Beyond the classical Cauchy-Born rule Andrea Braides SISSA via Bonomea 265 Trieste Italy

2025-04-27 0 0 2.11MB 117 页 10玖币

侵权投诉

Beyond the classical Cauchy-Born rule

Andrea Braides

SISSA, via Bonomea 265, Trieste, Italy

Andrea Causin and Margherita Solci

DADU, Universit`a di Sassari

piazza Duomo 6, 07041 Alghero (SS), Italy

Lev Truskinovsky

PMMH, CNRS - UMR 7636 PSL-ESPCI,

10 Rue Vauquelin, 75005 Paris, France

Abstract

Physically motivated variational problems involving non-convex energies are often for-

mulated in a discrete setting and contain boundary conditions. The long-range interactions

in such problems, combined with constraints imposed by lattice discreteness, can give rise

to the phenomenon of geometric frustration even in a one-dimensional setting. While non-

convexity entails the formation of microstructures, incompatibility between interactions

operating at diﬀerent scales can produce nontrivial mixing eﬀects which are exacerbated

in the case of incommensuration between the optimal microstructures and the scale of the

underlying lattice. Unraveling the intricacies of the underlying interplay between non-

convexity, non-locality and discreteness, represents the main goal of this study. While in

general one cannot expect that ground states in such problems possess global properties,

such as periodicity, in some cases the appropriately deﬁned ‘global’ solutions exist, and

are suﬃcient to describe the corresponding continuum (homogenized) limits. We interpret

those cases as complying with a Generalized Cauchy-Born (GCB) rule, and present a new

class of problems with geometrical frustration which comply with GCB rule in one range

of (loading) parameters while being strictly outside this class in a complimentary range.

A general approach to problems with such ‘mixed’ behavior is developed.

1 Introduction

Variational problems emerging from applications are often both discrete and non-convex. Im-

portant examples include one-dimensional boundary-value problems with translation-invariant

energy densities describing pairwise interactions. Such problems constitute the main subject

of this paper.

arXiv:2210.06147v2 [math.AP] 15 Oct 2022

The representative energies for this class of problems can be written in the following generic

form

F(w;k) = mink

i,j=0

fi−j(ui−uj) : u0= 0, uk=w,(1.1)

where for every nnatural number fnis a potentially nonconvex energy governing interactions

between the lattice points at distance n, and the minimum is searched among k+ 1-arrays

(u0, . . . , uk). We may assume that f0= 0. As the parameter kincreases and more interactions

are taken into account, a question arises about the behavior of minimal arrays (uk

0, . . . , uk

and of the corresponding minimal energy. One of the most important issues concerns the

existence of a continuum limit of the type Fhom(u) = RIfhom(u0)dt, with Ian interval in

which the nodes iin (1.1) are identiﬁed as a discrete subset (e.g., I= [0,1] where the discrete

subset is 1

kZ∩[0,1]). The single function fhom is expected to carry, in a condensed way, all

the relevant information about the inﬁnite set of functions fnfrom (1.1).

To track the asymptotic behavior of the minimum values in (1.1) we can use the average

derivative z=w/k as a parameter, and scale the energy by k. Then, under assumptions on a

suitably fast decay of fnwith respect to n, it can be shown that the limiting energy density

fhom exists and can be expressed by the formula

fhom(z) = lim

k→+∞

kmink

i,j=0

fi−j(ui−uj) : u0= 0, uk=kz.(1.2)

Moreover, it can be shown that the function fhom is convex in the parameter z. This result

represents a particular case of a more general variational theory for limits of lattice energies

(see e.g. [3]); it can be also seen as a zero-temperature limit of the analogous result in Statistical

Physics ([81, 79]). However, formula (1.2) is only a formal homogenization result in a discrete-

to-continuum setting which is usually non-constructive. In this paper we are raising the issue

of the actual computability of fhom(z).

Explicit formulas for fhom(z) in terms of fnare known only in few cases, most of which are

mentioned below. In general, it is known that the behavior of minimizing arrays (uk

0, . . . , uk

k) at

ﬁxed z, may be complex, including equi-distribution (‘crystallization’; see e.g. [66]), periodic

oscillations [24, 49], development of discontinuities (fracture in lattice models [84, 27]) or

defects (internal boundary layers [22]).

A robust approach to the computation of fhom(z) is known under the name of Cauchy-

Born (CB) rule and is applicable under some restrictive conditions ([41, 15]). It is based on

the assumption that the homogenized energy can be computed using the aﬃne interpolations

uj=zj and relying exclusively on problems with ﬁnite k. Various suﬃcient conditions for the

validity of the Cauchy-Born rule have been obtained by a number of authors mostly in the

context of local minimizers [40, 58, 67, 76, 85, 34, 86]. While those results are usually valid

only for subsets of loading parameters, they are often applicable for dimensions higher than

one. They are of considerable interest, ﬁrst of all, for the development of numerical methods

because the applicability of the classical CB rule makes such methods extremely eﬃcient, even

if for a limited set of boundary conditions. The diﬀerence of our approach to (1.2) is that

we are interested in global minimization (viewed as a zero temperature limit of a statistically

equilibrium response) and consider the possibility that the conventional CB rule is operative

only in a subset of the loading parameters while in the complementary subset the CB strategy

should be appropriately generalized or even completely ruled out.

The main reason for the failure of the classical Cauchy-Born rule is the geometrical frus-

tration caused by incompatible optimality demands imposed by (generically non-convex and

long range) potentials fnwith positive integer nand the discreteness of the lattice. More

speciﬁcally, while non-convexity entails the formation of microstructures, incompatibility be-

tween interactions operating at diﬀerent scales can produce nontrivial mixing eﬀects which

are exacerbated in the case of incommensuration between the optimal microstructures and the

scale of the underlying lattice. Unraveling the intricacies of the underlying interplay between

non-convexity, non-locality and discreteness, represents the main goal of this study.

If the classical Cauchy-Born rule fails, the natural task is to search for a nontrivial gen-

eralization of the Cauchy-Born rule. In this perspective, we pose the problem of ﬁnding the

conditions for which the minimal arrays in (1.1) have ‘global’ features in the sense that solv-

ing a ‘local’ problem on a ﬁnite domain opens the way towards describing the limit in (1.2).

More speciﬁcally, the question is whether the limiting energy fhom(z) can be approximately

computed by solving a ﬁnite set of ‘cell’ problems modeled on (1.1) and potentially producing

non-aﬃne optimal conﬁgurations. The validity of the so-interpreted generalized Cauchy-Born

(GCB) rule would then require that even if the implied ‘local’ problems could be solved only

on some subsets of parameters, the knowledge of the corresponding solutions would ensure the

recovery of the macroscopic (homogenized) energy in the whole range of loading parameters.

Note that in local problems like (1.1) the presence of interactions fnwith n∈ {1, . . . , k}

requires kboundary conditions on each side. By ﬁxing parametrically only the average strain

zin (1.2) we eﬀectively assume that the remaining boundary conditions are natural. This

simplifying assumption may stay on the way of acquiring, for the given ‘local’ problem, the

corresponding ‘global’ features. That is why we will understand the ‘local’ GCB problem

as having the right boundary conditions to ensure the recovery of the macroscopic energy

fhom(z). The simplest case is when the value of fhom(z) can be achieved on arrays such that

i7→ ui−zi is periodic with a given period, but in general one should be allowed to adjust

boundary conditions accordingly while keeping in mind that these changes should not aﬀect

the minimizers in an asymptotic sense.

We now illustrate the main diﬃculties on the way of generalizing the classical CB rule

with some known cases. We start with the simplest example where the conventional CB rule

works trivially. It is the case of convex nearest-neighbor (NN) interactions; i.e., when fn= 0

for all n≥2, and f1=fis a strictly convex function. In this case, the unique minimizer

of the problem in (1.2) is the aﬃne interpolation uk

j=zj. It is independent of kand hence

‘global’: in this case the classical Cauchy-Born rule is applicable in its simplest form, and

fhom(z) = f(z).

If we make the above example only a little more complex considering also convex next-

to-nearest-neighbour (NNN) interactions; i.e., fn= 0 for all n≥3, with f1and f2convex

functions, we loose this exact characterization of the minimal arrays. However, the discrepancy

between uk

jand zj decays fast away from the endpoints j= 0 and j=kof the array.

A slight adjustment of the boundary-value problems, say by imposing additional boundary

conditions u1=zand uk−1=z(k−1) (which do not inﬂuence the asymptotic value of

the minima in (1.2)) reestablishes the aﬃne interpolations uk

j=zj as minimizers, so that

fhom(z) = 2(f1(z) + f2(2z)). In this case the classical Cauchy-Born rule is applicable, given

that we modify boundary conditions in the ‘cell’ problem. Note that this analysis extends to

any suﬃciently fast decaying set of convex potentials fn, giving fhom(z) = 2 P∞

n=1 fn(nz).

Even if we abandon the convex setting, we may still easily describe the behavior of min-

imum problems in (1.2) in the case of nearest-neighbor interaction, with f1=f. It can be

shown that fhom in (1.2) is given by the convexiﬁcation f∗∗ of the NN potential [25]. However

the classical Cauchy-Born rule in this case has to be properly generalized. Suppose, for in-

stance, that the potential fhas a double-well form. In this case the relaxation points towards

conﬁgurations containing mixtures of the two energy wells. Since in this setting there are no

obstacles to simple mixing, the relaxation strategy providing fhom is straightforward. Indeed,

for each zthere exist z1,z2,θ∈[0,1] such that f∗∗(z) = θf(z1) + (1 −θ)f(z2). Hence, we

can construct a function uz:Z→Rwith uz

i−uz

i−1∈ {z1, z2},uz

0= 0 and |uz

i−iz| ≤ C.

Such uzmay be chosen periodic, if θis rational, or quasiperiodic (loosely speaking, as the

trace on Zof a periodic function with an irrational period) otherwise. In both cases we obtain

‘local’ minimizers with ‘global’ properties which allows one to talk about the applicability of

the GCB rule.

The situation is more complex in the case when non-convexity is combined with frustrated

(incompatible) interactions. To show this eﬀect in the simplest setting it is suﬃcient to

account for nearest-neighbor and next-to-nearest-neighbor interactions only and we make the

simplest nontrivial choice by assuming that f1is a ‘double-well’ potential and that f2is a

convex potential. In this case the homogenized potential fhom is also known explicitly [24, 77].

Its domain can be subdivided in three zones: two zones of ‘convexity’ where minimizers are

trivial (as for convex potentials) and a zone where (approximate) minimizers in (1.2) are

two-periodic functions with uz

i−uz

i−1∈ {z1, z2}and z1+z2=z(in a sense, a constrained

non-convex case as above). Hence, in these three zones we have minimizers with a ‘global’

form because the macroscopic energy can be obtained by solving elementary ‘cell’ problems.

One can say that in the two zones of ‘convexity’ the classical CB rule is applicable. In

the ‘two-periodic’ third zone we see that the homogeneity of the minimizers is lost but an

appropriately augmented GCB rule still holds. For the remaining values of zno ‘local’ GCB

rule is applicable since in those cases the unique (up to reﬂections) minimizer is a ‘two-phase’

conﬁguration with aﬃne and two-periodic minimizers coexisting while being separated by a

single ‘interface’ [22]. The frustration (incompatibility) manifests itself in this case through

the impossibility of the penalty-free accommodation of next-to-nearest interactions across such

an internal boundary layer. As a consequence, as kdiverges, such minimizers tend to an aﬃne

interpolation between the ‘convex’ and ‘oscillating’ zones which delivers the correct value of

fhom(z) without being a solution of any ﬁnite ‘cell’ problem. Eﬀectively, the ‘representative

cell’ in this case has an inﬁnite size and therefore no GCB-type ‘local’ description of the

macroscopic state is available. A somewhat similar situation is encountered in continuum

homogenization of both random [62] and strongly nonlinear [23, 72] elastic composites.

In what follows, we interpret the loss of ‘locality’ in homogenization problems, which was

illustrated above on the simplest example, as a failure of the GCB rule. To shed some light

on the mechanism of this phenomenon, we consider below a class of analytically transparent

discrete problems combining nonconvexity with geometrical frustration.

More speciﬁcally, given the complexity of a general asymptotic analysis for even one-

dimensional problems of this type, we limit our attention to a class of discrete functionals of

type (1.1) with f1(z) = 1

2f(z) + m1z2, where the function f(z) is non-convex, and quadratic

fn(z) = f−n(z) = mnz2for n≥2. The coeﬃcients mnwhich introduce nonlocality and frus-

tration, are assumed to be non negative and suﬃciently integrable. In other words, we suppose

that the non-convexity is ‘localized’ in the nearest-neighbor interactions, while all other inter-

actions are quadratic. The positivity of the inﬁnite sequence m={mn:n≥1}is chosen to

ensure that the implied quadratic ‘penalty’ is a measure of the distance of the conﬁguration

uifrom the aﬃne conﬁguration Lz(i) = zi and can be then seen as a non-local version of the

gradient of u−Lz. One can also say that such penalization brings anti-ferromagnetic inter-

actions; an alternative, ferromagnetic-type quadratic penalty, was considered, for instance, in

[80].

The advantage of this choice of fnis that the ensuing problem can exhibit both ‘local’

(GCB) and ‘global’ behavior depending on the structure of the sequence of scalar parameters

mn. Therefore our goal will be to use the chosen class of functionals to characterize the

diﬀerence between CB, GCB and non-GCB problems in terms of such sequences. We show

that in this naturally limited but still suﬃciently rich framework one can precisely specify

the factors preventing the GCB-type description of the macroscopic energy and pointing

instead towards the non-GCB nature of the minimizers. Moreover, the considered example

allow us to abstract some general technical tools which can facilitate the detection and the

characterization of the non-GCB asymptotic behavior in more general minimization problems.

We reiterate that even in the absence of an adequate ‘cell’ problem, the ensuing value of

fhom(z) is fully determined by the homogenization formula which in our case takes the form

fhom(z) = b

Qmf(z) where

Qmf(z) = lim

k→+∞

kminnk

i=1

f(ui−ui−1) +

i,j=0

mi−j(ui−uj)2:u0= 0, uk=kzo.(1.3)

The nontrivial part of the mapping b

Qmf, accentuating the nonlinearity of the problem, is

carried by the operator Qmf(z) = b

Qmf(z)−2Pn≥1mnn2z2.Thus, if fis convex, this

mapping, to which we refer as the m-transform of f, is the identity; actually, the same

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：

BeyondtheclassicalCauchy-BornruleAndreaBraidesSISSA,viaBonomea265,Trieste,ItalyAndreaCausinandMargheritaSolciDADU,UniversitadiSassaripiazzaDuomo6,07041Alghero(SS),ItalyLevTruskinovskyPMMH,CNRS-UMR7636PSL-ESPCI,10RueVauquelin,75005Paris,FranceAbstractPhysicallymotivatedvariationalproblemsinvolvingno...

展开>> 收起<<

Beyond the classical Cauchy-Born rule Andrea Braides SISSA via Bonomea 265 Trieste Italy.pdf

共117页,预览5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！

Beyond the classical Cauchy-Born rule Andrea Braides SISSA via Bonomea 265 Trieste Italy

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: