Nonlinear weakly curved rod by Γ -convergence

Transcription

1 Noname manuscript No. (will be inserted by te editor Nonlinear weakly curved rod by Γ -convergence Igor Velčić te date of receipt and acceptance sould be inserted later Abstract We present a nonlinear model of weakly curved rod, namely te type of curved rod were te curvature is of te order of te diameter of te cross-section. We use te approac analogous to te one for rods and curved rods and start from te strain energy functional of tree dimensional nonlinear elasticity and do not presuppose any constitutional beavior. To derive te model, by means of Γ -convergence, we need to propose ow is te order of strain energy related to te tickness of te body. We analyze te situation wen te strain energy (divided by te order of volume is of te order 4. Tat is te same approac as te one wen Föppl-von Kármán model for plates and te analogous model for rods are obtained. Te obtained model is analogous to Marguerre-von Kármán for sallow sells and its linearization is te linear sallow arc model wic can be found in te literature. Keywords weaky curved rod Gamma convergence sallow arc asymptotic analysis Matematics Subject Classification ( K20 74K25 Introduction Te study of tin structures is te subject of numerous works in te teory of elasticity. Tere is a vast literature on te subject of rods, plates and sells (see [5,8,9]. Te derivation and justification of te lower dimensional models, equilibrium and dynamic, of rods, curved rods, weakly curved rods, plates and sells Igor Velčić Faculty of Electrical Engineering and Computer Science, University of Zagreb, Unska 3, 0000 Zagreb, Croatia Tel: Fax: igor.velcic@fer.r

2 2 Igor Velčić in linearized elasticity, by using formal asymptotic expansion, is well establised (see [8, 9] and te references terein. In all tese approaces one starts from te equations of tree-dimensional linearized elasticity and ten via formal asymptotic expansion justify te lower dimensional models. One can also obtain te convergence results. In [3, 4] te linear model of weakly curved rod (or as it is called sallow arc is derived and te convergence result is obtained. We call weakly curved rods or sallow arces tose caracterized by te fact tat te curvature of teir centerline sould as te same order of magnitude as te diameter of te cross section, bot being muc smaller tan teir lengt. Formal asymptotic expansion is also applied to derive non linear models of rods, plates and sells (see [8, 9, 22] and te references terein, starting from tree-dimensional isotropic elasticity (usually Saint-Venant-Kircoff material. Hierarcy of te models is obtained, depending on te te order of te external loads related to te tickness of te body (see also [] for plates. However, formal asymptotic expansion does not provide us a convergence result. Te first convergence result, in deriving lower dimensional models from tree-dimensional non linear elasticity, is obtained applying Γ -convergence, very powerful tool introduced by Degiorgi (see [6, 0]. Using Γ -convergence, elastic string models, membrane plate and membrane sell models are obtained (see [,6,7]. It is assumed tat te external loads are of order 0. Te obtained models are different from tose ones obtained by te formal asymptotic expansion in te sense tat additional relaxation of te energy functional is done. Recently, ierarcy of models of rods, curved rods, plates and sells is obtained via Γ - convergence (see [2 5, 9, 24, 25, 30, 3]. Influence of te boundary conditions and te order and te type of te external loads is largely discussed for plates (see [3, 8]. Let us mention tat Γ -convergence results provide us te convergence of te global minimizers of te total energy functional. Recently, compensated compactness arguments are used to obtain te convergence of te stationary points of te energy functional (see [23, 28]. Here we apply te tools developed for rods, plates and sells to obtain weakly curved rod model by Γ -convergence. It is assumed tat we ave free boundary conditions and tat te strain energy (divided by te order of volume is of te order 4, were is te tickness of te rod. Tis corresponds to te situation wen external transversal dead loads are of order 3 (see Remark 8. Te order 4 of te strain energy gives Föppl-von Kármán model for plates, Marguerre-von Kármán model for sallow sells te analogous model for rods (see [4,25,32]. Te obtained model is non linear model of te lowest order in te ierarcy of models and its linearization is sallow arc model, obtained in [3, 4] for isotropic, omogenous case (see for comparison Remark 7 d. Here we do not presuppose any constitutional beavior and tus work in a more general framework. Te main result is stated in Teorem 5. Trougout te paper Ā or {A} denotes te closure of te set. By a domain we call a bounded open set wit Lipscitz boundary. I denotes te identity matrix, by SO(3 we denote te rotations in R 3, by so(3 te set of

3 Nonlinear weakly curved rod by Γ -convergence 3 antisymmetric matrices 3 3 and R 3 3 sym denotes te set of symmetric matrices. By sym A we denote te symmetric part of te matrix, sym A = 2 (A + AT. e, e 2, e 3 are te vectors of te canonical base in R 3. By we denote = e + e 2,e 3. f C (Ω stands for C norm of te function f : Ω R n R i.e. f C (Ω = max x Ω f + n i= max x Ω i f. denotes te strong convergence and te weak convergence. 2 Setting up te problem Let R 2 be an open set aving area equal to A and Lipscitz boundary. For all suc tat 0 < and for given L we define =, Ω = (0, L. (2. We sall leave out superscript wen =, i.e. Ω = Ω, =. Let us by µ( denote µ( = (x x 2 3dx 2 dx 3. (2.2 Let us coose coordinate axis suc tat x 2 dx 2 dx 3 = x 3 dx 2 dx 3 = For every we define te curve C of te form x 2 x 3 dx 2 dx 3 = 0. (2.3 C = {θ (x = (x, θ 2 (x, θ 3 (x R 3 : x (0, L}. (2.4 were θ k (x, for k = 2, 3, are given functions satisfying θ k C3 (0, L. Let (t, n, b be te Frenet triedron associated wit te curve C t = + ((θ ((θ 3 2 (, (θ 2, (θ 3, (2.5 n = (t (t, (2.6 b = t n. (2.7 We suppose n C (0, L wic is satisfied if (θ 2, (θ 3 do not vanis at te same time (wic is equivalent to te fact tat te curvature of C is strictly positive for any x (0, L. Te case were C as null curvature points can be treated in te same fasion, provided tat we suppose tat along tese points we ave te same degree of smootness as before wit t, n and b appropriately cosen (see Remark 3. We define te map Θ : Ω Θ ( Ω = { Ω } R 3, were Ω := Θ (Ω, in te following manner: Θ (x = (x, θ 2 (x, θ 3 (x + x 2n (x + x 3b (x (2.8

4 4 Igor Velčić and we assume tat Θ is a C diffeomorpism wic can be proved if is small enoug and θk, for k = 2, 3, are of te form considered ere. Namely, we take θ2 = θ 2, θ3 = θ 3 were θ k C 3 (0, L. Let us suppose ((θ 2 (x + ((θ 2 2 (x 0, (2.9 for all x (0, L. A generic point in Ω or { Ω } will be denoted by x = (x, x 2, x 3. Like in [24,25,2 5,9] we start from tree dimensional non linear elasticity functional of strain energy (see [7] for an introduction to non linear elasticity I (y := 2 Ω W (x, ydx. (2.0 It is natural to divide te strain energy wit 2, since te volume is vanising wit te order of 2. We are interested in finding Γ -limit (in some sense i.e. in caracterizing te limits of minimizers of te functionals I. Te reason 4 wy we divide wit 4 is tat we want to obtain teory analogous to Föppl-von Kármán for plates and rods (see [3, 4, 25] and Marguerre-von Kármán for sallow sells (see [32]. We do not look te total energy functional because te part wit te strain energy contains te igest order derivatives (at least for te external dead loads and tus makes te most difficult part of te analysis (see Remarks 8 and 0. We sall not impose Diriclet boundary condition and assume tat te body is free at te boundary. Te consideration of te oter boundary conditions is also possible. We rewrite te functional I on te domain Ω, i.e. we conclude I (y := W (Θ P (x, ( y Θ P det(( Θ P (xdx, (2. Ω were by P : R 3 R 3 we denote te mapping P (x, x 2, x 3 = (x, x 2, x 3. ( y Θ P denotes y evaluated at te point Θ (P (x. We assume tat for eac it is valid det(( Θ P (xw (Θ P (x, F = W (x, F, x Ω, F R 3 3, (2.2 were te stored energy function W is independent of and satisfies te following assumptions (te same ones as in [25]: i W : Ω R 3 3 [0, + ] is a Caratéodory function; for some δ > 0 te function F W (x, F is of class C 2 for dist(f, SO(3 < δ and for a.e. x Ω; ii te second derivative 2 W F is a Caratéodory function on te set Ω {F 2 R 3 3 : dist(f, SO(3 < δ} and tere exists a constant γ > 0 suc tat 2 W (x, F[G, G] F2 γ G 2 if dist(f, SO(3 < δ and G R 3 3 sym;

5 Nonlinear weakly curved rod by Γ -convergence 5 iii W is frame-indifferent, i.e. W (x, F = W (x, RF for a.e. x Ω and every F R 3 3, R SO(3; iv W (x, F = 0 if F SO(3; W (x, F C dist 2 (F, SO(3 for every F R 3 3, were te constant C > 0 is independent of x. Under tese assumptions we first sow te compactness result (Teorem 3 i.e. we take te sequence y W,2 ( Ω ; R 3 suc tat lim sup 0 4 I < + and conclude ow tat fact affects te limit displacement. In Lemma 2 we prove te lower bound, in Teorem 4 we prove te upper bound and tat enables us to identify limit functional (Teorem 5. First we start wit some basic properties of te mappings Θ wic are necessary for furter analysis. 3 Properties of te mappings Θ We introduce for k = 2, 3, Notice tat p k (x = Let us denote p = p 2 p 3 p 2p 3. Teorem Let te functions θ k θ k (x (θ 2 2 (x + (θ 3 2 (x. (3. p p 2 3 =, p 2 p 2 + p 3 p 3 = 0. (3.2 be suc tat θ k(x = θ k (x, for all x (0, L, k = 2, 3. were θ k C 3 (0, L is independent of. Ten tere exists 0 = 0 (θ > 0 suc tat te Jacobian matrix Θ (x, were te mappings Θ are defined wit (2.8, is invertible for all x Ω and all 0. Also tere exists C > 0 suc tat for 0 we ave and det Θ = + δ (x, (3.3 t (x = e + θ 2(x e 2 + θ 3(x e o (x, (3.4 n (x = p 2 (x e 2 + p 3 (x e 3 (θ 2p 2 + θ 3p 3 (x e + 2 o 2 (x, (3.5 b (x = p 3 (x e 2 + p 2 (x e 3 + (θ 2p 3 θ 3p 2 (x e + 2 o 3 (x, (3.6 Θ (x = R e (x + C(x + x 2D(x + x 3E(x + 2 O (x, (3.7 ( Θ (x = R T e (x C (x x 2D (x x 3E (x + 2 O 2(x, (3.8

6 6 Igor Velčić ( Θ L R e < C, (3.9 (Ω ;R 3 3 ( Θ R T e < C, (3.0 L (Ω ;R 3 3 were ( ( 0 p2 p R e =, R = 3, (3. 0 R p 3 p 2 C = 0 (θ 2p 2 + θ 3p 3 (θ 3p 2 θ 2p 3 θ 2 0 0, (3.2 θ D = p 2 0 0, E = p p 3 0 0, ( p θ 2 θ 3 C = θ 2p 2 + θ 3p 3 0 0, (3.4 θ 3p 2 θ 2p D = , E = p 0 0, (3.5 p and δ : Ω R, o i : (0, L R 3, i =, 2, 3, O k : Ω R 3 3, k =, 2 are functions wic satisfy sup 0< 0 max δ (x C 0, x Ω sup 0< 0 max x (0,L o i (x C 0, sup 0< 0 max x (0,L (o i (x C 0 for some constant C 0 > 0. Proof. It can be easily seen sup max max O k,ij(x C 0, k =, 2, 0< 0 i,j x Ω t (x = e +θ 2(x e 2 +θ 3(x e ((θ 2 2 +(θ 3 2 e + 3 o 4(x, (3.6 were o 4 C2 (0,L C. Te relations (3.5 and (3.6 are te direct consequences of te relation (3.6. Let us by u : (0, L R 3 denote te function It is easy to see u = (, θ 2, θ 3 T. (3.7 Θ (x = (u (x + x 2(n (x + x 3(b (x n (x b (x. (3.8

7 Nonlinear weakly curved rod by Γ -convergence 7 Te relations (3.3, (3.7, (3.9, (3.0 are te direct consequences of te relations (3.4-(3.6 and (3.8. Te relation (3.8 is te direct consequence of te fact tat, for a regular matrix A and arbitrary B, wic satisfies A B < ( is te operational norm, te matrix A + B is invertible and (A + B (A A BA A B 2 A A B. To end te proof observe tat C = R T e CR T e, D = R T e DR T e, E = R T e ER T e. Remark By a careful computation it can be seen tat o 2, o 3 and o 4 (defined in te relation (3.6 are dominantly in e 2, e 3 plane i.e. tat we ave for i = 2, 3, 4: o i (x = f 2 i (x e 2 + f 3 i (x e 3 + r i (x, (3.9 were f 2 i, f 3 i C (0, L, sup 0< 0 r i C (0,L C, for some C > 0. Remark 2 By a furter inspection it can be seen tat f4 2 = 2 θ 2((θ (θ 3 2, f4 3 = 2 θ 3((θ (θ 3 2, (3.20 f2 2 = p 2 (f 2 (θ, θ 2 ((θ (θ 3 2 θ 2(θ 2p 2 + θ 3p 3, (3.2 f2 3 = p 3 (f 2 (θ, θ 2 ((θ (θ 3 2 θ 3(θ 2p 2 + θ 3p 3, (3.22 f 2 3 = p 3 ((θ (θ 3 2 p 3 f 2 (θ, θ, (3.23 f 3 3 = p 2 ((θ (θ p 2 f 2 (θ, θ, (3.24 were f 2 (θ, θ C (0, L is te expression tat includes θ, θ : f 2 (θ, θ = ( (p 2 + p 3 ((θ (θ (θ 2 + θ 3(θ 2p 2 + θ 3p 3 (θ (θ 3 2 (θ 2p 2 + θ 3p 3 2 Remark 3 It is not necessary to impose te condition (2.9. All we need is te existence of te expansions given by (3.4-(3.6, were p 2, p 3 C (0, L, including te statement of Remark. Remark 4 Altoug Θ makes te small perturbation of te central line, (x, 0, 0, for x [0, L], it is not true tat Θ is close to te identity (like in te sallow sell model, see [32]. In fact, tere are torsional effects of order 0 on every cross section. Tis is te main reason wy is te cange of coordinates introduced in te next capter useful.

8 8 Igor Velčić 4 Γ -convergence We sall need te following teorem wic can be found in [2]. Teorem 2 (on geometric rigidity Let U R m be a bounded Lipscitz domain, m 2. Ten tere exists a constant C(U wit te following property: for every v W,2 (U; R m tere is an associated rotation R SO(m suc tat v R L2 (U C(U dist( v, SO(m L2 (U. (4. Te constant C(U can be cosen uniformly for a family of domains wic are Bilipscitz equivalent wit controlled Lipscitz constants. Te constant C(U is invariant under dilatations. Te following version of te Korn s inequality is needed. Lemma Let R 2 wit Lipscitz boundary and u L 2 (; R 2. Let us by e ij (u denote e ij (u = 2 ( iu + j u. Let us suppose tat for every i, j =, 2 we ave tat e ij (u L 2 (. Ten we ave tat u W,2 (; R 2. Also tere exists constant C(, depending only on te domain, suc tat we ave ( u W,2 (;R 2 C( udx dx 2 + (x u 2 x 2 u dx dx 2 + e ij (u L 2 (. (4.2 i,j=,2 Let us suppose tat te domains s are canging in te sense tat tey are equal to s = A s, were A s R 2 2, and tere exists a constant C suc tat A s, A s C. Ten te constant in te inequality (4.2 can be cosen independently of s. Proof. Te first part of te lemma (te fixed domain is a version of te Korn s inequality (see e.g. [29]. Te last part we sall prove by a contradiction. Let us suppose te contrary tat for eac n N tere exists s n and u n W,2 ( s n; R 2 suc tat we ave u n dx dx 2 + (x u n 2 x 2 u n dx dx 2 s n + i,j=,2 s n e sn ij (u n L2 ( s n n un W,2 ( s n ;R 2, (4.3 were we ave by e sn ij ( denoted te symmetrized gradient on te domain s n. Witout any loss of generality we can suppose tat u n W,2 ( s n ;R 2 =. Let us take te subsequence of (s n (still denoted by (s n suc tat A s n A and A s n A in R 2 2. Let us look te sequence u n c = u n A s n A. It is clear tat tere exist C, C 2 > 0 suc tat C u n c W,2 ( ;R 2 C 2, (4.4

9 Nonlinear weakly curved rod by Γ -convergence 9 were we ave put := A. Tus tere exists u W,2 ( ; R 2 suc tat u n c u weakly in W,2 ( ; R 2. Specially, by te compactness of te embedding L 2 W,2 (see e.g. [2], we also conclude te strong convergence u n c u in L 2 ( ; R 2. Since it is valid A s na I, it can be easily seen tat, from te weak convergence, it follows e sn ij (un (A s n A e ij (u, weakly in L 2 ( ; R 2, were we ave by e ij ( denoted te symmetrized gradient on te domain. From te weak convergence we can conclude tat e ij (u L 2 ( lim inf n esn ij (u n (A s n A L 2 ( = 0, (4.5 for every i, j =, 2. We can also from (4.3 conclude tat udx dx 2 = 0, (x u 2 x 2 u dx dx 2 = 0. (4.6 Applying te standard Korn s inequality on te domain, i.e. ( u u n c W,2 ( ;R 2 C( u u n c L 2 ( ;R 2 + e ij (u e ij (u n c L2 ( ;R 2, i,j=,2 we conclude tat u n c u strongly in W,2 ( ; R 2. But ten (4.4, (4.5, (4.6 make a contradiction wit te version of te Korn s inequality (4.2 on te domain. Remark 5 Te same proof can be done under te assumption tat s = F s (, were F s is te family of Bilipscitz mappings wose Bilipscitz constants we can control (i.e. te Lipscitz constants of F s and Fs are bounded by a universal constant, provided tat te family F s is strongly compact in W, (; R 2. It would require more analysis to conclude te same result only for Bilipscitz mappings wose Bilipscitz constants we can control. Let us by x : R 3 R 3 denote te cange of coordinates (x, x 2, x 3 = x (x, x 2, x 3 := R e (x x x 2. (4.7 x 3 By Ω we denote x (Ω and (x R 2 denotes x ({x }, for x [0, L]. Te generic point in Ω is denoted wit x = (x, x 2, x 3. Let us observe tat by (2.2 and (2.3 x 2dx 2dx 3 = x 3dx 2dx 3 = 0, (4.8 (x (x (x x 2x 3dx 2dx 3 = p 2 p 3 µ( = (x x 2 3dx 2 dx 3 = (x 2 2 x 2 3dx 2 dx 3, (4.9 (x ((x (x 3 2 dx 2dx 3, (4.0

10 0 Igor Velčić for all x [0, L]. By ( i y j Θ P we denote i y j evaluated at te point Θ (P (x. In te sequel we suppose 0 (see Teorem. If tis was not te case, wat follows could be easily adapted. Using teorem 2 we can prove te following teorem Teorem 3 Let y W,2 ( Ω ; R 3 and let E = 2 Ω dist 2 ( y, SO(3dx. Let us suppose tat lim sup 0 E < +. (4. 4 Ten tere exist maps R : [0, L] SO(3 and R : [0, L] R 3 3, wit R C, R W,2 ([0, L], R 3 3 and constants R SO(3, c R 3 suc tat te functions ỹ := (R T y c satisfy Moreover if we define ( ỹ Θ P R L2 (Ω C 2, (4.2 R R L 2 ([0,L] C 2, ( R L 2 ([0,L] C, (4.3 u = A vk = A w = Aµ( R I L ([0,L] C. (4.4 ỹ Θ P x 2 dx 2 dx 3, (4.5 ỹ k Θ P θ k dx 2 dx 3, for k = 2, 3, (4.6 x 2(ỹ 3 Θ P x 3(ỹ 2 Θ P 2 dx 2 dx 3 (4.7 ten, up to subsequences, te following properties are satisfied (a u u in W,2 (0, L; (b vk v k in W,2 (0, L, were v k W 2,2 (0, L for k = 2, 3. (c w w weakly in W,2 (0, L; (d ( ỹ Θ P I A, in L 2 (Ω, were A W,2 (0, L is given by 0 v 2 v 3 A = v 2 0 w. (4.8 v 3 w 0 (e sym R I 2 A2 2 uniformly on (0, L;

11 Nonlinear weakly curved rod by Γ -convergence (f te sequence γ defined by γ (x = ( (ỹ Θ P (x x 2 u (x +x 2((v 2 + θ 2(x + x 3((v 3 + θ 3(x, ((ỹ γ k(x = k Θ P (x θ k x k 2 vk (x (x k (x, for k = 2, 3, were (x := (0, x 3, x 2, is weakly convergent in L 2 (Ω to a function γ belonging to te space C, were C = {γ L 2 (Ω; R 3 : γ = 0, 2 γ, 3 γ L 2 (Ω; R 3, (x 3γ 2 (x, x 2γ 3 (x, dx 2 dx 3 = 0, for a.e. x (0, L}.(4.9 Moreover k γ k γ in L 2 (Ω for k = 2, 3, Proof. We follow te proof of Teorem 2.2 in [25]. Applying Teorem 2 as in te compactness result of [24] (using te boundedness of Θ and ( Θ we can find a sequence of piecewise constant maps R : [0, L] SO(3 suc tat ( y Θ P R 2 dx C 4, (4.20 and Ω I R (x + ξ R(x 2 dx C 2 ( ξ + 2, (4.2 were I is any open interval in (0, L and ξ R satisfies ξ dist(i, {0, L}. Let η C0 (0, be suc tat η 0 and 0 η(sds =. We set η = η( s and we define R (x := η (sr (x sds, were we ave extended R outside [0, L] by taking R (x = R (0 for every x < 0, R (x = R (L for every x > L. Clearly R C for every wile properties (4.3 follow from properties (4.2. Moreover since by construction (see [24] R (x + s R (x 2 C 3 Ω dist 2 ( y, SO(3 C 3,

12 2 Igor Velčić for every s we ave by Jensen s inequality tat R R 2 L ([0,L];R 3 3 C3 (4.22 By te Sobolev-Poincare inequality and te second inequality in (4.3, tere exist constants Q R 3 3 suc tat R Q L ([0,L];R 3 3 C. Combining tis inequality wit (4.22, we ave tat R Q L ([0,L];R 3 3 C. Tis implies tat dist(q, SO(3 C; tus, we may assume tat Q belongs to SO(3 and by modifying Q by order, if needed. Now coosing R = Q and replacing R by (Q T R and R by (Q T R, we obtain (4.4. By suitable coice of constants c R 3 we may assume tat (ỹ Θ P x = 0, (ỹ k Θ P θ k = 0, for k = 2, 3. (4.23 Ω Ω Let A = R I. By (4.4 tere exists A L ((0, L; R 3 3 suc tat, up to subsequences, A A weakly * in L ((0, L; R 3 3. (4.24 On te oter and it follows from (4.3 and (4.4 tat R I A weakly in W,2 ((0, L; R 3 3. (4.25 In particular, A W,2 ((0, L; R 3 3 and ( R I also converges uniformly. Using (4.22 we deduce tat A A uniformly. (4.26 In view of (4.2 tis clearly implies te convergence property in (d. Since R SO(3 we ave A + (A T = A (A T. Hence, A + A T = 0. Moreover, after division by 2 we obtain property (e by (4.26. For adapting te proof to te proof of Teorem 2.2 in [25] it is essential to see From (4.27 it follows and ( ỹ Θ P = ( (ỹ Θ P (( Θ P = (ỹ Θ P (( Θ P. (4.27 (( ỹ Θ P (( Θ P = (ỹ Θ P. (4.28 ( ỹ Θ P I = ( (ỹ Θ P Θ P (( Θ P R T e +( (ỹ Θ P Θ P R T e. (4.29

13 Nonlinear weakly curved rod by Γ -convergence 3 Let us notice tat from (2.8, (3.5, (3.6 we can conclude Θ k = θ k + x k + O k ( 3 for k = 2, 3, (4.30 were O k ( 3 C (Ω C 3. By multiplying (d wit ( Θ P = (Θ P and using (3.9, (4.28 we obtain (ỹ Θ P Θ P AR e in L 2 (Ω. (4.3 Property (b immediately from (4.3 by using (3.7, (4.23 and (4.30. Moreover, v k = A k for k = 2, 3 so tat v k W 2,2 (0, L since A W,2 (0, L. By using (e, (3.0, (4.2 and (4.3 from (4.29 we conclude tat 2 sym( (ỹ Θ P Θ P R T e C (4.32 L2 ((0,L;R 3 3 Te weak convergence of u follows from (3.7, (4.32 and te definition of R e. By using te convergence (4.3 and Poincare inequality on eac cut {x } we can conclude ỹ 2 Θ P Θ 2 P 2 2 A (ỹ 2 Θ 2 P Θ 2 P (AR e 22 x 2 + (AR e 23 x 3 in L 2 (Ω. (4.33 By using (2.3, (4.7 and (4.30 we conclude from (4.33 w2 := ỹ 2 Θ P x 2 2 ỹ 2 Θ P A 23 x 3 in L 2 (Ω. (4.34 A Let us note tat since te left and side of (4.33 i.e. (4.34 is bounded in W,2 (Ω te convergence in (4.34 is in fact weak in W,2 (Ω. Te only nontrivial ting to prove is te boundedness of w 2 in L 2 (Ω. By te cain rule we ave for i =, 2, 3 (ỹ i Θ P = (( ỹ i Θ P (( Θ P +(( 2 ỹ i Θ P (( Θ 2 P + (( 3 ỹ i Θ P (( Θ 3 P (4.35 and for k = 2, 3 [ k (ỹ i Θ P = (( ỹ i Θ P (( k Θ P ] +(( 2 ỹ i Θ P (( k Θ 2 P + (( 3 ỹ i Θ P (( k Θ 3 P (4.36 From (4.2, (4.34 and (4.35 we conclude tat te boundedness of w 2 in L 2 (Ω is equivalent to te boundedness of z2 = R 2 Θ + R 22 Θ 2 + R 23 Θ 3 (p 2x 2 p 3x A (R 2 Θ + R 22 Θ 2 + R 23 Θ 3, (4.37

14 4 Igor Velčić in L 2 (Ω. By using (2.3 and (3.8 we conclude z 2 = R 2(x 2 (n + x 3 (b + R 22(x 2 (n 2 + x 3 (b 2 + R 23(x 2 (n 3 + x 3 (b 3 (p 2x 2 p 3x 3. (4.38 Te boundedness of z 2 in L 2 (Ω is te consequence of (3.5, (3.6 and (4.4. Now we ave proved w 2 A 23 x 3 weakly in W,2 (Ω. Analogously we conclude w 3 := ỹ 3 Θ P x 3 2 ỹ 3 Θ P A 23 x A 2,. (4.39 weakly in W,2 (Ω. Now, since w can be written as w (x = (x Aµ( 2w3 x 3w2 dx 2 dx 3, (4.40 it is clear tat w converges weakly to te function w = A 23 = A 32 in W,2 (0, L. Let us define for β : Ω R 3, β = γ x. By te cain rule we ave β i = ( γ i (x + (p 2x 2 + p 3x 3( 2 γ i (x +( p 3x 2 + p 2x 3( 3 γ i (x, 2 β i = p 2 ( 2 γ i (x p 3 ( 3 γ i (x, 3 β i = p 3 ( 2 γ i (x + p 2 ( 3 γ i (x. (4.4 By differentiating β wit respect to x k, wit k=2,3, we ave 2 β = 3 2(ỹ Θ P (x + ((v 2 + θ 2, ( β = 3 3(ỹ Θ P (x + ((v 3 + θ 3. (4.43 Let us analyze only 2 β. We ave by (3.8, (4.36 and te cain rule 2 β = (( ỹ Θ P (x (p 2 n p 3 b 2 + (( 2ỹ Θ P (x (p 2 n 2 p 3 b (( 3ỹ Θ P (x (p 2 n 3 p 3 b ((v 2 + θ 2. (4.44

15 Nonlinear weakly curved rod by Γ -convergence 5 By using (3.5, (3.6, (3.8, (4.2, (4.4, (4.35 and te definition of v k we can conclude tat for proving te boundedness of 2 β it is enoug to prove te boundedness of δ,2 in L 2 (Ω were δ,2 := R θ 2 + R R 2 + θ 2 2 = R θ 2 + R 2 + R 2 2. (4.45 Te boundedness of δ,2 in L (Ω is ten te consequence of te property (e. In te same way we can prove te boundedness of 3 β. Using te Poincare inequality and te fact tat (x β dx 2, dx 3 = 0, we deduce tat tere exists a constant C > 0 suc tat (β (x 2 dx 2 dx 3 C [( 2 β (x 2 + ( 3 β (x 2 ]dx 2 dx 3 (x (x for a.e. x (0, L and for every. Altoug te constant C depends on te domain, since all domains are translations and rotations of te domain, te constant C can be cosen uniformly. Integrating bot sides wit respect to x, we obtain tat te sequence (β is bounded in L 2 (Ω so, up to subsequences β β and k β k β weakly in L 2 (Ω, for k = 2, 3. From te relations (4.4 it can be concluded tat γ γ and k γ k γ weakly in L 2 (Ω, for k = 2, 3, were γ = β x. For te sequences (β 2, (β 3, we ave by differentiation tat for j, k = 2, 3 ( j β k = 2 j(ỹ k Θ P (x δ jk w ( δ jk ( k. (4.46 By using te cain rule we see tat for k = 2, 3, ( 2 (ỹ k Θ P (x = (( ỹ k Θ P (x (p 2 n p 3 b + (( 2 ỹ k Θ P (x (p 2 n 2 p 3 b 2 + (( 3 ỹ k Θ P (x (p 2 n 3 p 3 b 3, ( 3 (ỹ k Θ P (x = (( ỹ k Θ P (x (p 3 n + p 2 b Now we want to ceck tat for j, k = 2, 3 + (( 2 ỹ k Θ P (x (p 3 n 2 + p 2 b 2 + (( 3 ỹ k Θ P (x (p 3 n 3 + p 2 b 3. e jk (β := 2 ( jβ k + k β j. (4.47 is bounded in L 2 (Ω. In te similar way as for β (relations (4.44 and (4.45 we can using (3.5, (3.6, (4.2, (4.4 and te property (e conclude tat for

16 6 Igor Velčić every j, k = 2, 3, e jk (β L 2 (Ω. By using Korn s inequality (Lemma we ave tat tere exists C > 0 suc tat β 2 2 W,2 ( (x + β 3 2 W,2 ( (x ( C β 2dx 2dx 3 + β 3dx 2dx 3 + (x + j,k=,2 e jk (β L2 ( (x (x (x (x 3β 2 x 2β 3dx 2dx 3, (4.48 for a.e. x (0, L. From te definition of vk and w we see tat te functions (β 2(x,, β 3(x, belong to te space B x = {β = (β 2, β 3 W,2 ( (x ; R 2 : βdx 2dx 3 = 0, (x (x (x 2β 3 x 3β 2 dx 2dx 3 = 0} (4.49 for every x. By integrating (4.48 wit respect to x we conclude tat β 2,β 3 are bounded in L 2 (Ω as well as teir derivatives wit respect to x 2, x 3. From tis we can conclude te same fact about γ 2, γ 3. Te fact tat te weak limit belongs to te space C can be concluded from te fact tat for every and a.e. x (β 2(x,, β 3(x, B x. Tis finises te proof of (f. 4. Lower bound Lemma 2 Let y, ỹ, E, R, u, v, w, γ, β = γ (x, γ, β = γ (x, A be as in Teorem 3 and let us suppose tat te condition (4. is satisfied and tat γ γ, 2 γ 2 γ, 3 γ 3 γ weakly in L 2 (Ω i.e. β β, 2 β 2 β, 3 β 3 β weakly in L 2 (Ω. Let us define ( η (x = (ỹ Θ P (x Θ P 2 u (x +x 2(v 2 (x + x 3(v 3 (x ( η k(x = (ỹ k Θ P (x Θ k P 2 v k (x (x k (x, (4.50, for k = 2, 3, (4.5 and κ = η (x. Ten we ave tat η η weakly in L 2 (Ω and k η k η weakly in L 2 (Ω i.e. κ κ, 2 κ 2 κ, 3 κ 3 κ weakly

17 Nonlinear weakly curved rod by Γ -convergence 7 in L 2 (Ω. Here η = γ, (4.52 η 2 = γ 2 + f 2 2 x 2 + f 3 2 x 3 = γ 2 + g 2 2x 2 + g 3 2x 3, (4.53 η 3 = γ 3 + f 2 3 x 2 + f 3 3 x 3 = γ 3 + g 2 3x 2 + g 3 3x 3, (4.54 κ = η (x, (4.55 f j k are defined in Remark 2 and gj k can be easily defined for te above identities to be valid i.e. for k = 2, 3, we define g 2 k = p 2 f 2 k p 3 f 3 k, g 3 k = p 3 f 2 k + p 2 f 3 k. (4.56 Te following strain convergence is valid G := (R T (( y Θ P I 2 G in L 2 (Ω; R 3 3. (4.57 and te symmetric part of G denoted by G, satisfies G = sym(j 2 A2 + K, (4.58 were J = u + v 2θ 2 + v 3θ wθ 3 v 2θ 2 v 2θ 3 (4.59 wθ 2 v 3θ 2 v 3θ 3 K = x 2v 2 x 3v 3 x 3w x 2w 2κ 3κ. (4.60 Moreover, lim inf 0 6 W (x, ŷ dx = lim inf Ω 0 4 W (x, ( ŷ Θ P dx Ω Q 3 (x, 2 G(xdx, were Q 3 is twice te quadratic form of linearized elasticity, i.e., Ω Q 3 (x, F = 2 W (I[F, F]. (4.6 F2 Proof. We follow te proof of Lemma 2.3 in [25]. Firstly, using Remark, it can be seen tat η = γ + o, η 2 = γ 2 + f 2 2 x 2 + f 3 2 x 3 + o 2, η 3 = γ 3 + f 2 3 x 2 + f 3 3 x 3 + o 3,

18 8 Igor Velčić were o i C (Ω C, for some C > 0. Te convergence of η is an easy consequence of te convergence of γ. Te estimate (4.2 implies tat te L 2 norm of G is bounded; terefore up to subsequences, tere exists G L 2 (Ω; R 3 3 suc tat (4.57 is satisfied. In order to identify te symmetric part of G we decompose R G as follows: so tat R G = ( ỹ Θ P I 2 R I 2, (4.62 F := sym ( ỹ Θ P I 2 = sym(r G + sym R I 2. (4.63 A2 Te rigt and side converges weakly to G+ 2 by (4.4, (4.57 and property (e of te Teorem 3. Terefore te sequence F as a weak limit in L 2 (0, L, satisfying F = G + A2 2. To conclude we only need to identify F. Consider te functions ϕ := ỹ Θ P x 2. (4.64 From property (f of Teorem 3 it follows tat te functions ϕ u +x 2((v 2 + θ 2 + x 3((v 3 + θ 3, wic are equal to γ converge strongly to 0 in L 2 (Ω. Tus by property (a and (b of Teorem 3 we conclude tat ϕ u x 2(v 2 + θ 2 x 3(v 3 + θ 3 in L 2 (Ω. (4.65 By using te cain rule, te property (d of Teorem 3, (3.5, (3.6, (4.30 we can conclude tat ϕ F x 2 (θ 2p 2 +θ 3p 3 +x 3 (θ 2p 3 θ 3p 2 v 2( x 2+θ 2 v 3( x 3+θ 3, (4.66 weakly in L 2 (Ω. From (4.65 and (4.66 we conclude tat u x 2(v 2 + θ 2 x 2(v 2 + θ 2 x 3(v 3 + θ 3 x 3(v 3 + θ 3 = F x 2 (θ 2p 2 + θ 3p 3 + x 3 (θ 2p 3 θ 3p 2 v 2( x 2 + θ 2 v 3( x 3 + θ 3. (4.67 After some calculation we obtain F = u + v 2θ 2 + v 3θ 3 x 2v 2 x 3v 3. (4.68 To identify F 2 we ave to do some straigt forward computations. By using te cain rule, (3.5, (3.6, Remark, property (d of Teorem (3 we can conclude 2 (ỹ 2 Θ P + ( 3 p 2 2 (ỹ Θ P p 3 3 (ỹ Θ P = 2F 2 + x 2 w(θ 3 + x 3 + O, (4.69

19 Nonlinear weakly curved rod by Γ -convergence 9 were lim 0 O L2 (Ω;R 3 3 = 0. On te oter and it can be easily seen tat 2 β = p 2 2 γ p 3 3 γ = (p (ỹ Θ P p 3 3 (ỹ Θ P + ((v 2 + θ 2. (4.70 From (4.34, (4.69, (4.70 we conclude 2F 2 = 2 (ỹ 2 Θ P x 2 ((v 2 + θ β + wθ 3 + w x 3 O = w β + wθ 3 + w x 3 O. (4.7 By using (4.34 we conclude tat te rigt and side of (4.7 converges in W,2 (Ω to ( wx β + wθ 3 + w x 3 = x 3w + wθ κ, (4.72 since β = κ. On te oter and we know tat te left and side of (4.7 converges strongly in L 2 (Ω to 2F 2 and tus we can conclude In te same way one can prove F 2 = 2 ( x 3w + wθ κ. (4.73 F 3 = 2 (x 2w wθ κ. (4.74 To identify F 22 let us observe tat by te cain rule, (3.5, (3.6 and te property (d of Teorem (3 we ave ( 3 p 2 2 (ỹ 2 Θ P Θ 2 P p 3 3 (ỹ 2 Θ P Θ 2 P = F 22 v 2θ 2 + O 2, (4.75 were lim 0 O 2 L2 (Ω;R 3 3 = 0. On te oter and we can conclude 2 κ 2 = p 2 2 η 2 p 3 3 η 3 = (p (ỹ 2 Θ P Θ 2 P p 3 3 (ỹ 2 Θ P Θ 2 P. (4.76 In te same way as before we conclude tat F 22 = v 2θ κ 2. (4.77

20 20 Igor Velčić Analogously we can conclude F 33 = v 3θ κ 3. (4.78 To identify F 23 = F 32 we, by using te cain rule, (3.5, (3.6 and te property (d of Teorem (3, can conclude: ( 3 p 2 2 (ỹ 3 Θ P Θ 2 P p 3 2 (ỹ 3 Θ P Θ 2 P = 2 ( 2ỹ 3 Θ P v 2θ 3 + O 3, (4.79 were lim 0 O 3 L2 (Ω;R 3 3 = 0. In te same way we conclude ( 3 p 3 2 (ỹ 2 Θ P Θ 2 P + p 2 3 (ỹ 2 Θ P Θ 2 P = 2 ( 2ỹ 3 Θ P v 3θ 2 + O 4, (4.80 were lim 0 O 4 L 2 (Ω;R 3 3 = 0. It can be also concluded p 2 2 η 3 p 3 3 η 3 = (p (ỹ 3 Θ P Θ 2 P p 3 2 (ỹ 3 Θ P Θ 2 P + 2 w, (4.8 p 3 2 η 2 + p 2 3 η 2 = (p (ỹ 2 Θ P Θ 2 P +p 2 2 (ỹ 3 Θ P Θ 2 P 2 w. (4.82 By summing te relations (4.79-(4.82 and letting 0 it can be concluded tat 2F 23 = v 2θ 3 + v 3θ κ κ 2. (4.83 To prove te lower bound we can continue in te same way as in te proof of Lemma 2.3 in [25], by using te Taylor expansion, te cutting and Scorza- Dragoni teorem. 4.2 Upper bound Teorem 4 (optimality of lower bound Let u, w W,2 (0, L and v k W 2,2 (0, L for k = 2, 3. Let γ be a function in C were C = {γ L 2 (Ω; R 3 : γ = 0, 2 γ, 3 γ L 2 (Ω; R 3, (x 3γ 2 (x, x 2γ 3 (x, dx 2 dx 3 = 0, x (0, L}. (4.84 Set G = sym(j 2 A2 + K. (4.85

21 Nonlinear weakly curved rod by Γ -convergence 2 Here A, J, K are defined by te expressions (4.8, (4.59 and (4.60 and η, κ are defined by te expressions (4.52-(4.55. Ten tere exists a sequence (ŷ W,2 ( Ω, R 3 suc tat for u, vk, w defined by te expressions (4.5-(4.7 te properties (a-(d of Teorem 3 are valid. Also we ave tat te property (f of Teorem 3 is valid (wic is equivalent tat for η defined by te expressions (4.50-(4.5 it is valid η η weakly in L 2 (Ω and k η k η weakly in L 2 (Ω. Also te following convergence is valid lim 0 6 W (x, ŷ dx = Q 3 (x, G(xdx (4.86 Ω 2 Ω Proof. Let us first assume tat u, w, v k, η are smoot. Ten we define for (x, x 2, x 3 Ω : ŷ (Θ (x, x 2, x 3 = Θ (x, x 2, x u(x v 2 (x v 3 (x x 2 (v 2p 2 + v 3p 3 (x x 3 (v 3p 2 v 2p 3 (x + 2 x 2 (p 3 w(x x 3 (p 2 w(x x 2 (p 2 w(x x 3 (p 3 w(x + 3 η(x, x 2, x 3, (4.87 were η : Ω R 3 is going to be cosen later. Te convergence (a-(d and tat η η weakly in L 2 (Ω and k η k η weakly in L 2 (Ω can easily seen to be valid for tis sequence. We also ave ŷ Θ = Θ + 2 u (v 2p 2 + v 3p 3 (v 3p 2 v 2p 3 v 2 p 3 w p 2 w v 3 p 2 w p 3 w + 2 x 2(v 2p 2 + v 3p 3 + x 3 (v 2p 3 v 3p 2 x 2 (p 3 w x 3 (p 2 w x 2 (p 2 w x 3 (p 3 w 2η 3η +O( 3. (4.88 From (4.88, by using (3.8, we conclude ŷ = I + 0 v 2 v 3 v 2 0 w v 3 w u + v 2θ 2 + v 3θ wθ 3 v 2θ 2 v 2θ 3 wθ 2 v 3θ 2 v 3θ x 2(v 2 p 2 + v 3 p 3 + x 3 (v 2 p 3 v 3 p 2 x 2 (p 3 w x 3 (p 2 w x 2 (p 2 w x 3 (p 3 w 2κ 3κ +O( 3. (4.89

22 22 Igor Velčić Using te identity (I + M T (I + M = I + 2 sym M + M T M we obtain ( ŷ T ( ŷ = I sym J sym K + 2 A T A + O( 3, were O( 3 L (Ω;R 3 3 C 3, for some C > 0. Taking te square root we obtain [( ŷ T ( ŷ ] /2 = I + 2 G + O( 3. (4.90 We ave det( ŷ > 0 for sufficiently small. Hence by frame-indifference W (x, ( ŷ Θ P = W (x, [ ŷ T ( ŷ ] /2 Θ P ; tus by (4.90 and Taylor expansion we obtain: 4 W (x, ( ŷ Θ P 2 Q 3(x, G(x a.e and by te property ii of W for small enoug 4 W (x, ( ŷ Θ P 2 C( J 2 + K 2 + A 4 + C. Te equality (4.86 follows by te dominated convergence teorem. Namely, we ave 6 W (x, ŷ dx = Ω 4 W (x, ( ŷ Θ P dx Ω Q 3 (x, 2 Gdx. In te general case, it is enoug to smootly approximate u, w in te strong topology of W,2, v k in te strong topology of W 2,2, and η, k η in te strong topology of L 2 and to use te continuity of te rigt and side of (4.86 wit respect to tese convergences. Remark 6 Notice tat 0 0 K = A x 2 2 κ 3 κ = L + A x x 2 2 β 3 β. 3 x 3 Here β = γ (x and Ω L = 0 g2 2 g2 3. (4.9 0 g3 2 g3 3 From te fact tat γ C we can conclude β B, were B = {β L 2 (Ω ; R 3 : β = 0, 2 β, 3 β L 2 (Ω ; R 3, (x 3β 2 (x, x 2β 3 (x, dx 2dx 3 = 0, for a.e. x (0, L}.(4.92 (x

23 Nonlinear weakly curved rod by Γ -convergence Identification of te Γ -limit Let Q : (0, L R so(3 [0, + be defined as Q(x, t, F = min α W,2 ( (x ;R 3 Q 3 (x 0 x, F x 2 + te 2 α x 3α dx 2dx 3, 3 (4.93 were Q 3 is te quadratic form defined in (4.6. For u, w W,2 (0, L and v 2, v 3 W 2,2 (0, L we introduce te functional L I 0 (u, v 2, v 3, w := Q(x, u + v 2 2θ 2 + v 3θ ((v (v 3 2, Adx, (4.94 were A W,2 ((0, L; so(3 is defined by (4.8. We sall state te result of Γ -convergence of te functionals I to I 0. Before stating te teorem we 4 analyze some properties of te limit density Q. Remark 7 By using te remarks in te beginning of capter 4 in [25] te following facts can be concluded: a Te functional Q 3 (x, G is coercive on symmetric matrices i.e. tere exists a constant C > 0, independent of x, suc tat Q 3 (x, G C sym G 2, for every G (tis is te direct consequence of te assumption iv on W. Te minimum in (4.93 is attained. Since te functional Q 3 (x, G depends only on te symmetric part of G, it is invariant under transformation α α+c +c 2 (x and ence te minimum can be computed on te subspace { V x := α W,2 ( (x, R 3 : α = 0, (x (x (x 3α 2 x 2α 3 dx 2dx 3 = 0 Strict convexity of Q 3 (x, on symmetric matrices ensures tat te minimizer is unique in V. b Fix x (0, L, t R and F so(3. Let α min V be te unique minimizer of te problem (4.93. We set g(x 2, x 3 = F 0 x 2 + te, x 3 } b k ij = 2 W F i F jk (x, I, and we call B k te matrix in R 3 3 wose elements are given by (B k ij = b k ij. Ten αmin satisfies te following Euler-Lagrange equation: (B k k α min, φdx 2dx 3 = (B g, φdx 2dx 3, (x,k=2,3 (x =2,3. (4.95

24 24 Igor Velčić for every φ W,2 ( (x ; R 3 3. From tis equation it is clear tat α min depends linearly on (t, F. Moreover Q is uniformly positive definite, i.e. Q(x, t, F C(t 2 + F 2, t R, F so(3, (4.96 and te constant C does not depend on x. c By mimicking te proof of Remark 4.3 in [25] it can be seen tat tere exists a constant C (independent of x, t and F suc tat 2 α min L2 ( (x ;R α min L2 ( (x ;R 3 3 C g 2 L 2 ( (x ;R 3 3, (4.97 for a.e. x (0, L. To adapt te proof we only need to ave tat te constant in te Korn s inequality k α min j 2 dx 2dx 3 C e jk (α min 2 dx 2dx 3 (x j,k=2,3 (x j,k=2,3 (4.98 can be cosen independently of x. Tis is proved in Lemma. d Wen Q 3 does not depend on x 2, x 3 we can find a more explicit representation for Q. More precisely Q can be decomposed into te sum of two quadratic forms were Q(x, t, F = Q (x, t + Q 2 (x, F, Q (x, t := min 3(x, (te a b, a,b R 3 (4.99 Q 2 (x, 0, F := Q(x, 0, F. (4.00 Te relations (4.8 are only needed for tis. If we assume te isotropic and omogenous case i.e. Q 3 (F = 2µ F + FT 2 + λ(trace F 2, 2 ten after some calculation (see Remark 3.5 in [24] it can be sown tat µ(3λ + 2µ Q (t = t 2 λ + µ µ(3λ + 2µ Q 2 (x, F = (F 2 (x λ + µ 2 2 dx 2dx 3 (x +2F 2 F 3 x 2x 3dx 2dx 3 + F 3 +µτf 23, (x (x (x 3 2 dx 2dx 3 were te constant τ is so-called torsional rigidity, defined as τ( (x = τ( = (x x 2 3 x 2 3 φ + x 3 2 φdx 2 dx 3,

25 Nonlinear weakly curved rod by Γ -convergence 25 and φ is te torsion function i.e. te solution of te Neumann problem { φ = 0 in ν φ = (x 3, x 2 ν on Te following teorem can be proved in te same way as Teorem 4.5 in [23] (we need Teorem 3, Lemma 2, Teorem 4, Remark 6 and Remark 7. Teorem 5 As 0, te functionals I are Γ -convergent to te functional I 0 given in (4.94, in te following 4 sense: i (compactness and liminf inequality if lim sup 0 4 I < + ten tere exists constants R SO(3, c R 3 suc tat (up to subsequences R R and te functions defined by ỹ := (R T y c, u = ỹ Θ P x A 2 dx 2 dx 3 vk = ỹ k Θ P θ k dx 2 dx 3 A w = x 2(ỹ 3 Θ P x 3(ỹ 2 Θ P Aµ( 2 dx 2 dx 3 satisfy (a ( ỹ Θ P I in L 2 (Ω. (b tere exist u, w W,2 (0, L suc tat u u and w w weakly in W,2 (0, L. (c tere exists v k W 2,2 (0, L suc tat v k v k strongly in W,2 (0, L for k = 2, 3. Moreover we ave lim inf 0 4 I (y I 0 (u, v 2, v 3, w. (4.0 ii (limsup inequality for every v, w W,2 (0, L, v 2, v 3 W 2,2 (0, L tere exists (ŷ suc tat (a-(c old (wit ỹ replaced by ŷ and lim 0 4 I (ŷ = I 0 (u, v 2, v 3, w (4.02 Remark 8 Let f 2, f 3 L 2 (0, L. We introduce te functional J 0 = I 0 (u, v 2, v 3, w L 0 k=2,3 f k v k, (4.03 for every u W,2 (0, L, v 2, v 3 W 2,2 (0, L, and w W,2 (0, L. Te functional J 0 can be obtained as Γ -limit of te energies 4 I by adding a term describing transversal body forces of order 3 (see [3], see also [32]. For longitudinal body forces see [8]. Te problem for longitudinal body forces arises because te longitudinal forces sould be of order 2, te same order as for

26 26 Igor Velčić te model in [24]. One needs to impose certain stability condition to see wic model describes te beavior of te body for te longitudinal forces of order 2. Remark 9 Te term u + v 2θ 2 + v 3θ ((v (v 3 2 in te strain measures te extension of te central line (wic is of te second order. Namely, if we approximate te deformation of te weakly curved rod by: φ (x, x 2, x 3 = x + 2 u + 2 x 2(v 2 + θ x 3(v 3 + θ 3 (4.04 φ k (x, x 2, x 3 = θ k + x k + v k + 2 (x k w, for k = 2, 3, (4.05 we see, tat it is valid φ(x, 0, 0 2 Θ (x, 0, 0 2 = 2( 2u + 2v 2θ 2 + 2v 3θ 3 +(v (v 3 2. Remark 0 Te existence of te solution for te functional J 0 under te Diriclet boundary condition for v k at bot ends of te rod can be proved directly. It is also enoug tat we impose v 2, v 2, v 3, v 3 at te one end. Te existence can also be proved for te free boundary condition under te ypotesis tat L 0 f kdx = 0, L 0 x f k dx = 0 for k = 2, 3. It can be done in te same way as te proof of Lemma 5 in [32]. References. Acerbi, E., Butazzo, G., Percivale, D.: A variational definition of te strain energy for an elastic string, Journal of Elasticity, 25, (99 2. Adams, R.A.: Sobolev spaces, Academic press, New York Alvarez-Dios, J.A., Viano, J.M.: A bending and stretcing asymptotic teory for general elastic sallow arces, ESAIM: Proc., Vol. 2, ( Alvarez-Dios, J.A., Viano, J.M.: Matematical Justication of a One-dimensional Model for General Elastic Sallow Arces, Matematical Metods in te Applied Sciences, Volume 2, Issue 4, ( S.S. Antman, Nonlinear problems of elasticity. Second edition, Applied Matematical Sciences, 07, Springer, New York, A. Braides: Γ -convergence for Beginners, Oxford University Press, Oxford, Ciarlet, P.G.: Matematical elasticity. Vol. I, Tree-dimensional elasticity, Nort- Holland Publising Co., Amsterdam, Ciarlet, P.G.: Matematical elasticity. Vol. II. Teory of plates. Studies in Matematics and its Applications, 27. Nort-Holland Publising Co., Amsterdam ( Ciarlet, P.G.: Matematical elasticity. Vol. III. Teory of sells. Studies in Matematics and its Applications, 29. Nort-Holland Publising Co., Amsterdam ( Dal Maso, G.: An introduction to Γ -convergence, Progress in Nonlinear Differential Equations and Teir Applications, Birkäuser, Basel (993.. Fox, D.D., Raoult A., Simo, J.C.: A justification of nonlinear properly invariant plate teories, Arc. Rational Mec. Anal., 24, p ( Friesecke, G., James R.D., Müler, S.: A teorem on geometric rigidity and te derivation of nonlinear plate teory from tree-dimensional elasticity, Comm. Pure Appl. Mat. 55, (2002.

27 Nonlinear weakly curved rod by Γ -convergence Friesecke, G., James R.D., Müler, S.: A Hierarcy of Plate Models Derived from Nonlinear Elasticity by Γ -Convergence, Arcive for Rational Mecanics and Analysis 80, no.2, ( Friesecke, G., James R.D., Müler, S.: Te Föppl-von Kármán plate teory as a low energy Γ -limit of nonlinear elasticity, Comptes Rendus Matematique 335, no. 2, ( Friesecke, G., James R., Mora, M.G., Müller, S.: Derivation of nonlinear bending teory for sells from tree-dimensional nonlinear elasticity by Γ -convergence, C. R. Mat. Acad. Sci. Paris, 336, no. 8, ( Le Dret, H., Raoult, A.: Te nonlinear membrane model as a variational limit of nonlinear tree-dimensional elasticity, Journal de Matématiques Pures et Appliquées 74, ( Le Dret, H., Raoult, A.: Te membrane sell model in nonlinear elasticity: A variational asymptotic derivation, Journal of Nonlinear Science 6, Number, ( Lecumberry, M., Müller, S.: Stability of slender bodies under compression and validity of te von Kármán teory, Arcive for Rational Mecanics and Analysis, Volume 93, Number 2, ( Lewicka, M., Mora, M.G., Pakzad, M.: Sell teories arising as low energy Γ -limit of 3d nonlinear elasticity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5, Vol. IX, 43 ( Lewicka, M., Mora, M.G., Pakzad, M.: Te matcing property of infinitesimal isometries on elliptic surfaces and elasticity of tin sells, accepted in Arc. Rational Mec. Anal. 2. Lewicka, M., Pakzad, M.: Te infinite ierarcy of elastic sell models: some recent results and a conjecture, accepted in Fields Institute Communications ( Marigo, J.J., Meunier, N.: Hierarcy of One-Dimensional Models in Nonlinear Elasticity, Journal of Elasticity, 83, 28 ( Mora, M.G., Scardia, L.: Convergence of equilibria of tin elastic plates under pysical growt conditions for te energy density, submitted paper. 24. Mora, M.G., Müller, S.: Derivation of te nonlinear bending-torsion teory for inextensible rods by Gamma-convergence, Calc. Var., 8, ( Mora, M.G., Müller, S.: A nonlinear model for inextensible rods as a low energy Gamma-limit of tree-dimensional nonlinear elasticity, Ann. Inst. H. Poincar Anal. Nonlin., 2, ( Mora, M.G., Müller, S., Scultz, M.G.: Convergence of equilibria of planar tin elastic beams, Indiana Univ. Mat. J., 56, ( Mora, M.G., Müller, S.: Convergence of equilibria of tree-dimensional tin elastic beams, Proc. Roy. Soc. Edinburg Sect. A 38, ( Müller, S., Packzad, M.R.: Convergence of equilibria of tin elastic plates : te von Kármán case, Communications in Partial Differential Equations 33, Number 6, ( Oleinik, O.A., Samaev, A.S., Yosifian, G.A.: Matematical problems in elasticity and omogenization, Nort-Holland, Scardia, L.: Te nonlinear bending-torsion teory for curved rods as Gamma-limit of tree-dimensional elasticity, Asymptot. Anal., 47, ( Scardia, L.: Asymptotic models for curved rods derived from nonlinear elasticity by Γ -convergence, submitted paper. 32. Velcic, I.: Sallow sell models by Γ -convergence, submitted paper. Preprint ttp://web.mat.r/ ivelcic 33. Ziemer, W.: Weakly Differentiable Functions, Springer-Verlag: New York (989.