASETRANSFORMATIONANGKANARULANDANDTHERESAM.SIMONAbstract.Weclassifyallexactlystress-freesolutionstothecubic-to-trigonalphasetrans-formationwithinthegeometricallylinearizedtheoryofelasticity,showingthatonlysimplelaminatesandcrossing-twins...

展开>> 收起<<

ON RIGIDITY FOR THE FOUR-WELL PROBLEM ARISING IN THE CUBIC-TO-TRIGONAL PHASE TRANSFORMATION ANGKANA R ULAND AND THERESA M. SIMON.pdf

共19页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

ON RIGIDITY FOR THE FOUR-WELL PROBLEM ARISING IN THE CUBIC-TO-TRIGONAL PHASE TRANSFORMATION ANGKANA R ULAND AND THERESA M. SIMON

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: