A GENERAL ASYMPTOTIC DYNAMIC MODEL FOR LIPSCHITZIAN ELASTIC CURVED RODS

The main task of this paper is to relax regularity assumptions on a shape of elastic curved rods in a general asymptotic dynamic model and to derive this asymptotic model from a linear evolution equation of three dimensional elasticity by asymptotic technique. We use the asymptotic approach presented by Aganovic and Tutek [17] for straight rods, which was modified by Jurak and Tambača [8, 9] (see also Trabucho and Viaño [16]) and by Ignat, Sprekels, and Tiba [7] for curved rods generated by a function Φ ∈ Ck([0, l])3, k ≥ 3, and k ≥ 2, respectively. The approach from [8, 9] was applied to a dynamic model of curved rods in [13]. Following an idea from Blouza and Le Dret [3], a new formulation of the equations of elasticity which required Φ∈ C2([0, l])3 was found in Tiba and Vodák [14]. In addition, the general asymptotic model obtained in [14] was well-defined for a Jordan unit speed curve with Lipschitzian parameterization which led to a special construction of its approximations with smooth Jordan unit speed curves. The whole construction was valid for Φi > 0 a.e. on (0, l) for some i, i = 1,2,3. This restrictive condition was excluded by the modification of the construction in [15]. In this paper, we want to extend the theory presented in [14, 15] on the dynamic model for the curved rods. Among other papers concerning with the dynamic models for thin curved structures we mention here Raoult [10], Xiao [18], and Sprekels and Tiba [12]. Further we recommend the reader Álvarez-Dios and Viaño [1], Bermúdez and Viaño [2]. Finally, we give a brief outline of the paper. In Section 2, we introduce the basic notations and notions that will be further needed. Section 3 contains auxiliary propositions, which are used throughout the paper. Section 4 is devoted to a weak formulation of the linear elasticity equation and its transformation. Section 5 deals with basic estimates. Section 6 gives us a basic overview about behaviour of the displacements if → 0 and about qualitative properties of their limit state. In Section 7, the passage to the limit → 0 is performed and the main existence and uniqueness result is proved.


Introduction
The main task of this paper is to relax regularity assumptions on a shape of elastic curved rods in a general asymptotic dynamic model and to derive this asymptotic model from a linear evolution equation of three dimensional elasticity by asymptotic technique.
We use the asymptotic approach presented by Aganovic and Tutek [17] for straight rods, which was modified by Jurak and Tambača [8,9] (see also Trabucho and Viaño [16]) and by Ignat, Sprekels, and Tiba [7] for curved rods generated by a function Φ ∈ C k ([0,l]) 3 , k ≥ 3, and k ≥ 2, respectively.The approach from [8,9] was applied to a dynamic model of curved rods in [13].Following an idea from Blouza and Le Dret [3], a new formulation of the equations of elasticity which required Φ ∈ C 2 ([0,l]) 3 was found in Tiba and Vodák [14].In addition, the general asymptotic model obtained in [14] was well-defined for a Jordan unit speed curve with Lipschitzian parameterization which led to a special construction of its approximations with smooth Jordan unit speed curves.The whole construction was valid for Φ i > 0 a.e. on (0,l) for some i, i = 1,2,3.This restrictive condition was excluded by the modification of the construction in [15].In this paper, we want to extend the theory presented in [14,15] on the dynamic model for the curved rods.Among other papers concerning with the dynamic models for thin curved structures we mention here Raoult [10], Xiao [18], and Sprekels and Tiba [12].Further we recommend the reader Álvarez-Dios and Viaño [1], Berm údez and Viaño [2].
Finally, we give a brief outline of the paper.In Section 2, we introduce the basic notations and notions that will be further needed.Section 3 contains auxiliary propositions, which are used throughout the paper.Section 4 is devoted to a weak formulation of the linear elasticity equation and its transformation.Section 5 deals with basic estimates.Section 6 gives us a basic overview about behaviour of the displacements if → 0 and about qualitative properties of their limit state.In Section 7, the passage to the limit → 0 is performed and the main existence and uniqueness result is proved.
The main result of this paper can be summarized in the following theorem.

Preliminaries
We denote by R 3 the usual three-dimensional Euclidean space with scalar product (•,•) and norm | • |.Let S ⊂ R 2 be a bounded simply connected domain of class C 1 satisfying the "symmetry" condition We denote by Ω = (0,l) × S, Ω = (0,l) × S open cylinders in R 3 , where l > 0 and > 0 "small," are given.Let a function Φ : [0,l] → R 3 , Φ ∈ W 1,∞ (0,l) 3 , be a parametrization of a Jordan unit speed curve Ꮿ in R 3 and let t, n, b denote its tangent, normal and binormal vectors.Let Φ : [0,l] → R 3 be a smoothing of Φ such that it remains a Jordan unit speed curve (i.e., |Φ (y 1 )| = 1, ∀y 1 ∈ [0,l]) and t , n , b be the associated local frame.Alternative ways, how to construct local frames under low regularity assumptions, may be found in [12].The whole construction of the local frame associated with the function Φ and its smoothing can be found in [14,15].Here we mention only the needed properties of the approximation: The whole construction can be found in [14,15].

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The orthogonality properties (t ,t ) = 0, (n ,n ) = 0, (b ,b ) = 0 lead to so called "laws of motion" of the local frame We adopt the usual notation for the standard Sobolev and Lebesgue spaces, that is, H 1 (Ω), H 1 0 (Ω) and L p (Ω), p ∈ [1,∞] for the spaces and • 1,2 , • p for their norms.We will use the same notation of the norms also for vector or tensor functions in the case that all their components belong to the above mentioned Sobolev or Lebesgue spaces.H −1 (Ω) and [X] stand for the dual space to H 1 0 (Ω) or X, respectively.The notation C m (Ω), with m ∈ N 0 , means the usual spaces of continuous functions whose derivatives up to the order m are continuous in Ω, and we denote by C ∞ 0 (Ω) the space of all functions which have derivatives of any order on Ω and whose supports are compact subsets of Ω.The symbols L p (I;X), p ∈ [1,∞] and C(I;X), where X is a Banach space and I is a bounded interval, stand for the Bochner spaces endowed with the norms v L p (I;X) and v C(I;X) , respectively.We say that v ∈ C([0,T];L 2 weak (Ω)) if the function Ω v(t)w dx is continuous on [0, T] for an arbitrary function w ∈ L 2 (Ω).
Except for the standard definition of the weak convergence in X or L p (I;X), p ∈ (1,∞), and * -weak convergence in L ∞ (I,[X] ), we say without danger of confusion that v n v in L 2 (0,l;H −1 (S)), if where [X] •, • X denotes the dual pairing of [X] and X, and for any ψ ∈ L 1 (I,L 2 (Ω)).Further, we introduce the space (2.9) We refer the reader to [14] for the proof that ᐂ t,n,b 0 (0,l) is a nontrivial Hilbert space endowed with the norm (2.10) Let v ∈ L 1 loc (0,T) and ϕ ∈ C ∞ 0 (0,T).Then we denote v ϕ = T 0 v(t)ϕ(t)dt.Now, we introduce the mappings R and P , (y 1 , y 2 , y 3 ) ∈ Ω = (0,l) × S, where the second one gives the parametrization of the curved rod Ω = P (Ω ).Furthermore, We can suppose that d (y) > 0 for all y ∈ Ω and for "small" (see (2.6) and the definition of Ω ).Then P : In the sequel, we will write Thus, in (2.13), ∇ = ( ∂1 , ∂2 , ∂3 ).In the case that a function v depends only on t or x 1 (or y 1 ), we denote its first (second) derivation by v (v) and v (v ), respectively.Sometimes it is more convenient to use the notation (d/dt)v instead of v.The definition of the domain Ω enables us to introduce the following spaces: In an analogous way as above, we denote by V a function defined on Ω , V a function defined on Ω , and V a function defined on Ω.

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After substitution y = R (x), we adopt the notation where x ∈ Ω. Analogously as above, we can find the contravariant tensor o i j, for the mapping P • R having the form By "×" we will denote the Cartesian product of two spaces and by •, • any ordered pair.In the text, the symbol |A| will also denote the Lebesgue measure of some measurable set A, without danger of confusion.The summation convention with respect to repeated indices will be also used, if not otherwise explicitly stated.We use for constants the symbols C or C i , for i ∈ N 0 = {0, 1,2,...}.

Weak formulation of an evolution equation for the curved rods and its transformation
We consider Ω defined by mapping P • R (see (2.11)-(2.12))for ∈ (0,1) arbitrary but fixed as a three-dimensional homogeneous and isotropic elastic body with the Lamé constants λ ≥ 0, µ > 0 and with mass density ρ .Let F be the body force and G the surface traction acting on the curved rod Ω such that F ∈ L 2 (0,T;L 2 ( Ω ) 3 ) and G ∈ W 1,1 (0,T;L 2 (( P • R )((0,l) × ∂S)) 3 ), for ∈ (0,1).Let Ω be clamped on both bases P ({0} × S) and P ({l} × S).The equilibrium displacement U is a (weak) solution of the equation Rostislav Vodák 431 for all V ∈ C ∞ 0 (0,T;V ( Ω ) 3 ), where S = ( P • R )((0,l) × ∂S), A i jkl = λδ i j δ kl + µ(δ ik δ jl + δ il δ jk ) and (e i j ( V)) 3  i, j=1 stands for the symmetric part of the gradient of the function V.The solution U satisfies the initial state Using the fact that the functions t ,n ,b ∈ C ∞ ([0,l]) 3 together with (2.11), (2.12), it is easy to see that the mapping P • R is the parametrization of the smooth threedimensional curved rod.
We transform now (4.1).Denoting For the transformation of other terms we refer the reader to [14].It is easy to see that if we can rewrite the model (4.1)-(4.2) using (4.3) and the transformation from [14] as for all V ∈ C ∞ 0 (0,T;V (Ω) 3 ), where the solution U satisfies the initial state ) , are the components of the unit outward normal to (0,l) × ∂S, (o i j, ) 3 i, j=1 was introduced in (2.21), and where the symmetric tensor ω (V) has the form The individual nonzero components of the symmetric tensors θ and κ are defined by ) ) Assumptions.The following assumptions will be needed throughout the paper: (1) ρ = 2 ρ, where ρ ∈ L ∞ (Ω) and where the constant C is independent of , and for → 0, where Q 0 ∈ H 1 0 (0,l) 3 and Q 1 ∈ L 2 (0,l) 3 , that is, the functions Q 0 , Q 1 are the constant functions in the second and third variable.The reason for the choice of these scalings can be found in the inequalities (3.7) and (5.1).We are not able to guarantee boundedness of the functions U in appropriate spaces without the scalings, which means that the curved rod can be broken when the diameter converges to zero.

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Before we start to prove Proposition 5.1, we construct a finite dimensional approximation of the weak solution to our problem using analogous arguments as in [5,6], and we prove an auxiliary lemma, which enable us to prove Proposition 5.1.
Let ∈ (0,1) be arbitrary but fixed.Since the space V (Ω) is a separable Hilbert space with the scalar product ((V ,W)) ρd ,Ω = Ω ρV Wd dx + Ω ρ(∇V ,∇W)d dx, we can select smooth functions W k , k = 1,2,..., such that W k ∞ k=1 is a basis of V (Ω) and an orthonormal basis of L 2 (Ω) (5.3) in the sense of the scalar product (V ,W) ρd ,Ω = Ω ρV W d dx.The proof that the above mentioned scalar products are well-defined follows from (3.11) and (4.10).Now, we fix a positive integer m, and we write ) where , we will study the system of equations for a.a.t ∈ (0,T), completed with the initial states Since the proof of the existence and uniqueness of a solution to the problem (5.6)-(5.7) is very close to the proofs in [5] or [6], we omit it.Lemma 5.2.Under the assumptions of Section 4, the solution to the problem (5.6)-(5.7)satisfies the estimates ) where the constant C is independent of .
If we fix t ∈ (0,T), we can prove the second part of the above assertion in the same way as in [14].The proof of (6.23) follows from the expression from (4.6)-(4.9),(6.2) and from the fact that the function U depends only on t and x 1 (see Remark 6.2).Applying Proposition 3.4 together with (6.22)-(6.23),we get

The main result
In this section, we pass from the three-dimensional model (4.13)-(4.14) to the asymptotic model and our main result is stated and proved.Let us mention for the reader's convenience that we have proved in Corollary 5.3 that 1 for n → 0, where U ∈ L ∞ (0,T;H 1 0 (0,l) 3 ) according to Proposition 6.1.Now, we mention without proofs two propositions and one corollary, because the proofs can be obtained similarly as in [14].
Proposition 7.3.Under the assumptions on the domain S from Section 2, the system (7.9),(7.10) has the unique solution in L ∞ (0,T;L 2 (Ω) 2 ) given by where the function p ∈ H 1 (S) is the unique solution to the Neumann problem for all r ∈ H 1 (S).Now, we derive the asymptotic model.First we introduce some constants: where p ∈ H 1 (S) is the unique solution to the Neumann problem (7.12).
446 An asymptotic dynamic model for curved rods Taking ϕ V as a test function in (4.13), and using (7.18)-(7.23),lead to the equation (7.38) Equation ( 7.38) yields that the function Ω ρ(∂ t U (t), V )d dx belongs to the space W 1,∞ (0,T), which together with (4.14), (5.1), (5.2) enable us to rewrite (7.38) as We leave to the reader the proof that the right-hand side of (7.39) is convergent in C([0, T]).Hence, from (7.26) and from the second convergence in (7.1), we get that The rest of the proof is obvious.

Transformation of the limit equation
In this section, we go back to the original curve Ꮿ described by the parametrization Φ.