Adhesive flexible material structures

We study variational problems modeling the 
adhesion interaction with a rigid substrate for elastic strings and 
 rods. 
 We produce conditions characterizing bonded and detached states as well as 
optimality properties with respect to loading and geometry. 
 We show Euler equations for minimizers of the total 
energy outside self-contact and secondary contact points with the 
substrate.


Introduction.
At the fundamental level of some recent fields of research such as nanoscale engineering and biophysics there is the need of a fine understanding of the behaviour of thin flexible material structures involved in complex interactions. We consider one-dimensional nanostructures governed by surface-tension forces and adhesive forces, like nanotubes, nanowires and biopolymers adhering on different material substrates. The small scale interactions of these material components are crucial in the study of biological adhesion and the development of nanoelectronics and nanocomposites as well as MEMS (micro electronic mechanical systems) and NEMS (nano electronic mechanical systems) devices ( [29], [32]): e.g. super coiled DNA molecules, bacteria filaments, gecko inspired materials, actuators, etc. ( [10], [15], [16], [24]). It has been shown that the peeling of long slender molecules and nanostructures from substrates involves a strong coupling between elasticity, friction, and adhesive forces at the nanoscale. At these scales, if carbon nanotubes can adhere to each other under the influence of capillary forces, fluid-regulated forces are not the only factors that must be examined and dispersion or van der Waals forces may become more important than at larger scales, as well as the microscopic intermolecular forces ( [13]) of extended media start to have a macroscopic effect on structural stability ( [8], [2], [22], [23]).
The previous considerations sketch the physical framework in which we move from the mathematical perspective with the aim of establishing sufficiently general and, as far as possible, simplified mathematical models capturing the essentials of the involved phenomena. In particular, here we intend to develop the ideas exposed in [17], [18], [19] (where only linear elastic behaviour was considered) [7], [21], [25], 554 FRANCESCO MADDALENA, DANILO PERCIVALE & FRANCO TOMARELLI [26] and [27] by focusing on nonlinear models of the structural behaviour. The aim is a variational scheme for the study of the adhesion interactions of one-dimensional non linear elastic filaments and curved rods.
In Section 1 we study elastic models whose bulk energy is characterized by shear deformations, under the simplifying assumption that the rigid substrate boundary is a graph: we study the adhesion regime and focus the attention on the main features regulating the mechanical behaviour. In particular we show that the debonded state depends on the constitutive parameters and on the length of the curve representing the substrate boundary, but it does not depend on the shape of such a curve (Theorems 1. 3, 1.4).
In Section 2 we study a geometrically nonlinear adhesion model governed by curvature elasticity: the bulk energy density is a measure of the curvature gap between the rod and the rigid substrate, precisely we focus our analysis on the minimization of the functional where u * and u denote respectively the unloaded and loaded rod, κ is the scalar curvature, the flexural rigidity of the rod is given by the product EJ of the Young modulus E times the moment of inertia J of the cross-section of the rod, f is a given load acting at the endpoint , adhesion energy W ψ and load potential W f are expressed respectively by where Ω ⊂ R 2 is an open set. The rigid substrate is given by the set Ω \ R 2 in this notation.
We emphasize that A contains also non-simple curves, nevertheless self-crossing of the rod is always forbidden in A, while self-contact of the rod without interpenetration may take place (see Lemma 2.3, Definition 2.7, Definition 2.10, Theorem 2.11): coincidence of the tangents must hold true (up to the sign) at any multiple point (Theorem 2.2). The set A allows also configurations undergoing secondary contact with the rigid substrate at detached points of the rod. We analyze general conditions regulating bonded and debonded states of the rod and, in particular, we deduce precise relationship governing the case of strong adhesion (Theorem 2.12, Corollary 1, Remark 6) in which the whole rod remains bonded to the substrate. This suggests a shape optimization problem (Remark 7) in view of finding the unloaded curve realizing the strongest adhesion. Several properties of functional (1) are proven in the last sections: in Section 3 we derive necessary conditions of minimality, precisely we deduce the Euler-Lagrange equation (75) of a detached solution in a general geometry; such equation retrieves Euler elastica equation when the substrate is flat and the rod is compressed; in Section 4 we show explicit conditions (Theorems 4.3, 4.4) for detachment of rectilinear rods by exploiting an auxiliary rescaled functional.
About motivations for taking into account only scalar curvature in functional (1) we refer to [1] and to a forthcoming paper [20] where justification of this assumption is deduced by a dimension reduction via scaling arguments. We refer to a forthcoming paper also for the analysis of local minimizers (related to buckling phenomenon), which is motivated by data of type described in Example 2.13 and Example 3.2 and can be performed by exploiting the Euler equation (75) itself.
1. Adhesion of shearable elastic strings to a rigid substrate. In this section we study a shearable elastic string modeling a viscous fluid filament. The filament is bonded to a rigid substrate through a thin adhesive layer and undergoes prescribed displacements at the endpoints. We assume that the rigid substrate is given by the subgraph of a given scalar function h ∈ C 1 ([0, 1]) and that the reference configuration Γ of the string is the graph of h: hence Γ is a C 1 regular curve. We denote by γ the parametrization of Γ and by L its length: The unit normal vector to the curve is inward oriented with respect to the rigid substrate: Let u : Γ → R 2 be the displacement field of the string which can be represented by . By setting w = v · n we study the following expression of the elastic (shearing) energy of the string: where the density of elastic energy is given by k denotes the stiffness of the string and D t represents the tangential derivative on the curve Γ. The admissible displacements are confined in the epigraph of h: explicitely they belong to the set v ∈ H 1 ((0, 1); Since we suppose that the displacements are small, the nonlinear and nonconvex constraint (6) is equivalent, up to higher order terms, to The adhesion interaction of the string with the substrate adds a contribution W (u) to the energetic competition: the length of the detached set J u , that is where the detached set is represented by and is a given constitutive parameter. Therefore the total energy is given by the functional Alternatively, if I w is the set of parameters related to the detached set, then we may minimize the functional in the admissible set Actually and minimizing G over w in W γ is equivalent to the minimization of F over the set Theorem 1.1. Assume (10), (12), (13). Then the functional G achieves a nonnegative minimum in the set W γ , whenever we fix the C 1 regular graph γ.
Proof. Notice that min |γ| ≥ 1 since γ is a graph. Then the elastic term in (13) is lower semicontinuous in H 1 (0, 1). If w k w in H 1 (0, 1) then w k uniformly converges to w hence, by w k ≤ 0, lim inf k 1 Iw k ≥ 1 Iw ; therefore Then G is sequentially lower semicontinuous in H 1 ((0, 1)). Since G is coercive and nonnegative, we get the thesis by applying a standard compactness argument.
Theorem 1.4. Assumptions (10), (12), (13) entail that only one of the following two alternatives hold true. If then the detachment parameter ξ is the unique solution in (0, L) of and then the detachment point is ξ = 0, i.e. we obtain a complete debonding, and Proof. By Theorem 1.3 there is detachment at ξ < 1. Then the theorem follows by minimizing over ξ ∈ [0, 1[ the function 2. Adhesion of elastic rods to a rigid substrate. We focus our attention on the adhesion of an Euler rod which is glued to a rigid substrate, clamped at one end and loaded at the other one. The aim of this section is to give some condition on the load in order to avoid the detachment. We study the adhesion phenomenon in the context of non linear elasticity by considering the bulk energy density as the curvature gap between the rod and the support.
We denote the standard basis of R 2 by {e 1 , e 2 } so that the clockwise rotation of π/2 is expressed by

FRANCESCO MADDALENA, DANILO PERCIVALE & FRANCO TOMARELLI
We fix a C 2 -regular open subset Ω of R 2 . Ω is the region where the obstacle geometry allows the rod to undergo deformations. The unstressed bonded configuration Γ of the rod is a (not necessarily flat) portion of ∂Ω such that H 1 (Γ) = L, with 0 < L < ∞. ψ is the cost function to detach a unit length of the road. We assume We use the notation and we introduce the following parametrization u * of Γ with respect to the arc length and related regularity assumptions: The above parametrization is chosen in such a way thatu * provides the standard positive orientation of the boundary ∂Ω and n Ω = Wu * = n u * is the unit outward vector normal to ∂Ω. We describe the admissible region Ω as the non-positivity set of the signed distance ϕ from Ω itself, say and we assume In order to describe peeling the elastic rod Γ off the substrate, we fix f ∈ R 2 as the given concentrated load acting at the endpoint of the rod. For every u ∈ H 2 (0, L; Ω) such that |u| = 1 a.e. in [0, L] we label κ the scalar curvature, say κ(u) =ü · n u =ü · Wu (36) so that κ fulfills the identity |κ(u)| = |ü|.
It is worth noticing that for any such u we haveü ·u = 0 a.e. henceü = κ(u)Wu. The elastic rod is clamped at s = 0 and confined in Ω. We define the set A of the admissible configurations of the rod, by setting A is the closure of A in the weak topology of H 2 ((0, L); Ω).
We emphasize that, in contrast with the notation adopted in Section 1, for the rest of the paper u represents the deformation (and not the displacement) of the rod. The total energy of the rod (at equilibrium under the adhesion force and the given load acting at the endpoint s = L) is expressed by the functional Theorem 2.1. Assume (30)-(42). Then the functional F admits minimizers.
Proof. Since the rod is clamped at . Hence F is bounded from below and coercive. Then every minimizing sequence, say (u n ) n∈N , is bounded in H 2 (0, L; R 2 ) hence, up to subsequences, bothu n and u n are uniformly convergent in [0, L]. Lower semi-continuity of F follows by convexity, (36) and (37).
From now on (Sections 2, 3, 4) we shall use the short notation argmin F in place of argmin A F , always referring to the functional F defined by (40).
The set A 0 will be shortly denoted by A.
The curves in A τ may lack injectivity in (0, L), nevertheless the self contact is allowed only without crossing, as it is clarified in the sequel by Definitions 2.7, 2.10, and Theorems 2.2, 2.11 .
Definition 2.4. We will denote by A the subset of u ∈ A such that there exists a value ξ u ∈ [0, L) with

Remark 1. By virtue of Lemma 2.3 we get argmin A F ⊂ A and min
We emphasize that this value ξ u coincides with the one introduced in Lemma 2.3. This is the reason why they are labeled in the same way. The detachment parameter ξ u will be shortly denoted by ξ whenever there is no risk of confusion.
Definition 2.6. We say that x ∈ ∂Ω is a secondary contact point of u ∈ A with the substrate if there exist s ∈ (ξ u , L] with x = u(s) and ϕ(x) = 0.
We notice that self-contact points may have multiplicity bigger than 2: this happens whenever the cardinality of u −1 (x) is bigger than 2.  Remark 3. The property u ∈ A does not exclude self-contact points x, moreover Theorem 2.2 entails that all oriented tangent vectors at a self-contact point x must coincide up to the sign if x = u(L). Nevertheless self-crossing is forbidden for any u in the set of admissible configurations A even when self-contact takes places, as it is clarified by the following statements.
It is easy to show that if (33), (34), (38) and (39) hold true, then u does not undergo any isolated self-contact: it does not exist any x = u(s) = u(s) such that s =s and x is the only self-contact point of u in a small ball B δ (x).
In general self-crossing of a C ∞ curve may take place in a more complicate situation than the case of an isolated self-contact point. For instance the crossing may take place at a point x where an infinite set of self-contact points accumulate. In order to show that no kind of self-crossing ever occurs in the admissible set of configurations A , we introduce a general definition of self-crossing (Definition 2.10), then we show that self-crossing cannot occur: see Theorem 2.11 which excludes also the simple case of isolated self-contact points.
Definition 2. 8. Given an open bounded set V ⊂ R 2 and points x, y, z ∈ ∂V such that ∂V is a Jordan curve and x = y = z = x, we say that x < y < z if the path connecting x with z and passing through y on ∂V has the positive orientation induced by ∂V . Lemma 2.9. Assume V ⊂ R 2 is an open bounded set, ∂V is a Jordan curve, x 1 , x 2 , x 3 , x 4 ∈ ∂V with x 1 < x 2 < x 3 < x 4 and γ 1 , γ 2 are continuous curves in V joining respectively x 1 to x 3 and x 2 to x 4 , with u((s 1 , s 3 )) ⊂ V and u((s 2 , s 4 )) ⊂ V . Then γ 1 meets γ 2 in V : say, ∃ x ∈ V ∩ u((s 1 , s 3 )) ∩ u ((s 2 , s 4 )).
Proof. Up to an homeomorphism we can assume V is a square and x j are the midpoints of the four sides. V \ γ 1 is an open set. It is enough showing that V \γ 1 is disconnected and both x 2 , x 4 belong to the boundary of different connected components but not to their intersection. By contradiction, if V \ γ 1 is connected, then there exists a polygonal curve η connecting x 2 and x 4 .
We claim that η disconnects V and x 1 , x 3 are in different connected components, hence η ∩ γ 1 ∩ V = ∅. The fact that η disconnects V is a consequence of Jordan Curve Theorem applied to the simple closed polygonals defined by η together with the paths x 4 , x 1 , x 2 and x 2 , x 3 , x 4 along ∂V . For the sake of completeness we provide an elementary direct proof of disconnectedness by exploiting well known classical tools ( [6]) adapted to the easy context of polygonal curves. First, assume no horizontal segment is contained in η. Then for any x ∈ V \ η set ϕ(x) = 0 or ϕ(x) = 1 if the horizontal right half-line starting at x crosses η respectively an even or odd numbers of times (crossing at a vertex of η counts zero if the two segments lie on the same side of the half-line and one otherwise). For j = 0, 1 both A j = ϕ −1 (j) are open sets and nonempty (in a small horizontal strip between x 2 and the second lower vertex of η ϕ takes both values 0, 1). Moreover, ϕ(x) = ϕ(y) whenever the closed segment from x to y belongs to V \ η; hence A 0 ∩ A 1 = ∅ and ϕ(x 1 ) = ϕ(x 3 ). If an horizontal segment belongs to η, then we modify V into a parallelogram V by gluing two triangles at lower and upper basis in such a way the oblique edges are not parallel to any segment in η and modify η in η by adding two vertical segment from x 2 and x 4 reaching ∂ V . Then the previous technique can be applied with half-lines equi-oriented and parallel to oblique edges. ii)  Proof. Assume by contradiction that u undergoes self crossing, hence we choose the notation as in Definition 2.10. Then there exists a sequence of simple curves (u k ) k∈N such that u k weakly converges to u in H 2 (0, L), hence it converges uniformly in [0, L]. We assume u(s 2 ) = x 2 , u(s 4 ) = x 4 (the opposite case can be dealt exactly in the same way). Without loss of generality we can assume V ∩u k ((s 1 , s 3 )∪(s 2 , s 4 )) = ∅ for k > 4. So we can define s k , t k as follows, for k > 4 : By iii) in Definition 2.10 and u k (s) → u(s) ∈ V for all s ∈ (s 1 , s 3 ), we get If then only one of the following mutually exclusive four cases may occur.
for large k and u k (s k , t k ) ∩ u k (s 2 , s 4 ) = ∅ by Lemma 2.9. This is a contradiction since u k is a simple curve. II) s k = s 1 , u k (s 1 ) ∈ V and u k (t k ) ∈ ∂V for infinitely many k ∈ N. By referring to this subsequence and without relabeling we consider the curve v k whose image is the union of u k ([s 1 , t k )) and the segment σ k joining u k (s 1 ) with By construction v k is a curve whose endpoints belong to ∂V and are, for k large enough, close to x 1 and x 3 respectively; moreover its inner part is contained in V , hence by Lemma 2.9 it intersects u k ((s 2 , s 4 )). Since u k is simple , the intersection belongs to the support of σ k and therefore there exists τ k ∈ [s 2 , s 4 ] such that u k (τ k ) belongs to the support of σ k , hence u k (τ k ) → x 1 . Up to subsequences τ k → τ . It is readily seen that neither τ ∈ {s 2 , s 4 } (otherwise III) t k = s 3 , u k (s 3 ) ∈ V and u k (s k ) ∈ ∂V for infinitely many k ∈ N. This case can be dealt as II) by interchanging the role of s k and t k .
IV) u k ([s 1 , s 3 ]) ⊂ V for infinitely many k ∈ N. In this case we exploit the curve w k whose support is the union of u k ([s 1 , s 3 ]) and of the segments σ k , η k (defined as in cases II), III)) which join respectively u k (s 1 ) and u k (s 3 ) with suitable points in the segments from u k (s 1 ) to x 1 and from u k (s 3 ) to x 3 . Then to proceed as in the cases II), III). V) Otherwise, if (47) fails then, up to subsequences, there exists θ k ∈ (s k , t k ) such that u k (θ k ) ∈ ∂V . Then at least one of the two sets We get s k < t k in any case. Since u k (s) → u(s) ∈ V ∀s ∈ (s 1 , s 3 ) we get s k → s 1 , t k → s 3 . Then by substituting s k to s k and t k to t k we may repeat the discussion in I), II), III), IV).

Remark 5.
Let us observe that the proof of Theorem 2.11 dos not make any use of the C 1 -regularity of (u k ) k∈N and u since we have preferred to set the problem of self-crossing into a more general context than the one strictly needed to prove the theorem: indeed only continuity and uniform convergence of the curves has been employed in the proof.
We are now in a position to show conditions which exclude the detachment of the rod. Explicit conditions entailing detachment will be given in Section 4.
and let u ∈ argmin F. Then u ≡ u * .
Proof. An integration by parts and the conditionu(ξ) =u * (ξ) show that and since by using (44), (45) and |u| = 1, we get and the proof is achieved.
Remark 6. We underline the dependence of the right hand side of (52) on the physical and geometrical characteristics of the structure: in particular the dependence on the ratio EJ/L α (α > 1) which is crucial in the study of elastic stability, while the dependence on the ratio µ/ κ(u * ) L p (0,L) says that the constitutive property of adhesion material and the substrate curvature determine the overall adhesion strength.
Remark 7. The right-hand side of (52) can be thought of as a measure of the global adhesion strength of the rod glued in the configuration u * . This perspective leads to the formulation of the following optimization problem: find a curve maximizing the global adhesion strength among the closed curves Γ which enclose a connected region with fixed area", via the minimization of functionals of type Similar minimization problems are studied also in image segmentation and image inpainting: we refer to [4], where the relaxed formulation of (53) in the class of varifolds is studied.
When the force field has the same direction of the inner normal to the rigid substrate, then intuition suggests that minimizers coincide with the fully bonded rod, since admissible deformations are allowed to stay only in the complementary region of the rigid obstacle. Indeed this is not true in general: precisely the following statement shows that, if Ω is convex (say, the substrate is concave), then this intuition is correct; on the other hand Example 2.13 shows that, if Ω is concave then it fails to be true. Proposition 1. Assume (30)-(42), u ∈ argmin F , ϕ is a convex function and Then u ≡ u * .

FRANCESCO MADDALENA, DANILO PERCIVALE & FRANCO TOMARELLI
Unfortunately the above result is true only in the case the admissible deformations take place in a convex set, as we can show in the following Example 2.13. We choose ϕ(x) = 1 − |x| 2 , Ω = R 2 \ B 1 (0) , u * (s) = (cos s, sin s), s ∈ [− π 2 , π], and f = f e 1 . By assumingv(s) ≡ e 1 with v(− π 2 ) = u * (− π 2 ) = −e 2 we get ϕ(v(s)) ≤ 0 with strict inequality for −π/2 < s ≤ L . Then, by taking f sufficiently large, the energy of v becomes strictly negative, therefore u * cannot be a minimizer: The previous example suggests that a more accurate description of the problem requires a careful analysis of the local minimizers besides the study of global minimizers which we are considering in the present work.
3. Euler equations for a detached rod. In this section we assume a general geometry of the substrate as described by (33)-(35) and look for necessary conditions fulfilled by an optimal configuration u in case of detachment state. Under the assumptions (30)-(42) we fix a detached state u ∈ argmin F. Then, by Lemma 2.3, we can assume that the detachment parameter ξ = ξ u is such that Let M ∈ SO(2) be an orthogonal matrix, then by Euler Formula there is ϑ ∈ [−π, π) s.t. M represents a rotation of angle ϑ in R 2 : where W is given by (29) and I is the identity matrix. We can represent any admissible configuration v ∈ A of the rod as followṡ by selecting a continuous branch ϑ v of the multi-valued function Θ v (oriented angle between v and u * ) so that (2)).
The restriction of u to the interval [ξ u , L] minimizes We look for necessary conditions of minimality. So we study variations of F ξ around a curve u, whose restriction in [ξ, L] is a global minimizer in A ξ . In order to perform these variations correctly, since u may undergo self-contact and/or secondary contact with the substrate, we can perform bilateral variations only in the last interval avoiding these interactions. Here the last interval refers to the one with endpoint L: such interval exists only if secondary contact and self-contact points do not accumulate at u(L).
By (36) we have (72) By taking into account (71) and (36) we evaluate the functional (61) at v ε , we get where the functional I ξu of the angular function is defined as follows:
Remark 8. The right-hand side of (65) is equal to where ϕ * (s) denote the positively-oriented angle between f and u * (s) .
Hence the Euler equation (65) reads as follows: we emphasize that whenever ϕ * (s) ≡ kπ, k ∈ Z, by equation (75) we retrieve the well known Euler elastica equation.
we must consider only ξ and related v ϑ ξ such that v ϑ ξ ∈ A and v ϑ ξ does not undergo neither self-contact nor secondary contact points in (ξ, L], that is v ϑ ξ (s) ∈ Ω ∀s ∈ (ξ, L] and v ϑ ξ is injective in [ξ, L]. Proof. By multiplying for ϑ both the terms in (65), after integrating and taking into account (66) and (75) we get (79). After a simple substitution (80) follows.
A slight modification of Example 2.13 provides a simple explicit solution of the nonlinear equation (75) fulfilling boundary conditions (66), as shown by the following example.
Indeed we have ξ w = 0 and ϑ w (s) = ϕ * (s) for every s. Then, by taking f sufficiently large we have F(w) < F(u * ). It is easy to verify that u * is a strict local minimizer for F in the weak topology of H 2 (0, L; Ω), moreover u * seems to be the physical solution since u * cannot snap to w without over leaping a potential wall. It seems reasonable also that w is a global minimizer for F in A, though we are not able to prove this point.

4.
Explicit conditions for rod detachment from a flat substrate. In this section we suppose the reference configuration of the rod is glued to a flat substrate. More precisely, we assume (38)-(42) together with We introduce an auxiliary problem for a re-scaled version of the functional F. First we define an auxiliary functional J as follows: where B is the closure in the weak topology of H 2 ((0, 1); R 2 ) of the set B, and In order to prove that J admits global minimizers via direct method in the calculus of variations it is enough showing that J is lower semicontinuous in the product of [0, L] and H 2 ((0, 1); Ω) endowed with euclidean and weak convergence respectively. This property is proved by the following Lemma. Such relationship follows by Poincarè and Young inequalities: Then the thesis follows by We prove now that minimization of J and F are equivalent problems. • If u has a detachment parameter ξ u < L, then by setting we have that (ξ u , w) ∈ argmin J . Conversely let (ξ, v) ∈ argmin J then belongs to argmin F. In addition if ξ < L we get u(t) · e 2 > 0 in (ξ, L) and v(t) · e 2 > 0 in (0, 1).   thus contradicting minimality of u. The case ξ = L can be treated analogously.
In order to prove the converse we may notice that we have only to show that v 2 > 0 in (0, 1) whenever ξ < L: if this were not true, then by proceeding as in the proof of Lemma 2.4, we may show that there exists a unique 0 < τ < 1 such that v(t) = te 1 in [0, τ ] and v(t) = te 1 for every t ∈ (τ, 1]. Then we may choose 0 < δ < τ and by setting w(t) = (1 − δ) −1 v(δ + t(1 − δ)) we get, by taking into account that v(t) = te 1 in [0, δ], a contradiction that completes the proof.
The equivalence Theorem 4.2 provides additional informations on the structure of global minimizers of F. For instance, in the present context of flat substrate, if ψ grows slowly enough then either the rod stays bonded to the substrate or it is fully detached, as stated by the following Theorem. (93) Then either u ≡ u * or the detachment parameter fulfills ξ u = 0.
A necessary condition for a complete peeling of the rod is given by the following Theorem.