Asymptotic variational analysis of incompressible elastic strings

Starting from three-dimensional nonlinear elasticity under the restriction of incompressibility, we derive reduced models to capture the behavior of strings in response to external forces. Our $\Gamma$-convergence analysis of the constrained energy functionals in the limit of shrinking cross sections gives rise to explicit one-dimensional limit energies. The latter depend on the scaling of the applied forces. The effect of local volume preservation is reflected either in their energy densities through a constrained minimization over the cross-section variables or in the class of admissible deformations. Interestingly, all scaling regimes allow for compression and/or stretching of the string. The main difficulty in the proof of the $\Gamma$-limit is to establish recovery sequences that accommodate the nonlinear differential constraint imposed by the incompressibility. To this end, we modify classical constructions in the unconstrained case with the help of an inner perturbation argument tailored for $3$d-$1$d dimension reduction problems.


Introduction
Modern mathematical approaches to applications in materials science result in variational problems with non-standard constraints for which the classical methods of the calculus of variations do not apply. Constraints involving non-convexity, differential expressions and/or nonlocal effects are known to be particularly challenging.
In the context of the analysis of thin objects, interesting effects may occur due to the interaction between restrictive material properties and the lower-dimensional structure of the objects. We mention here a few selected examples: thin (heterogenous) films and strings subject to linear firstorder partial differential equations, which are general enough to cover applications in nonlinear elasticity and micromagnetism at the same time, are studied in [26,27,28], cf. also [24,25]; pointwise constraints on the stress fields appear naturally in models of perfectly plastic plates [15,17]; for work on lower-dimensional material models that involve issues related to non-interpenetration of matter and (global) invertibility, we refer for instance to [29,35,37,41]; physical growth conditions, which guarantee orientation preservation of deformation maps, have been taken into account in models of thin nematic elastomers [2] and von Kármán type rods and plates [16,33].
This paper is concerned with 3d-1d dimension reduction problems in nonlinear elasticity with incompressibility -a determinant constraint on the deformation gradient, which ensures local volume preservation, and is ideal to model e.g. rubber-like materials [34]. To be more specific, we provide an ansatz-free derivation of reduced models for incompressible thin tubes by means of Γ-convergence techniques (see [8,14] for a comprehensive introduction). We take the limit of vanishing cross section, considering external loading of the order of magnitude that gives rise to string type models.
The analogous problem in the 3d-2d context, meaning for incompressible membranes, was solved independently by Trabelsi [38] and Conti & Dolzmann [11] based on different approaches. To overcome the difficulty of accommodating the nonlinear differential constraint when constructing recovery sequences, [11] involves the construction of suitable inner variations. This idea has been applied in the analysis of incompressible Kirchhoff and von Kármán plates [12,32], and lends inspiration to this paper, where we adapt it for 3d-1d reductions.
The first results in the literature to use Γ-convergence techniques to deduce reduced models for thin objects go back to the 1990s, with the seminal works by Acerbi, Buttazzo & Percivale [1] on strings and Le Dret & Raoult [31] on membranes. Notice that in both papers, the authors start from unconstrained energy functionals whose energy densities satisfy standard growth. Before that, common techniques for gaining quantitative insight into thin structures relied mostly on asymptotic expansion methods, and were applied in the setting of linearized elasticity, see e.g. [10,39].
Over the last two decades, the fundamental results in [1,31] have been generalized in multiple directions. This includes for instance the study of membrane theory with Cosserat vectors [6,7], curved strings [36], inhomogeneous thin films [9], thin structures made of periodically heterogeneous material [4,5,30], or junctions between membranes and strings [22].
1.1. Problem formulation. For small ε > 0, let Ω ε := (0, L) × εω with L > 0 and a bounded Lipschitz domain ω ⊂ R 2 represent the reference configuration of a thin unilaterally extended body. Up to translation, we may always assume that the origin lies in ω.
The starting point of our analysis is a three-dimensional model in hyperelasticity with an energy functional (per unit cross section) of the form here, f ε ∈ L 2 (Ω ε ; R 3 ) are external forces and W is a constrained stored elastic energy density enforcing incompressibility, precisely, a continuous function with suitable growth behavior. We give more details on the exact assumptions on W 0 in Section 2.2, see (H1)-(H3). In this model, the observed deformations of the thin object in response to external forces correspond to minimizers (or, if the latter do not exist, low energy states) of E ε . To derive reduced one-dimensional models that capture the asymptotic behavior of these minimizers, it is technically convenient to work with functionals defined on ε-independent spaces, which can be achieved by a classical rescaling argument in the cross section. Indeed, let u ε (x) := v(y) for v ∈ H 1 (Ω ε ; R 3 ) with the parameter transformation y = (x 1 , εx 2 , εx 3 ) for x ∈ Ω := Ω 1 , and suppose for simplicity that f ε is independent of the cross-section variables. Then, the rescaled gradient of u. In analogy to the well-known facts in the context of compressible materials (see e.g. [23]), here as well, the scaling behavior of I ε depends strongly on the external forces f ε . Whenever f ε is of order ε α for some α ≥ 0, then inf u∈H 1 (Ω;R 3 ) I ε (u) behaves like ε β with β = α if α ≤ 2 and β = 2α − 2 if α ≥ 2. Depending on these scalings, one has to expect qualitatively different limit models, falling into the categories of string theory (α = 0), rod theories (α = 2 and α = 3) or other intermediate theories.
Since this article deals with the regimes α < 2 (the cases α ≥ 2 are addressed in a different work, see [19]), it is natural to consider in the following the rescaled functionals I α ε : notice that one may, without loss of generality, omit here the term describing work due to the external forces, for it is merely a continuous perturbation of the total (rescaled) elastic energy.
1.2. Statement of the main results. The new contribution of this paper is a complete characterization of the Γ-limits of sequences (I α ε ) ε as in (1.2) for ε → 0. To be more precise, we prove that under suitable assumptions, (I α ε ) ε Γ-converges with respect to the weak topology in H 1 (Ω; R 3 ) to I α : H 1 (Ω; R 3 ) → [0, ∞] given for α = 0 by and for α ∈ (0, 2) by cf. (2.2), while the convexification W c of W reflects a relaxation process.
The representation formulas (1.3) and (1.4) indicate that the two regimes α = 0 and α ∈ (0, 2) give rise to qualitatively different reduced one-dimensional models. Whereas the latter admits only restricted deformations of the thin object, which can however be obtained with zero energy, the former allows us for any deformation of the string at finite energetic cost.
Despite their differences, both cases share a feature that may seem surprising at first. In fact, the incompressibilty constraint imposed on the three-dimensional elasticity models does not carry over to the reduced ones, in the sense that admissible deformations are not length preserving in general, but can undergo compression and/or stretching. For a similar observation in the context of incompressible membranes, see [11].
1.3. Approach and techniques. The proofs for the cases α = 0 and α ∈ (0, 2) can be found in Section 3 and Section 4, respectively. Overall, our idea is to combine tools from [11] on 3d-2d dimension reduction for incompressible membranes and the references [1,36], where the authors derive one-dimensional models for strings without volumentric constraints.
In both regimes, compactness and the liminf-inequalities are straightforward to show, as they follow immediately from the corresponding results for the unconstrained problems. However, the construction of recovery sequences is more delicate.
The difficulty is to accommodate the incompressibility condition, while approximating the desired limit deformation in an energetically optimal way. To achieve this, we take the recovery sequences from the compressible case -i.e., the ones from [1] if α = 0, and from [36] for α ∈ (0, 2) -as a basis, and modify them with the help of an inner perturbation argument tailored for 3d-1d dimension reduction. The latter, which is stated in Lemma 2.3, is a key ingredient of the proof.
In order to apply Lemma 2.3, though, one needs sequences that are sufficiently regular and whose rescaled deformation gradients have determinant close to 1 up to a small, quantified error. Especially in the string regime α = 0, this requires some technical effort. Indeed, with the help of Bézier curves, we establish a mollification argument for piecewise affine functions of one variable, which, amongst other useful properties, yields uniform bounds on the derivatives, see Lemma 2.4. Moreover, we construct tailored moving frames along the resulting smooth curves in order to guarantee that fattening them to tubes results in deformed configurations that are almost (with controlled errors) locally volume-preserving.

Preliminaries
To start with, we introduce notations and collect a few technical tools.
2.1. Notation. The following notational conventions are used throughout the paper: Let a · b be the standard inner product of two vectors a, b ∈ R 3 , and e 1 , e 2 , e 3 the standard unit vectors in R 3 . On the space R m×n of real-valued m × n matrices, we denote the Forbenius norm by | · |.
Moreover, the closure of a given set U ⊂ R n is denoted by U , whereas U c stands for the convex hull of U . Accordingly, the convex envelope of a function f : R n → R is f c . For the zero level set of f , we use L 0 (f ).
The partial derivative of v : U → R m , where U ⊂ R n is open, with respect to the i-th variable is denoted by ∂ i v, and gradients by ∇v. If v depends only on one real variable, meaning if n = 1, we simplify ∂ 1 v to v . The rescaled gradient of v : U ⊂ R 3 → R 3 is given by (2.1) We employ standard notation for Lebesgue and Sobolev spaces as well as for spaces of continuous and k-times continuously differentiable functions; in particular, L 2 (U ; R m ), H 1 (U ; R m ) and C k (U ; R m ) with k ∈ N 0 for an open set U ⊂ R n . We shall equip the latter with the norm To shorten notation, let L 2 (a, b; R m ) : If Ω = (0, L)×ω as in the introduction, without explicit mention, we identify any u : (0, L) → R 3 with its trivial extension to a function on Ω; in particular, given sufficient regularity, ∂ 1 u and u are used interchangeably.
Finally, O(·) is the well-known Landau symbol.

2.2.
Hypotheses and properties of W 0 and W . Consider the following regularity and growth assumptions for the energy density W 0 : (H3) there are constants C 3 , c 3 > 0 such that Clearly, (H3) implies (H2).
Recalling the definition of W in (1.1), we define W : R 3 → [0, ∞] by minimizing out the cross-section variables, that is, for ξ ∈ R 3 . Notice that the hypotheses (H1) and (H2) guarantee that the infima in (2.2) are attained.
As we detail next, the convexification W c of W inherits growth properties from W 0 .
The following lemma collects a few basic properties of the zero level sets of W and W c .
Remark 2.2. Suppose that W 0 satifies (H1) and (H2). a) The growth assumption (H3) implies immediately that c) A frame-indifferent single-well energy density W 0 vanishing at the identity, has SO(3) as zero level set of W 0 . Hence, L 0 (W ) = {ξ ∈ R 3 : |ξ| = 1} in light of b), and after convexification, 2.3. Technical tools. The following auxiliary result on inner perturbations is a key ingredient for the construction of recovery sequences, both in the regimes α = 0 and α ∈ (0, 2), since it allows us to modify sequences subject to the incompressibility constraint in an approximate sense into ones that satisfy it exactly. Analogous techniques applicable to 3d-2d dimension reduction problems were first introduced in [11] and later exploited in [12,32]. We adapt the method to the 3d-1d context, where perturbing one of the cross-section variables, instead of both, is enough to realize the desired determinant condition.
Proof. We subdivide the construction of a sequence (ϕ ε ) ε ∈ C 1 (Q L ; J ) satisfying the conditions (2.6) and (2.7) into two steps. The arguments are strongly inspired by the ideas and techniques of [11,12].
Note that according to (2.5), there is a constant l > 0 such that for all ε > 0 sufficiently small.
Step 1: Implementation of the determinant constraint. Recalling the definition of the rescaled gradients in (2.1), we deduce from the chain rule that .7) is fulfilled if ϕ ε solves the following initial value problem: For each notice that in view of (2.8), the denominator on the right-hand of the differential equation is in particular non-zero; also, the choice of initial conditions is indeed admissible, considering that 0 ∈ J. The existence of a unique solution ϕ ε ∈ C 1 (Q L ; J ) to the initial value problem in (2.9) with continuously differentiable dependence on the parameters x 1 and x 2 follows from standard ODE theory. More precisely, the argument is based on Banach's fixed point theorem, see e.g. [40,III, §13, Satz II and IV]; note that in our case, the contraction may be defined on C 0 (Q L ; J ), since (2.5) implies that for any φ ∈ C 0 (Q L ; J ) and x ∈ Q L , provided ε is small enough.
Step 2: Estimates for ϕ ε . To verify (2.6) for the previously constructed ϕ ε , we are going show that Indeed, (2.10) follows from in combination with (2.5), and it suffices for (2.11) to observe that for any x ∈ Q L . For the proof of (2.12), it is convenient to rewrite (2.9) equivalently in terms of the integral equation (2.14) By the Leibniz integral rule, differentiating (2.14) with respect to x 1 leads to for x ∈ Q L , and hence, along with (2.8), In view of (2.13) and (2.5), this gives the first part of (2.12). The second part involving ∂ 2 ϕ ε follows in the same way.
The next lemma provides a technical tool that, intuitively speaking, allows us to round off the corners of a piecewise affine curve in such a way that the resulting mollification is a regular curve and still piecewise affine on most of its domain. This can be achieved with the help of Bézier curves, see e.g. [20]. Although there is a substantial literature on the subject, we have not been able to track down the specific statement needed for the construction of a recovery sequence in Theorem 3.1. We present a self-contained proof in the appendix. Then there exists a sequence (u i ) i ⊂ C k ([0, L]; R 3 ) with the following three properties: 3. The regime α = 0 The first main result derives an effective one-dimensional model for incompressible elastic strings via a Γ-convergence analysis of the energies I α ε with α = 0 in the limit of vanishing ε. Its two main characteristics reflect an optimization over all deformations of the cross section at finite thickness and a relaxation procedure minimizing the energy over possible microstructures.
Proof. The overall idea of the proof follows along the lines of [11], but the arguments need to be suitably modified for this setting of 3d-1d reduction. This involves a taylored mollification and frame construction for piecewise affine regular curves, as well as elements from [1], see also [36]. The key ingredient for realizing the volumetric constraint in the construction of the recovery sequence is Lemma 2.3.
Part I: Lower bound and compactness. Let (u ε ) ε ⊂ H 1 (Ω; R 3 ) be a sequence of functions with zero mean value and uniformly bounded energy regarding (I 0 ε ) ε . Due to W ≥ W 0 and the fact that, by assumption, W 0 satisfies the necessary properties to apply the compactness of the corresponding compressible problem (see e.g. [1, Theorem 2.1]), we conclude the existence of a subsequence of (u ε ) ε converging weakly in H 1 (Ω; R 3 ) to some u ∈ H 1 (0, L; R 3 ). Since W c is convex, continuous and bounded from below, the functional which is the desired liminf-inequality. Part II: Upper bound. Let u ∈ H 1 (0, L; R 3 ). For easier reading, we subdivide the argument into several steps.
Then, in light of the continuity of W c and its quadratic growth (2.3), we can pass to the limit via the Vitali-Lebesgue convergence theorem to obtain Step 2: Relaxation. Next, we will construct a sequence in particular, this means that v j = 0 a.e. in (0, L) for j ∈ N.
For j ∈ N, letṽ j be the function from Step 1, and denote the finitely many disjoint open subintervals of (0, L) on whichṽ j is constant byĨ (n) j with n = 1, ...,Ñ j ; notice that The idea is to modifyṽ j suitably on eachĨ the add-on (3.4) follows via a simple refinement argument, just replace φ (n) j with a piecewise affine function that consists of multiple scaled copies of the latter.
Define v j :=ṽ j + Hence, letting j → ∞ implies (3.3) in view of (3.2), as well as v j → u in L 2 (0, L; R 3 ) due to (3.1). The observation that (v j ) j is uniformly bounded in H 1 (0, L; R 3 ) as a consequence of the lower bound in (2.4), allows us to conclude the desired weak convergence v j u in H 1 (0, L; R 3 ).
Step 3: Mollification of the piecewise affine approximations. With the help of Lemma 2.4 applied to each v j from Step 2 and a diagonalization argument, we obtain a sequence of functions (u j ) j ⊂ C 3 ([0, L]; R 3 ) with these properties: (i) for every j ∈ N there are 0 < l j ≤ L j such that l j ≤ |u j | ≤ L j in [0, L]; without loss of generality, we may assume that l j ≤ 1; (ii) for every j ∈ N there are finitely many disjoint open intervals I , and hence by Step 2, Step 4: Taylored frame. For any curve u j with j ∈ N as in the previous step, let n j ∈ C 2 ([0, L]; R 3 ) be a normal unit vector field along u j , meaning u j · n j = 0 and |n j | = 1 everywhere in [0, L]; we may assume without restriction that n j is constant whenever u j is. Moreover, define b j := u j × n j |u j × n j | 2 ∈ C 2 ([0, L]; R 3 ); (3.6) indeed, the denominator in (3.6) is non-zero, because n j is orthogonal to u j and u j a regular curve by Step 3 (i). By definition, the triple (u j , n j , b j ) forms an orthogonal moving frame along the trajectory given by u j . Our aim in this step is to modify this moving frame into a version that is well-suited for the construction of an approximating sequence for u along which the energies converge as well, cf.
To this end, recall that u j is affine on each I (n) j with n = 1, . . . , N j according to Step 3 (ii), that is, in particular, det(ξ we remark that the last two quanities are well-defined due to (3.10) and the fact that j e 2 = 0. Next, we collect a few useful properties of the newly constructed moving frames (u j ,n j,δ ,b j,δ,η ). Setting F j,δ,η := (n j,δ |b j,δ,η ) ∈ C 2 ([0, L]; R 3×2 ), we observe that with a constant C > 0 independent of j, δ and η; see again Step 3, where the constants L j and l j have been introduced. Indeed, to see (3.12), we infer from (3.10) together with the estimate for all n = 1, . . . , N j and j ∈ N. Hence, in view of (3.7) and Step 3 (i), combining (3.15) with (3.13) and (3.14) eventually implies (3.12).
Proof of Theorem 4.1. Under consideration of Lemma 2.3, the proof of the upper bound comes down to a modification and generalization of the construction in [36,Theorem 4.5]. The compactness and lower bound follow as an immediate consequence of the respective results for the case α = 0.
Part II: Upper bound. Let u ∈ H 1 (0, L; R 3 ) and assume that u ∈ L 0 (W ) c a.e. in (0, L), otherwise there is nothing to prove. We proceed in two steps.
Since β < 1 2 , the sequence (∇ ε v ε ) ε is bounded in C 0 (Q L ; R 3×3 ). Moreover, the function v ε satisfies the incompressibility condition exactly except on sets of small measure, where det ∇ ε v ε is close to 1. To quantify this statement, we compute det ∇ ε v ε (x) = 1 + ε det P ε (x 1 )(x 2 e 2 + x 3 e 3 )|P ε (x 1 )e 2 |P ε (x 1 )e 3 for x ∈ Q ε := Γ ε × J × J , and observe that Thus, it follows in view of (4.2) that Now, with (4.5) at hand, we are in the position to apply Proposition 2.3 to the sequence (v ε ) ε with γ = 1 − 2β to obtain a modified sequence (u ε ) ε ⊂ C 1 (Ω; R 3 ) that satisfies det ∇ ε u ε = 1 everywhere in Ω, namely with ϕ ε ∈ C 1 (Q L ; J ) such that (2.6) holds. Notice that the inner perturbation defining u ε corresponds to the identity map on Q L \ Q ε , since, due to (4.4), the ordinary differential equation in (2.9) reduces to ∂ 3 ϕ ε = 1 on this set; thus, along with (4.1), on Ω \ Q ε . (4.6) Step 1: The case without reversions. First, we will prove the statement under the assumption that neither two ξ (n) and ξ (n+1) from (4.9) are anti-parallel. Without loss of generality, it suffices to detail the case N = 2, where u takes only the two values ξ (1) and ξ (2) . For general N , one can simply repeat the same construction.
Step 1a: Definition of suitable Bézier curves. For η > 0 sufficiently small, we choose 2k + 1 control points around u(t (1) ) by Then, (4.11) Based on the control points in (4.10), we consider the Bézier curve B η : R → R 3 by where b q,p : R → R are the Bernstein polynomials, cf. Lemma 4.3. After suitable reparametrization, (4.12) provides a mollification of u via see Figure 2.
Similar calculations, invoking again the properties of Bernstein polynomials, in particular Lemma 4.3 d), give (4.15).
Step 2: The general case with reversions. The idea is to reduce the argument to the situation of Step 1 via a loop construction and to conclude with a diagonalization argument.
In the following, let I stand for the index set consisting of all n ∈ {1, . . . , N − 1} such that ξ (n) and ξ (n+1) are anti-parallel, that is, for some ν n > 0.
Step 2a: Loop construction. Without loss of generality, I is a singleton, say I = {1}; otherwise the argument below can be performed analogously for all (finitely many) elements in I. Besides, as in Step 1, we take N = 2 to keep notations simple.
The next lemma gathers some basic facts about Bernstein polynomials, which were an important ingredient in the definition of Bézier curves in the previous proof. For more details, we refer the reader e.g. to [18,20].