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HEMIHELICAL LOCAL MINIMIZERS IN PRESTRAINED ELASTIC BI-STRIPS MARCOCICALESE,FRANCESCOSOLOMBRINO,ANDMATTHIASRUF Abstract. Weconsideradoublelayeredprestrainedelasticrodinthelimitofvanishingcrosssection. For the resulting limit Kirchoff-rod model with intrinsic curvature we prove a supercritical bifurcation 7 result, rigorously showing the emergence of a branch of hemihelical local minimizers from the straight 1 configuration, at a critical force and under clamping at both ends. As a consequence we obtain the 0 existenceofnontriviallocalminimizersofthe 3-dsystem. 2 n a J 9 1. Introduction 1 The derivation of the elastic energy of a thin object as limit of the elastic energy of the three- ] dimensional body when its thickness vanishes has recently gained increasing attention in the case of P prestrained bodies. Many authors contributed to this topics, both in the case of 3-d to 2-d dimension A reduction as in [4, 11, 12, 15] and in the case of 3-d to 1-d dimension reduction as in [5, 8, 10] (see also . h [1, 2] for a similar problem in the theory of nematic elastomers). t a As a model example in the 3-d to 1-d case in [5], motivated by recent experiments in [14], we have m considered the following model: given two strips of elastomers of the same initial width, but unequal [ length,onestretchestheshortoneuniaxiallytobeequalinlengthtothelongeroneandthengluesthem together side-by-side along their length. In such a way a bi-strip system is formed in which the initially 1 shorter strip is under a uniaxial prestrain. The bi-strip is made flat by the presence of a terminal load v which is gradually released so that it starts to bend and twist out of plane. According to [14] the system 9 6 may evolve towards either a helical or a hemihelical shape, more complex structure in which helices with 4 different chiralities seem to periodically alternate, leading to the formation of the so-called perversions. 5 Inthepaper[5]wehaveanalyzedthe 3-dto 1-dlimitofthebi-stripsystemabovevia Γ-convergence, 0 rigorously proving that for small prestrains the Kirchoff-rod energy with intrinsic curvature is the right . 1 variationalapproximationofthemodel.Amongotherthingswehavealsoprovedapreliminarybifurcation 0 result, showing theemergenceof a branch ofnontrivial stationary pointsfrom the straight configuration, 7 at a critical force f and under clamping at both ends. Furthermore, along the lines of a result by 1 crit Kohn and Sternberg ([9]), we have proved the existence of local minimizer for the 3d-model arbitrarily : v close to the limiting ones. Such a result however could not be applied to the branch of critical points we i X found, since we had no information about their stability. r a Inthepresentpaperwefillthisgapbyathoroughanalysisestablishingthatthebranchofstationary points found in [5] is locally made of exactly two non trivial strict local minimizers, close to the straight configuration, for any given value of the force in a left neighborhood of the critical force f (see the crit bifurcation diagram in Fig 1). Taking into account our sign convention, motivated by the mechanical experiment we have in mind, this corresponds to a supercritical bifurcation diagram. As we discuss in Remark 4.5, these local minimizers have a so-called hemihelical structure. The emergence of such configurations is a well-known experimental fact in the mechanical literature (see [13, Introduction] for an heuristic description of the phenomenon) which finds now a full mathematical justification. We additionally prove that for those values of the forces, the local minimizers have less energy than the straight configuration,aninformationthatcanbeimportanttostudyevolutionaryproblems.Furthermore,thanks to our result in [5], existence of 3-d nontrivial local minimizers arbitrarily close to these configurations immediatelyfollows(seeTheorem4.6).Comingbacktotheexperimentin[14],itisworthnoticing,aswe discuss in Remark 3.5, that our result do not cover the case of multiple perversions for which a different 1 2 MARCOCICALESE,FRANCESCOSOLOMBRINO,ANDMATTHIASRUF analysis seems to be needed (see also [10]). R (f ,I) f crit Figure 1. A local bifurcation diagram of the energy. The bold lines indicates stable configurations. The main idea behind the proof of our main result (Theorem 4.4) is a careful combination of variational techniques with classical results about eigenvalue problems in Banach spaces. In particular we showthatiftheseconddifferentialoftheenergyalongacurveofstationarypointsisnotpositivedefinite, then a nontrivial solution to an eigenvalue problem (see (4.10)) appears. The behaviour of the smallest eigenvalue of such a problem can be determined using a classical result by Crandall and Rabinowitz (Theorem2.5).Eventuallythisallowsustofindacontradictionandthenshowthatourstationarypoints areindeedstrictlocalminimizers.Itisworthpointingoutthattheabstractapproachtobifurcationprob- lems(see[3])isusuallyformulatedinspacesofsmoothfunctionsinordertohavehighdifferentiabilityof the involved functionals, while local minimality of our configurations is meaningful in the natural energy spaceofSobolevfunctions.WefillthisgapthankstotheregularityofthesolutionsoftheEuler-Lagrange equation. From a technical point of view, in order to prove these results we need to describe our energy functional in terms of Cardan angles (see Section 3), while for the existence result in [5, Theorem 5.6] we introducedadifferentauxiliaryfunctional.Thislatter,however,wouldbenotusefultoperformthemore detailed analysis contained in the present paper, since its differentials coincide with those of the energy only at first order, while for some auxiliary computations as those in Lemma 3.6 and for the arguments in Theorem 4.4 we need to use differentials of the energy of higher order. Moreover, it is only thanks to the use of Cardan angles that already in Proposition 4.1 we can obtain more information than in [5, Theorem 5.6], namely the local behaviour of the force component of the bifurcation curve, as well as the energy inequality (iii) in the statement. 2. Notation and preliminaries 2.1. Basic notation. We denote by {e ,e ,e } the standard basis in R3, by M3×3 the set of all real- 1 2 3 valued 3×3 matricesandby I theidentitymatrix.Givenamatrix M wedenoteby MT itstransposed matrix (this convention will be also used for row and column vectors). We let SO(3) ⊂ M3×3 be the submanifoldofallrotations,while M3×3 denotesthelinearspaceofthe 3×3 skew-symmetricmatrices. skew Given A∈M3×3 we define ω as the unique vector such that Av =ω ×v for all v ∈R3. skew A A All euclidean spaces will be endowed with the canonical Euclidean norm. The symbol (cid:104)·,·(cid:105) indicates HEMIHELICAL LOCAL MINIMIZERS IN PRESTRAINED ELASTIC BI-STRIPS 3 duality products in euclidean or Banach spaces. For an operator F between Banach spaces, we will denote by N its nullspace and by Rg its range. The derivative of one-dimensional absolutely continuous functions will be denoted by the prime symbol (cid:48). We will use the standard notation D (resp. D2, D3...) for first (resp. second, third...) order (partial) differentials of operators on Euclidean or Banach spaces, while the symbol ∂ will be used for partial derivatives in Rn. Only in this last context, with a slight abuseofnotationwewillsometimesidentifydifferentialswiththecorrespondingderivativesfornotational simplicity. 2.2. A simple experiment with two-layer prestrained beam and its mathematical modeling. The situation which we aim to describe in this paper is the following. We deform a two-layer elastic beam of length L and cross section hS, where S satisfies some symmetry conditions (see (2.1)) and h > 0 is a small parameter. We suppose the physical system (0,L)×hS to be such that the upper layer (0,L)×hS+ is prestrained with a stretching of order hχ > 0 in order to match the lower one. We consider the system in its straight configuration (0,L)×hS to be already at equilibrium under the action of a terminal load fe applied at {L}×hS. Moreover both ends of the beam are kept at a fixed 1 twist by means of a suitable torsional moment (a similar situation is described in [13] under the slightly more general assumption that the total twist of the filament is constant). As h tends to 0, at the onset of instability for the straight configuration, one expects that the mid-fiber of the beam deforms into a new stable configuration showing at least one inversion of curvature. The intrinsic curvature induced by the prestrain would indeed prefer an helical-like configuration which is now forbidden by the boundary conditions. Namely, in order to obtain a helix, one end of the beam must be left free to rotate. We below set the mathematical notation of the problem. Given a small parameter h>0, a stripe of thickness h, mid-fiber (0,L) and cross section S ⊂ R2 is denoted by Ω := (0,L)×hS. On S we will h assume that it is a bounded open connected set having unitary area and Lipschitz boundary. We set S+ :=S∩{x >0}. 3 We moreover assume that S satisfies the following symmetry properties: (cid:90) (cid:90) (cid:90) (cid:90) (2.1) x x dx dx = x dx dx = x dx dx = x dx dx =0. 2 3 2 3 2 2 3 3 2 3 2 2 3 S S S S+ We consider a hyperelastic material and assume a multiplicative decomposition for the strain (see [11]). Denoting by W : M3×3 → [0,+∞] the strain energy density, the stored energy of a deformation u:Ω →R3 is expressed by h (cid:90) E (u)= W(∇u(x)A (x))dx. h h Ωh where the prestrain A :Ω →M3×3 is of the form h h (cid:40) diag(1+hχ,√ 1 ,√ 1 ) if x∈S+, (2.2) A (x):= 1+hχ 1+hχ h I otherwise, and χ>0 can be thought of as the effective strength of the stretching. Throughout the paper we make the following standard assumptions on the density W: (i) W(RF)=W(F) ∀R∈SO(3) (frame indifference), (ii) W(F)≥cdist2(F,SO(3)) and W(I)=0 (non-degeneracy), (iii) W is C2 in a neighborhood U of SO(3) (regularity), (iv) W(FR)=W(F) ∀R∈SO(3) (isotropy). In addition to the stored energy we consider an external boundary force that is meant to describe the loading at one end. We set Γ :={L}×hS. Given a force field f :Γ →R3 we define the total energy h h h as (cid:90) (cid:90) E (u)= W(∇u(x)A (x))dx− (cid:104)f (x),u(x)(cid:105)dH2. h h h Ωh Γh 4 MARCOCICALESE,FRANCESCOSOLOMBRINO,ANDMATTHIASRUF As it is customary when dealing with the variational analysis of thin objects, we perform a change of variables to rewrite the energy on a fixed domain. Setting Ω = Ω and Γ = Γ , we define a rescaled 1 1 deformation field v :Ω→R3, a rescaled prestrain A :Ω→M3×3 and a rescaled force f :Γ→R3 as h h v(x)=u(x ,hx ,hx ), A (x)=A (x ,hx ,hx ), f (x)=f (x ,hx ,hx ). 1 2 3 h h 1 2 3 h h 1 2 3 Introducing the rescaled gradient ∇ v = ∂ v⊗e + 1(∂ v⊗e +∂ v⊗e ), the energy takes the form h 1 1 h 2 2 3 3 E (u)=h2E (v), where h h (cid:90) (cid:90) (2.3) E (v)= W(∇ v(x)A (x))dx− (cid:104)f (x),v(x)(cid:105)dH2. h h h h Ω Γ We are interested in the case where there exists f ∈R such that f (2.4) h (cid:42)fe in L2(Γ) h2 1 as h→0, suggesting a meaningful scaling of E to be E /h2. h h On the thin rod we prescribe the following clamped-clamped boundary conditions at both ends, namely   (cid:90) 0 (2.5) v(0,x2,x3)=v(L,x2,x3)− v(L,x2,x3)dH2 =hx2. S hx 3 To save notation, we introduce the class of admissible deformations as A ={v ∈W1,2(Ω,R3): v satisfies (2.5) in the sense of traces}. h In[5]weused Γ-convergencetechniquestoderiveaneffectiveone-dimensionallimitenergywhen h→0. It has been proved that, due to (2.5), if R ∈ W1,2((0,L),SO(3)) is an L2-limit of the rescaled strains ∇ v , then R has to satisfy the following one-dimensional version of the clamped Dirichlet boundary h h conditions: (2.6) R(0)=R(L)=I; asaconsequenceweintroducethefamilyofthoselimitingconfigurationswithfiniteenergytobedefined as A:={R∈W1,2((0,L),SO(3)): R satisfies (2.6) in the sense of traces}. Notethatwiththisnotationthelimitdeformation v ∈W2,2((0,L),R3) canberecoveredviatheformula v(x )=(cid:82)x1R(t)e dt. In the setting introduced above, up to an additive constant the limit energy takes 1 0 1 the form 1(cid:90) L (2.7) Ef(R)= c a (t)2+c (a (t)−k)2+c a (t)2−2f(cid:104)e ,R(t)e (cid:105)dt, 0 2 12 12 13 13 23 23 1 1 0 where A(t)=RT(t)R(cid:48)(t). The constants c ,c ,c depend on the coefficients of the quadratic form of 12 13 23 linearized elasticity for the energy density W and on the geometry of S, while k encodes the intrinsic curvature caused by the two-layer structure of the prestrain (see Proposition 3.8 in [5]). We now recall one of the main results of [5] that connects isolated local minimizers of the reduced energy functional to local minimizers of the full three-dimensional model. In the statement below local minimality of a deformation v has to be understood in the following sense: we say that v is a local minimizerof E ifthereexists δ >0 suchthat E (v)≤E (w) forall w suchthat (cid:107)∇ v−∇ w(cid:107) ≤δ. h h h h h L2 Notethat,asshownin[5,Proposition3.3],suchadefinitionarisesnaturallyonthesublevelsetsof 1 E . h2 h Theorem 2.1. Assume that the functional E defined in (2.3) is lower semicontinuous with respect to h weak convergence in W1,2(Ω,R3). Moreover let Ef be defined as in (2.7) and R ∈ W1,2((0,L),SO(3)) 0 be a strict local minimizer of Ef in A with respect to the strong W1,2-topology. Then there exists a 0 sequence v of local minimizers of E in A such that v →v strongly in W1,2(Ω,R3) and ∇ v →R h h h h h h strongly in L2(Ω,M3×3). HEMIHELICAL LOCAL MINIMIZERS IN PRESTRAINED ELASTIC BI-STRIPS 5 In [5] we also analyzed the local minimality of the straight configuration for the limit model (2.7). Here we recall also this result under the additional assumption that the following inequality holds: (c k)2 4π2c (2.8) 13 − 12 >0. c L2 23 Such an inequality, which we assume to be true throughout the paper, implies that the critical force, at which local minimality of the straight configuration gets lost, is positive (a natural condition in the experiment described at the beginning of the section). Theorem 2.2 ([5], Theorem 5.2). Let E0f be as in (2.7), set fcrit = (c1c32k3)2 − 4πL22c12 and assume (2.8). Then for f > f the straight configuration R(t) ≡ I is a strict local minimizer of Ef in the L2- crit 0 topology with the boundary conditions (2.6). If instead f < f the straight configuration is not a local crit minimizer. 2.3. Abstract bifurcation results. We now recall two abstract results concerning existence and stability of bifurcation branches, that are fundamental to our analysis. The first one is an existence result proved in [6] which we state below: Theorem 2.3 (Crandall-Rabinowitz, 1971). Let X,Y be Banach spaces, V a neighbourhood of (0,λ ) 0 in X×R and F :V →Y have the properties (i) F(0,λ)=0, (ii) the partial derivatives D F,D F,D2 F exist and are continuous, ϕ λ ϕ,λ (iii) D F(0,λ ) is a Fredholm operator with zero index and N(D F(0,λ ))=span{v}, ϕ 0 ϕ 0 (iv) D2 F(0,λ )v ∈/ Rg(D F(0,λ )). ϕ,λ 0 ϕ 0 If Z is any complement of N(D F(0,λ )) in X, then there is a neighbourhood U of (0,λ ) in X×R, ϕ 0 0 an interval (−a,a) and continuous functions λ : (−δ,δ) → R and ψ : (−δ,δ) → Z such that λ(0) = 0, ψ(0)=0 and (2.9) F−1(0)∩U ={(sv+sψ(s),λ +λ(s)): |s|<δ}∪{(0,s): (0,s)∈U}. 0 If F ∈Cn(V), then ψ,λ,∈Cn−1((−δ,δ)). In order to state the second one, proved in [7, Corollary 1.13 and Theorem 1.16], we need to recall the following Definition. Definition 2.4. Let X,Y beBanachspacesandlet T,K ∈L(X,Y) bebounded,linearoperators.Then µ ∈ R is called a K-simple eigenvalue of T if dim(N(T −µK)) = codim(Rg(T −µK)) = 1 and, if N(T −µK)=span{v }, then Kv ∈/ Rg(T −µK). 0 0 Theorem 2.5 (Crandall-Rabinowitz,1973). In the setting of Theorem 2.3, let K ∈L(X,Y) and assume that 0 is a K-simple eigenvalue of D F(0,λ ). Then there exist open intervals A,B ⊂ R such that ϕ 0 λ ∈ A and 0 ∈ B and continuously differentiable functions γ : A → R, µ : B → R, u : A → X and 0 w :B →X such that D F(0,λ)u(λ)=γ(λ)Ku(λ) for all λ∈A, ϕ (2.10) D F(sv +sψ(s),λ +λ(s))w(s)=µ(s)Kw(s) for all s∈B. ϕ 0 0 It holds that γ(λ )=µ(0)=0, u(λ )=w(0)=v and u(λ)−v ∈Z as well as w(s)−v ∈Z. 0 0 0 0 0 Moreover we have γ(cid:48)(λ ) (cid:54)= 0 and for |s| small enough the functions µ(s) and −sλ(cid:48)(s)γ(cid:48)(λ ) have the 0 0 same zeros and the same sign in the sense that −sλ(cid:48)(s)γ(cid:48)(λ ) lim 0 =1. s→0 µ(s) µ(s)(cid:54)=0 The curve (µ(s),w(s)) satisfying the eigenvalue problem (2.10) is locally uniquely determined by the operator, in a sense made precise by the following Lemma, also proved in [7]. 6 MARCOCICALESE,FRANCESCOSOLOMBRINO,ANDMATTHIASRUF Lemma 2.6 ([7], Lemma 1.3). Let T ,K ∈L(X,Y) be bounded, linear operators and µ be a K-simple 0 0 eigenvalue of T , with N(T −µ K) = span{v }. Then there exist a neigborhood U of T in L(X,Y) 0 0 0 0 1 0 and a neighborhood U of µ in R such that, for all T ∈U , T −µK is singular for a unique µ∈U . 2 0 1 2 Furthermore, µ depends smoothly on T and is itself a K-simple eigenvalue. If Z is a complement of span{v } in X , given (T,µ) ∈ U ×U such that T −µK is singular, there 0 1 2 exists a unique vector w such that w−v ∈Z and 0 Tw =µKw. Also w is a smooth function of T. 2.4. Regularity of stationary points. In our analysis, the functional F appearing in the previous abstractresultswillbe,roughlyspeaking,givenbythefirstvariationoftheenergy (2.7)(uptorewriting it in suitable local coordinates) and the sign of λ(cid:48)(s) will be determined by using the implicit function theorem along the lines of [3, Section 5.4]. This will require some additional differentiability of the functional, which can be ensured in stronger topologies than the one of W1,2. To this end, we need to show a priori that local minimizers of the energy are indeed regular. To see this, we first recall the formula for the first variation of the energy as proved in [5]. Using the short-hand C=diag(c ,c ,c ), stationary points of (2.7) satisfy the integral equality 23 13 12 (cid:90) L 0= (cid:104)Cω (t)−c ke ,RT(t)ω (t)(cid:105)−f(cid:104)e ,(ω (t)×R(t)e )(cid:105)dt A 13 2 B(cid:48) 1 B 1 0 (cid:90) L (2.11) = (cid:104)Cω (t)−c ke ,RT(t)ω (t)(cid:105)+f(cid:104)RT(t)e ×e ,RT(t)ω (t)(cid:105)dt A 13 2 B(cid:48) 1 1 B 0 for all B ∈W1,2((0,L),M3×3 ) This gives C∞-regularity of stationary points, which we state and prove 0 skew in the next lemma for the sake of completeness. Lemma 2.7. Let R ∈ W1,2((0,L),SO(3)) satisfy (2.6) and (2.11). Then R ∈ Wk,1((0,L),SO(3)) for every k ∈N. In particular R∈C∞([0,L],SO(3)). Proof. Withtheadmissibleansatz ω =Rϕ with ϕ∈W1,2((0,L),R3),usingthat R(cid:48) =RA,theintegral B 0 equality (2.11) becomes (cid:90) L 0= (cid:104)Cω (t),ϕ(cid:48)(t)(cid:105)+(cid:104)c kA(t)e −A(t)Cω (t)+fRT(t)e ×e ,ϕ(t)(cid:105)dt. A 13 2 A 1 1 0 NotethatbyHölder’sinequalitytheleftentryinthesecondscalarproductbelongsto L1((0,L)),sothat by definition of distributional derivatives it follows that ω ∈W1,1((0,L),R3) and, again by R(cid:48) =RA, A we deduce R ∈ W2,1((0,L),SO(3)). Inductively we conclude that R ∈ Wk,1((0,L),SO(3)) for every k and by the Sobolev embedding we conclude that R∈C∞([0,L],SO(3)). (cid:3) By the previous lemma stationary points and local minimizers must be regular and solve the system of ODEs (2.12) ω(cid:48) (t)=c kC−1A(t)e −C−1A(t)Cω (t)+fC−1(RT(t)e ×e ). A 13 2 A 1 1 3. The energy in local coordinates In order to work in a linear space, instead than on the manifold W1,2((0,L),SO(3)), we will pre- liminarily rewrite the energy in local coordinates in an L∞ neighborhood of I. To this end we recall the notion of Cardan angles that we use as parameters. We namely define G :R3 →SO(3) as G =G(α,β,γ)   cos(β)cos(γ) −cos(β)sin(γ) sin(β) =sin(α)sin(β)cos(γ)+cos(α)sin(γ) cos(α)cos(γ)−sin(α)sin(β)sin(γ) −sin(α)cos(β). sin(α)sin(γ)−cos(α)sin(β)cos(γ) sin(α)cos(γ)+cos(α)sin(β)sin(γ) cos(α)cos(β) HEMIHELICAL LOCAL MINIMIZERS IN PRESTRAINED ELASTIC BI-STRIPS 7 Itiswell-knownthat G,whenrestrictedto U =(−π,π)×(−π/2,π/2)×(−π,π),isadiffeomorphismfrom U onto an open neighbourhood of I in SO(3). With a slight abuse of notation we will also denote with G the induced mapping from W1,2((0,L),R3) to W1,2((0,L),SO(3)). Its properties are summarized in the following lemma. Lemma 3.1. There exists δ > 0 such that for each R ∈ W1,2((0,L),SO(3)) with (cid:107)R−I(cid:107) < δ there ∞ exists ϕ ∈ W1,2((0,L),R3) with R = G(ϕ). The function ϕ = G−1(R) inherits the differentiability properties of R. If R additionally satisfies (2.6), then ϕ∈W1,2((0,L),R3). 0 Proof. Since G is invertible with smooth inverse in a neighbourhood of I, the claim follows by the chain rule for Sobolev functions. (cid:3) Withthepreviouslemmaathand,werewritetheenergyintermsofCardanangles.Adirectcompu- tation yields that, for a vector-valued function ϕ=(ϕ ,ϕ ,ϕ ), the components of the skew-symmetric 1 2 3 matrix GT(ϕ(t))G(ϕ(t))(cid:48) are given by a (t)=−ϕ(cid:48)(t)sin(ϕ (t))−ϕ(cid:48)(t), 12 1 2 3 a (t)=−ϕ(cid:48)(t)cos(ϕ (t))sin(ϕ (t))+ϕ(cid:48)(t)cos(ϕ (t)), 13 1 2 3 2 3 a (t)=−ϕ(cid:48)(t)cos(ϕ (t))cos(ϕ (t))−ϕ(cid:48)(t)sin(ϕ (t)). 23 1 2 3 2 3 Hence, the one-dimensional energy in (2.7) can be written in terms of Cardan angles as 1(cid:90) L Ef(ϕ)= c (ϕ(cid:48)(t)sin(ϕ (t))+ϕ(cid:48)(t))2+c (ϕ(cid:48)(t)cos(ϕ (t))sin(ϕ (t))−ϕ(cid:48)(t)cos(ϕ (t))+k)2 dt 0 2 12 1 2 3 13 1 2 3 2 3 0 1(cid:90) L (3.1) + c (ϕ(cid:48)(t)cos(ϕ (t))cos(ϕ (t))+ϕ(cid:48)(t)sin(ϕ (t)))2−2fcos(ϕ (t))cos(ϕ (t)) dt. 2 23 1 2 3 2 3 2 3 0 For notational convenience we introduce the integrand g :R3×R3 →R setting f c c g (u,ξ)= 12(ξ sin(u )+ξ )2+ 13(ξ cos(u )sin(u )−ξ cos(u )+k)2 f 2 1 2 3 2 1 2 3 2 3 c + 23(ξ cos(u )cos(u )+ξ sin(u ))2−fcos(u )cos(u ). 2 1 2 3 2 3 2 3 Noticethatthisintegrandisquadraticin ξ andsatisfiesalltheassumptionsin[5,Lemma4.5].Therefore, the same proof yields the following differentiability property: (3.2) Ef ∈C2(W1,2((0,L),R3),R). 0 In order to study the behaviour of the energy close to the critical force, we start with a bifurcation analysis of the angular-energy (3.1). To this end, we need the associated Euler-Lagrange equation given by (3.3) (∇ g (ϕ(t),ϕ(cid:48)(t)))(cid:48) =∇ g (ϕ(t),ϕ(cid:48)(t)) ξ f u f Remark 3.2. Let us observe that if ϕ ∈ C2([0,L],R3) is a strong solution of the system above, then 0 the function G(ϕ) is a stationary point of the functional Ef in (2.7), that is it satisfies (2.11). Indeed, 0 any curve of admissible deformations that is tangential to G(ϕ) can be transformed via Lemma 3.1 to a tangential curve of Cardan angles. Conversely, by the Lemmata 2.7 and 3.1, any stationary point of the functional(2.7)whichissufficientlyclosetotheidentityyieldsCardananglesthatarearegularsolutions of (3.3). We now set f = λ as bifurcation parameter, and study the operator F : C2([0,L],R3) × R → 0 C([0,L],R3) defined as (3.4) F(ϕ,λ)=(∇ g (ϕ(t),ϕ(cid:48)(t)))(cid:48)−∇ g (ϕ(t),ϕ(cid:48)(t)). ξ λ u λ Note that, by definition, for any ϕ∈C2([0,L],R3) and every w ∈W1,2((0,L),R3) it holds 0 0 (3.5) (cid:104)F(ϕ,λ),w(cid:105)=−(cid:104)DEλ(ϕ),w(cid:105). 0 8 MARCOCICALESE,FRANCESCOSOLOMBRINO,ANDMATTHIASRUF On its domain this functional is very regular in the sense of Fréchet-differentiability as stated in the lemma below. Lemma 3.3. The operator F :C2([0,L],R3)×R→C([0,L],R3) is C∞. 0 Proof. The proof is left to the reader. (cid:3) We set (c k)2 4π2c (3.6) λ = 13 − 12 0 c L2 23 and c13kL(cid:0)1−cos(2πt)(cid:1) (3.7) w∗(t)=2c23π 0 L . −sin(2πt) L Weremarkthat,becauseof (2.8), λ >0.Wealsonotethatthethirdcomponentof w∗ hasachange 0 of sign in [0,L], which is due to considering clamped boundary conditions in the eigenvalue problem for D F and will eventually lead to an inversion of curvature of the bifurcating stable configurations (see ϕ Fig 2). Asafirststepofouranalysis,weshowthattheoperator F fulfillstheassumptionofthebifurcation theorem for λ=λ . 0 Lemma 3.4. Let λ and w∗ ∈C2([0,L],R3) begivenby (3.6),and (3.7),respectively.Thentheoperator 0 0 F defined in (3.4) satisfies the assumptions of Theorem 2.3 with v =w∗. Proof. By a direct computation ∇ g (0,0) = ∇ g (0,0) = 0 for all λ. Hence F(0,λ) = 0. Moreover, ξ λ u λ by Lemma 3.3 the operator fulfills the differentiability assumptions (ii). Next, let us calculate the first derivative. We have that (3.8) D F(ϕ,λ)w =(cid:0)∂ ∂ g (ϕ,ϕ(cid:48))w+∂2g (ϕ,ϕ(cid:48))w(cid:48)(cid:1)(cid:48)−∂2g (ϕ,ϕ(cid:48))w−∂ ∂ g (ϕ,ϕ(cid:48))w(cid:48). ϕ ξ u λ ξ λ u λ u ξ λ Plugging in ϕ=0 we obtain by a straightforward calculation that D F(0,λ)w =c w(cid:48)(cid:48)e +c w(cid:48)(cid:48)e +c kw(cid:48)e −c kw(cid:48)e +c w(cid:48)(cid:48)e −λw e −λw e . ϕ 12 3 3 13 2 2 13 3 1 13 1 3 23 1 1 2 2 3 3 Note that the bounded, linear operator T : C2([0,L],R3) → C([0,L],R3) defined by Tw = Cw(cid:48)(cid:48) is 0 bijective. Hence D F(0,λ ) is a compact perturbation of a bijective operator, whence a Fredholm- ϕ 0 operator of index zero. A direct computation shows that w∗ given by (3.7) satisfies D F(0,λ )w∗ =0. ϕ 0 In order to determine the dimension of the kernel, we note that the equation D F(0,λ )w = 0 is a ϕ 0 system of second order linear differential equations. The second component w must satisfy 2 c w(cid:48)(cid:48)−λ w =0. 13 2 0 2 Since we assume λ > 0, by the Dirichlet boundary conditions we immediately get w ≡ 0. For the 0 2 remaining components one can write the equation as a first order four-dimensional system with the constant matrix   0 −c k/c 0 0 13 23 A=c13k/c12 0 0 λ0/c12.  1 0 0 0  0 1 0 0 (cid:26) (cid:113) (cid:27) This matrix as the eigenvalues 0,± λ0 − (c13k)2 , where 0 has algebraic multiplicity 2. Hence the c12 c12c23 solutions are of the form w (t)=a +a t+a exp(2πit/L)+a exp(−2πit/L), 1 1 2 3 4 w (t)=b +b t+b exp(2πit/L)+b exp(−2πit/L), 3 1 2 3 4 HEMIHELICAL LOCAL MINIMIZERS IN PRESTRAINED ELASTIC BI-STRIPS 9 with a ,b ∈C.Pluggingthisansatzintotheequationandcomparingthecoefficientsoftheindependent i i functions, together with the boundary conditions we obtain a 8-dimensional linear system that can be solved explicitly for a one-dimensional kernel spanned by the function w∗ in (3.7). To show the transversality condition D2 F(0,λ )w∗ ∈/ Rg(D F(0,λ )), we argue by contradiction. ϕ,λ 0 ϕ 0 Then there exists a solution of the system (cid:18) (cid:19) 2π c w(cid:48)(cid:48)e +c w(cid:48)(cid:48)e +c kw(cid:48)e −c kw(cid:48)e +c w(cid:48)(cid:48)e −λ w e −λ w e =λ sin t e . 12 3 3 13 2 2 13 3 1 13 1 3 23 1 1 0 2 2 0 3 3 0 L 3 Integrating the first component we infer that c w(cid:48) =−c kw −C for some constant C ∈R. With this 23 1 13 3 formula we can rewrite the third component via (cid:18) (cid:19) c k 2π c w(cid:48)(cid:48)+ 13 (c kw +C)−λ w =λ sin t . 12 3 c 13 3 0 3 0 L 23 Multiplying the equation with the right hand side and integrating twice by parts over (0,L) we obtain (cid:90) L 4π2c (cid:18)2π (cid:19) (cid:18)(c k)2 (cid:19) (cid:18)2π (cid:19) 0<λ − 12w sin t + 13 −λ w sin t dt=0, 0 L2 3 L c 0 3 L 0 23 where we used the definition of λ and the boundary conditions on w . This gives the desired contradic- 0 3 tion. (cid:3) Remark3.5. Withasimilaranalysisastheoneinthepreviouslemma,onecanalsofindothereigenvalues oftheoperator D F,correspondingtosmallerforcesthanthecriticalone.Thecorrespondingeigenstates ϕ show more than one sign change in the third component, that would lead to multiple inversions of curvature (see [14] for experimental evidence). On the other hand, possible bifurcation curves starting from the straight configuration along those directions are, at least close to the identity, not made of local minimizers. This can be proved exploiting that, according to Theorem 2.2, the second differential of Ef 0 at the identity is not positive semidefinite for f <f , together with a lower semicontinuity argument. crit This is not the case for the bifurcation branch from the largest eigenvalue λ = f . Indeed we will 0 crit show in Theorem 4.4 that such a branch in a neighborhood of the straight configuration consists of local minimizers. The previous lemma and the Crandall-Rabinowitz Theorem 2.3 entail the existence of a branch of solutions bifurcating the identity at the critical force. A precise statement with additional properties will be given in the next section. In order to investigate further the behaviour of the non-trivial branch, we will follow the general approach described in [3, Section 5.4]. To this end, we first notice that (3.9) Rg(D F(0,λ ))={w ∈C([0,L],R3): (cid:104)w∗,w(cid:105)=0}, ϕ 0 where we identify the function w∗ in (4.6) with an absolutely continuous vector-valued measure as usual. The range has indeed codimension 1 by the previous lemma, while a direct computation based on integration by parts gives for all w ∈C2([0,L],R3) 0 (cid:90) L (cid:90) L (3.10) (cid:104)w∗,D F(0,λ )w(cid:105)= w∗(t)TD F(0,λ )w(t)dt= w(t)TD F(0,λ )w∗(t)dt=0. ϕ 0 ϕ 0 ϕ 0 0 0 In order to determine the type of bifurcation, in the next lemma we compute the following terms: (3.11) a:=(cid:104)w∗,D2 F(0,λ )w∗(cid:105), ϕ,λ 0 1 (3.12) b:= (cid:104)w∗,D2 F(0,λ )[w∗,w∗](cid:105), 2 ϕ,ϕ 0 1 (3.13) c:=− (cid:104)w∗,D3 F(0,λ )[w∗,w∗,w∗](cid:105). 3a ϕ,ϕ,ϕ 0 10 MARCOCICALESE,FRANCESCOSOLOMBRINO,ANDMATTHIASRUF Lemma 3.6. For F as in (3.4) and λ as in (3.6), it holds 0 L a=− , 2 b=0, (cid:18)3(c −c )(c k)2 9(c k)2 λ (cid:19) (cid:18)(3c + 5c )(c k)2 π2c (cid:19) c=− 13 23 13 + 13 − 0 =− 13 4 23 13 + 12 . c2 2c 4 c2 L2 23 23 23 In particular c<0. Proof. First note that D2 (0,λ )w∗ =−w∗e . Then it holds that ϕ,λ 0 3 3 (cid:90) L (cid:90) L (cid:18)2π (cid:19) L a=− |w∗(t)|2 dt=− sin2 t dt=− . 3 L 2 0 0 In order to calculate b and c, we first write the operator F in components. Note that we need only the first and the third component since w∗ =0. By linearity of differentiation, it holds that 2 (cid:16) (cid:17)(cid:48) (cid:104)D2 F(0,λ)[w∗,w∗],e (cid:105)= D2∂ g (0,0)[(w∗,(w∗)(cid:48)),(w∗,(w∗)(cid:48))] , ϕ,ϕ 1 ξ1 λ where the symbol D2 on the right hand side denotes the Hessian of the scalar function ∂ g . Observe ξ1 λ that ∂ g (u,ξ) reads as ξ1 λ (cid:16) (cid:17) (cid:16) (cid:17) ∂ g (u,ξ)=c ξ sin(u )+ξ sin(u )+c ξ cos(u )sin(u )−ξ cos(u )+k cos(u )sin(u ) ξ1 λ 12 1 2 3 2 13 1 2 3 2 3 2 3 (cid:16) (cid:17) +c ξ cos(u )cos(u )+ξ sin(u ) cos(u )cos(u ). 23 1 2 3 2 3 2 3 Note that due to the fact that w∗ = 0, any higher order derivative of ∂ g (u,ξ) with at least one 2 ξ1 λ derivative with respect to the variables u or ξ along the direction w∗ vanishes. Since ∂ ∂ g (u,ξ)= 2 2 ξ3 ξ1 λ c sin(u ), the same reasoning allows to neglect any higher order derivatives with at least one derivative 12 2 with respect to ξ . Finally, the function ∂ g (u,ξ) is independent of the variable u . Summarizing we 3 ξ1 λ 1 needtotakeintoaccountonlythederivativesof ∂ g (u,ξ) withrespectto u and ξ .Astraightforward ξ1 λ 3 1 calculation shows that ∂2 ∂ g (0,0)=∂ ∂2 g (0,0)=∂3 g (0,0)=0, u3 ξ1 λ u3 ξ1 λ ξ1 λ so that it follows directly that (3.14) (cid:104)D2 F(0,λ)[w∗,w∗],e (cid:105)=0 ϕ,ϕ 1 for all λ. For the third component we need to compute (cid:16) (cid:17)(cid:48) (cid:104)D2 F(0,λ)[w∗,w∗],e (cid:105)= D2∂ g (0,0)[(w∗,(w∗)(cid:48)),(w∗,(w∗)(cid:48))] −D2∂ g (0,0)[(w∗,(w∗)(cid:48)),(w∗,(w∗)(cid:48))]. ϕ,ϕ 3 ξ3 λ u3 λ Using the explicit expressions ∂ g (u,ξ)=c (ξ sin(u )+ξ ), ξ3 λ 12 1 2 3 −∂ g (u,ξ)=−fcos(u )sin(u ) u3 λ 2 3 −c (ξ cos(u )sin(u )−ξ cos(u )+k)(ξ cos(u )cos(u )+ξ sin(u )) 13 1 2 3 2 3 1 2 3 2 3 −c (ξ cos(u )cos(u )+ξ sin(u ))(−ξ cos(u )sin(u )+ξ cos(u )) 23 1 2 3 2 3 1 2 3 2 3 and arguing as for the first component, for second or higher order derivatives along the direction w∗ it suffices to consider partial derivatives with respect to the variables u and ξ . Again we obtain 3 1 (3.15) (cid:104)D2 F(0,λ)[w∗,w∗],e (cid:105)=0. ϕ,ϕ 3 for all λ. Combining (3.14) and (3.15) we obtain that b=0.