Convergence of Thin Vibrating Rods to a Linear Beam Equation

We show that solutions for a specifically scaled nonlinear wave equation of nonlinear elasticity converge to solutions of a linear Euler-Bernoulli beam system. We construct an approximation of the solution, using a suitable asymptotic expansion ansatz based upon solutions to the one-dimensional beam equation. Following this, we derive the existence of appropriately scaled initial data and can bound the difference between the analytical solution and the approximating sequence.


Introduction
The relation between solutions of the equations of nonlinear elasticity and solutions for lower dimensional models is of great interest since the lower dimensional models are often easier to analyse and to use for numerical simulations. A general introduction to this topic can be found in [5] or for continuum mechanics see [9]. In dependence of the size of the deformation and the applied forces different lower dimensional models can occur. Therefore a rigoros derivation of the lower dimensional models are of great interest. In the case of the time independent case there are many results, cf. e.g. Friesecke, James and Müller [7,8] for the case of plates and Mora and Müller [11,12] in the case of rods and Scardia [14,15] for curved rods. But there are only few results in the time independent case so far.
In this contribution we investigate the relation between solutions of an appropriately scaled wave equation of nonlinear elasticity and solutions of a linear Euler-Bernoulli beam system. More precisely, let Ω := [0, L] × S be the reference configuration of a three dimensional rod, where L > 0 and S ⊂ R 2 is the cross section. Then we consider the following nonlinear system (u h , ∂ t u h )| t=0 = (u 0,h , u 1,h ), (1.4) where T > 0 andW is some elastic energy density chosen later. Existence of strong solutions for large times of this system was shown in [1]. More details on how to justify the scaling can be found there as well. The limit system as h → 0 is given by whereṽ 0 ,ṽ 1 and I 2 , I 3 are appropriately chosen initial values and weights, respectively. It is the goal of this manuscript to prove convergence of the solutions of the system (1.1)-(1.4) to solutions of the limit system (1.5)-(1.7) with appropriate convergence rates for well-prepared initial data, cf. Theorem 3.3 below.
In an energetic setting the relations between higher dimensional models and lower dimensional ones, using the notion of Γ-convergence a fundamental contribution was given in [7]. There the classical geometric rigidity is proven. Using this result it was possible to prove a lot of convergence results in different geometrical situations and scaling regimes in the static setting, see for instance [8,12,11]. In the dynamical case for plates a convergence result can be found in [3] and in [6] in the case of viscelasticity. The large times existence and a first order asymptotic for plates was shown in [2].
In the following we want to explain the main novelties and difficulties of this contribution. In a first step we construct an approximation using the solution of the lower dimensional system. This approximation is constructed such that it solves the linearisation around zero of the nonlinear, three dimensional equation up to an error of order h 3 . This is done explicitly by determining suitable prefactor functions as solutions of systems on S. Thereafter the main difficulty of this work is to establish existence of suitable initial data in order to ensure large times existence for the solution of the nonlinear problem. This is done in Section 3.2. Here we use the nonlinear equations for the initial data from the compatability conditions. These are solved via a fixed point argument on precisely chosen function spaces. Finally in Section 3.3 the convergence properties are proven. For this we use a general result for solutions of the linearised equation. Moreover we have to carefully treat the rotational parts of the initial data, as the spaces for the fixed point argument do not cover them. For this we use a decomposition and the fact that the elastic energy density is chosen asW (F ) = dist(Id + F ; SO(3)).

Notation
We use standard notation; in particular N and N 0 := N ∪ {0} denote the natural numbers with and without zero, respectively. Moreover, the norm on R and absolute value in R n , R n×n is denoted by |.| for all n ∈ N. For p, k ∈ N, we denote the classical Lebesgue and Sobolev spaces for some bounded, open set M ⊂ R n , by L p (M ), W k p (M ) and H k (M ) := W k 2 (M ). A subscript (0) on a function space will always indicate that elements have zero mean value, e.g., The cross section of the rod is always denoted by S ⊂ R 2 and is assumed to be a smooth and bounded domain. Furthermore be Ω h := (0, L) × hS ⊂ R 3 for h ∈ (0, 1] and L > 0 and for convenience we write Ω := Ω 1 . We assume that S satisfies This is no loss of generality, as it can always be achieved via a translation and rotation. The scaling shell be such that we can assume |S| = 1. Furthermore, we denote with ∇ h the scaled gradient defined as The respective gradient in only x ′ := (x 2 , x 3 ) direction is denoted by The standard notation H k (Ω) and H k (Ω; X) is used for L 2 -Sobolev spaces of order k ∈ N with values in R and some space X, respectively. The space of all n-linear mappings G : V n → R for a vector space V is denoted, throughout the paper by L n (V ), n ∈ N. We deploy the standard identification of a ij x ij is the usual inner product on R n×n . Analogously, for G ∈ L 2 (R n×n ) we use the identification withG : R n×n → R n×n defined bỹ . As we will work with periodic boundary condition in This space can equivalently defined in the following way, which is in some situations more convenient In various estimates we will use an anisotropic variant of H k (Ω), as we will have more regularity in lateral direction. Therefore we define where m 1 , m 2 ∈ N 0 , the inner product is given by . Furthermore we will use the scaled norms for A ∈ H m (Ω; R n×n ) and B ∈ H m 1 ,m 2 (Ω; R n×n ) and n ∈ N. As an abbreviation we denote for u ∈ H k (Ω; R 3 ) the symmetric scaled gradient by ε h (u) := sym(∇ h u) and ε(u) = ε 1 (u) = sym(∇u).
The following lemma provides the possibility to take traces for u ∈ H 0,1 (Ω):

The Strain Energy Density W and Korn's Inequality
We investigate the mathematical assumptions and resulting properties of the strain-energy density W we use in this contribution. We assume to have W : where SO(3) denotes the group of special orthogonal matrices. This energy density clearly satisfies the following general assumptions (i) W ∈ C ∞ (B δ (Id); [0, ∞)) for some δ > 0; (ii) W is frame-invariant, i.e. W (RF ) = W (F ) for all F ∈ R 3×3 and R ∈ SO(3); (iii) there exists c 0 > 0 such that W (F ) ≥ c 0 dist(F, SO(3)) 2 for all F ∈ R 3×3 and W (R) = 0 for every R ∈ SO(3).

Remark 2.2.
We note that W has a minimum point at the identity, as W (Id) = 0 and W (F ) ≥ 0 for all F ∈ R 3×3 . Hence, we have forW (F ) : (2.5) The following lemma provides an essential decomposition of D 3W in the general form.

Lemma 2.3.
There is some constant C > 0, ε > 0 and A ∈ C ∞ (B ε (0); L 3 (R n×n )) such that for all G ∈ R n×n with |G| ≤ ε we have Proof: The inequalities follow directly from Lemma 2.3 and Hölder's inequality. In order to bound the full scaled gradient ∇ h g of some function g ∈ H 1 per (Ω) by the symmetric one, we need a sharp Korn's inequality for thin rods. As rigid motions x → αx ⊥ for α ∈ R arbitrary are admissible functions in H 1 per (Ω) we can not expect that the full scaled gradient is bounded by ε h (g). Precisely we obtain the following results. Lemma 2.5. There exists a constant C = C(Ω) > 0 such that for all 0 < h ≤ 1 and u ∈ H 1 per (Ω; R 3 ) we have

Proof:
The proof is similar to [ Proof: A proof can be found in [1].

First Order Expansion in a Linearised Regime
We construct an approximation to the unique solution of the non-linear system for k = 2, 3. Moreover we assume that max σ=0,1,2 where M > 0 is chosen later. Without loss of generality we can assume L 0 gdx 1 = 0. Otherwise we substract from u h analogously as in the proof of [1, Theorem 3.1].

Construction of the ansatz function
For the ansatz function we consider the following system of one-dimensional beam equations and Then we obtain with standard methods, as e.g. in [13,Theorem 11.8], the existence of a unique solution Moreover, due to the assumptions for g and the periodicity of v it follows where a, b, c : S → R 2 are chosen later. Then Moreover for the boundary condition it holds We choose now a : S → R 2 as the solution of the following system Such a solution exists, because we can apply the Lax-Milgram Lemma for the weak Laplacian on H 1 (0) (S; R 2 ). Thereby, the coercivity follows from Poincaré's inequality. With well known regularity result, e.g. Theorem 4.18 in [10], we obtain a ∈ C ∞ (S, R 2 ). The systems for b and c decouple to Defining the matrix of coefficients (p αβ ij ) α,β=2,3 i,j=1,2 in the following way be arbitrary. Then it holds The approximating solutionũ h solves then the following system where r h is chosen as above, , and the initial data is given bỹ with v j := ∂ j t v| t=0 and j = 0, . . . , 4. For the remainder it holds

Existence of and Bounds on Initial Values
Define now .
Proof: Using a Taylor series expansion for DW (∇ h w) we obtain (3.14) Thus (3.11) is equivalent to The idea is now to use the contraction mapping principle in order to prove the existence of a solution for (3.11), i.e., with the later equivalence For f ∈ L 2 (Ω; R 3 ) and F ∈ L 2 (Ω; R 3×3 ) we obtain with the Lemma of Lax-Milgram the existence of a unique solution w ∈ B for for all ϕ ∈ B. The solution satisfies If now f ∈ H 0,k (Ω; R 3 ) and F ∈ H 0,k (Ω; R 3×3 ) for k = 1, 2, it follows by a different quotient argument that w ∈ H 0,k (Ω; R 3 ) holds and for all ϕ ∈ H 1 (0),per (Ω; R 3 ). Hence, if f ∈ H 1 per (Ω; R 3 ) and F ∈ H 2 per (Ω; R 3×3 ), then w solves the system in a weak sense. Thus with elliptic regularity theory it follows w ∈ H 3 per (Ω; R 3 ) ∩ B. By Theorem A.3 in the appendix, we obtain where we have exploited Using that tr ∂S : With this L −1 h : Y h → X h is a bilinear, bijective and bounded operator, mapping a tuple (f, F ) ∈ Y h to the corresponding solution w ∈ X h of (3.15). In order to close the proof we have to show that G h is a contraction with respect to the relevant norms.
In a first step we assume that w i ∈ X h with for i = 1, 2 and M 1 > 0 to be chosen later. Then The definition of G implies that for k = 1, 2, 3 it holds Hence, analogously as above With the aid of (3.18) it follows for j, k = 1, 2, 3 Thus we obtain in the same manner as above Then we obtain with the Therefore (3.12) and (3.13) hold with the H 1,1 (Ω)-norm on the left hand side replaced by the X h -norm. Using the decomposition B ⊕ span{x → x ⊥ } = H 1 (0),per (Ω; R 3 ) it follows that for for all ϕ ∈ H 1 (0),per (Ω; R 3 ). If now f ∈ H 1,1 per (Ω; R 3 ) we obtain, with a difference quotient argument, that w ∈ H 3 per (Ω; for all ϕ ∈ H 1 (0),per (Ω; R 3 ). Thus with Theorem A.3 the claimed inequalities follow. We define the initial values for the analytical problem as for j = 1, 2 and v 2+j = ∂ 2+j t v| t=0 as above. (3.7),ũ j,h for j = 0, 1, 2 as in (3.10), u 3,h , u 4,h and f h be as above. Then for sufficiently small h 0 ∈ (0, 1] and M > 0 there exist solutions (u 0,h , u 1,h , u 2,h ) of

Lemma 3.2. Letũ h be as in
The solution satisfies for all h ∈ (0, h 0 ] and C > 0 independent of h.

Proof: We can equivalently formulate (3.19)-(3.21) via
and deploying (3.13) we obtain for for all ϕ ∈ B. Defining now the relevant function spaces by with the respective norms defined by .

With this we define the linear operator
Hence L −1 h is a bijective, linear and bounded operator. For the nonlinearity we define 2 and Ω f dx = 0 and G is defined as in (3.14).
We deduce the contraction properties of Q h similar as in the proof of Lemma 3.1. For this we assume that (u 1 , where we used (3.27). Similarly one deduces that for j, k = 1, 2, 3. Analogously we deduce for Q 2,h where we used again Corollary 2.4, |P | h = |P |, |γ h (u 0,h )| ≤ Ch 2 and Finally from Choosing now M 2 ∈ (0, 1] small enough we obtain that We can use an analogous argument as in Lemma 3.1. First it holds, due to (3.6) and (3.5), for M > 0 sufficiently small and with the 1 2 -contraction property we obtain the self mapping of F h,f 0 ,f 1 ,f 2 . Moreover due to the norm on X h and W h we obtain (3.22) and (3.23), respectively.
Finally, the construction ofũ h implies thatũ j,h satisfies for j = 0, 1, 2 and all ϕ ∈ B. This implies with (3.19)-(3.21) for all ϕ ∈ B, where we defined With this it follows max j=1,2 r j,h C 0 (0,T ;L 2 (Ω)) ≤ Ch 3 because of the definition of u 2+j,h and the bound on ∂ t r h . Additionally we have due to Lemma 2.3 and Corollary 2.4, the bounds on (u 0,h , u 1,h , u 2,h ) and ϕ ∈ B as well as .
Regarding the boundary terms we use that tr ∂S : H 1 (S) → H 1 2 (∂S) is linear and bounded. Hence for j = 0, 1, 2 where we used that r N,h C 2 ([0,T ];H 1 (Ω)) ≤ Ch 5 and the Poincaré and Korn inequality for ϕ. Choosing ϕ = u j,h −ũ j,h it follows with an absorption argument max j=1,2 The definition of G implies now because of the bounds for u 0,h and Corollary 2.4. Using (3.30) it follows

Main Result
Moreover, if u h solves (3.1)-(3.4), then Using the structure of u 3,h we obtain Altogether we obtain that u 3,h and u 4,h satisfy (A.1)-(A.3), the necessary conditions for the large times existence result in the appendix. The assumptions on g and the structure of f h imply that (A.4) and (A.5) are fulfilled. Applying Lemma 3.1 and 3.2 we obtain for h 0 sufficiently small, the existence of (u 0,h , u 1,h ,ū 2,h ) such that for all ϕ ∈ B. We use the ansatz for all ϕ ∈ H 1 per (Ω; R 3 ). Then it holds |γ h 2 | ≤ Ch 2 as and |γ h 3 | ≤ Ch 2 with a similar calculation. Lastly, we need to find γ h 4 such that 1 for all ϕ ∈ H 1 per (Ω; R 3 ). Therefore we choose The first and last term can be bounded easily, using Corollary 2.4 For the second part of γ h 4 we use the following equality .
Utilizing the inequality for the initial values (3.24), we deduce Lastly due to the symmetry properties of D 3W , the structure of ∇ hũ0,h and (2.5) it follows Due to the structure of Q and P = ∇x ⊥ it follows Thus, altogether, it follows with |γ h We obtain for h 0 sufficiently small, the existence of (u 0,h , u 1,h , u 2,h ) such that (A.1)-(A.3) are satisfied and max j=0,1 per,(0) (Ω; R 3 )) with ϕ| t=T = 0 and with w j,h := u j,h −ũ j,h , j = 0, 1. Hence with (A.16) we obtain an upper bound for w. For this we use that, due to the structure of r h and r N,h , it follows as a, b, c and v are sufficiently regular. Moreover, using (3.24) Lastly we have to deal with the rotational term. Using the momentum balance law, u 0,h , u 1,h ∈ B and the structure of g, we obtain with Hence it follows ≤ Ch 3 as due to (A.6) for δ = 0, 2. Thus with (A.16) it follows

A Large Times Existence for the Non-linear Problem
The existence of solutions follows from We want to show h-independent estimates for solutions of the linearised system. For this we assume that u h satisfies for 0 < h ≤ 1 sup |α|≤1,k=0,1,2 where R ∈ (0, R 0 ], with R 0 chosen later appropriately small. Lemma A.2. Assume that (A.11) holds, t ∈ [0, T ] and 0 < R ≤ R 0 . Then (A.14) Proof: See [1, Theorem 3.5].
Using the later inequalities and applying the supremum over T ′ ∈ [0,T ] such that RT ≤ κ, κ ∈ (0, 1] it follows Hence, with Young's inequality and κ, thusT , small enough, we can conclude with an absorption argument that . Applying now the Lemma of Gronwall we obtain (A.16) for all 0 < T < ∞ such that RT ≤ κ holds.