A 3D Griffith peeling model to unify and generalize single and double peeling theories

It has been shown in recent years that many species in Nature employ hierarchy and contact splitting as a strategy to enhance the adhesive properties of their attachments. Maximizing the adhesive force is however not the only goal. Many animals can achieve a tunable adhesive force, which allows them to both strongly attach to a surface and easily detach when necessary. Here, we study the adhesive properties of 3D dendritic attachments, which are structures that are widely occurring in nature and which allow to achieve these goals. These structures exploit branching to provide high variability in the geometry, and thus tunability, and contact splitting, to increase the total peeling line and thus the adhesion force. By applying the same principles presented by A.A. Griffith 100 years ago, we derive an analytical model for the detachment forces as a function of their defining angles in 3D space, finding as limit cases 2D double peeling and 1D single peeling. We also develop a numerical model, including a nonlinear elastic constitutive law, for the validation of analytical calculations, allowing additionally to simulate the entire detachment phase, and discuss how geometrical variations influence the adhesive properties of the structure. Finally, we also realize a proof of concept experiment to further validate theoretical/numerical results. Overall, we show how this generalized attachment structure can achieve large variations in its adhesive and mechanical properties, exploiting variations of its geometrical parameters, and thus tunability. The in-depth study of similar basic structural units and their combination can in future lead to a better understanding of the mechanical properties of complex architectures found in Nature.


Introduction
Nature provides many examples of biological adhesive surfaces functioning over a wide range of scales and environmental conditions. Natural adhesives often display extraordinary mechanical properties, such as high strength, tunability and self-cleaning. A typical example of such an adhesive structure is a climbing animal toe-pad [1][2][3]. Adhesive pads exploit van der Waals and/or capillary forces to achieve strong attachment and easy detachment, thus enabling locomotion without an excessive wear of their contact elements [4,5]. Another interesting case is the spider web attachment disc [6][7][8]. Depending on the desired functionality, spiders spin their silks in different architectures to form anchorages with varying mechanical properties [7]. Glycoprotein glues are secreted together with the silk to form a fibrous material that adheres to the substrate [8]. Despite the differences between the many examples of adhesives found in Nature, some recurring features can be traced in their geometrical features, in that their structures are often based on the complex organization of fibrils or tape-like elements with sizes that can span across several length scales [9,10]. In recent years, the mechanical response of these biological structures has been studied in depth, leading to experimental evidence of the correlation between their adhesive properties and their geometries [11][12][13][14][15][16].
One of the first models used to describe adhesion is Kendall's peeling theory [17] for the detachment of an elastic tape from a rigid substrate. By applying Griffith's criterion [18], presented 100 years ago, adhesive theories for tape-like elements provide an explanation for the presence in biological adhesives of contact splitting, i.e. the procedure of dividing a contact in n sub-contacts to better distribute the load, enhancing the maximal pull-off force by increasing the total width of the peeling front [2,3]. Moreover, multiple-peeling theory has proven that Kendall's model still holds for complex geometries [19,20]. This approach has led to many works investigating tape-like hierarchical contacts [1,21], staple-pin, double tape geometries [7,22] and conical membranes [23]. Furthermore, studies have been conducting on the effect of different mechanical properties on the peeling forces, i.e. pre-tension [24], bending resistance [25], plasticity [26], viscoelasticity [27], etc. However, when it comes to the description of biological structures, all of these geometries are often considered separately, and each of them is described by a distinct model. This choice is often unrealistic, since many biological adhesive structures have complex geometries which lead to a rich mechanical behaviour that cannot be explained using only one of the afore-mentioned models [12].
In this work, we show how, starting from a single branched element in a generic 3D configuration, we can unify the previous models and obtain highly tunable pull-off forces. The angles determining this simple geometry lead in limit cases to mechanical responses typical of both the single or double peeling theories. We compare the analytical solutions to the results obtained by the numerical model presented in [28], and to those obtained in a proof of concept experiment.

Theoretical model
We consider the structure shown in Fig. 1, which can be considered as a basic building block for 3D branched attachments. Two identical tapes are represented as wires, i.e. with negligible width and thickness compared to their length. The tapes are connected at a hinge-like point and form a symmetrical v-shape. Each tape forms an angle with the substrate plane. The two tapes form an isosceles triangle of base angle . The projection of the detached length of the two tapes on the substrate forms an angle 2 . If the attached part is aligned with the detached one, the contact angle becomes equal to : � = . If a misalignment exists between the two portions, we define a misalignment angle = − � . An external force is applied to the hinge point at a pull-off angle ′ . The angle gives rise to a symmetrical structure. In this symmetrical case, it can be demonstrated through energy balance considerations [19,29] and Griffith [18] theory that the total delamination can be obtained as the composition of two single peeling forces. We first write the peeling force (or, equivalently, pull-off force) of the corresponding single tape peeling problem = 2 , = � , = 0 , shown in Fig. 1c: where the subscript SP stands for "Single Peeling", is the adhesion energy, E is the tape Young's (1) modulus, t is its thickness and w its width. We now consider the case for ≠ 2 . For a v-shaped symmetrical double peeling structure, it is known that the peeling force is [19] where is the angle between each tape and the substrate, and the subscript DP stands for "Double Peeling". As shown in Fig. 1a, in a generalized 3D structure, for each branch, the force must be first projected along the tape angle and then again along the misalignment angle . By applying single tape peeling theory, the critical tape tension at which delamination occurs thus is where the subscript P stands for "Peeling". Notice that here we are assuming, as done in previous works, that the delamination force is equal to the tension along the tape. In the three-dimensional structure shown in Fig. 1a it can be seen that the two tapes form a triangle of base angle . We thus sum the two tensions of the tape, as done in [19], to obtain the overall pull-off force. By noticing from Fig. 1a that the angle corresponding to in Eq. (2) is , we obtain The angles , , and ′ are not independent. From geometrical considerations, we have the following independent relationships showing that only two angles are needed to uniquely define the structure. Using Eq. (4), we can rewrite Eq.   From Eq. (4), it is clear that for given tape mechanical properties, the three angles , , are the parameters determining the peeling force. We analyse the dependency of the pull-off force by solving Eq. (4) for different values of the coefficient = Et∕ , which represents the ratio between elastic and adhesive terms. In this study we consider values of included between 1 and 100. These values can be roughly estimated by the available data in literature concerning biological adhesives [11,[30][31][32]. Results in Fig. 2 show that a maximum in peeling force is reached for = 0 , i.e. when attached parts of the tape are aligned with the detaching tape. Both the single tape peeling problem and the symmetrical double peeling problem appear as limiting cases: the single peeling equation for the same pull-off angle ′ occurs for = ∕2, = 0 ; the double peeling equation for a splitting angle is found when � = ∕2 , and thus by applying Eq. (5), the relationship cos ⋅ cos = cos holds. The latter is represented as a red curve in Fig. 2.
We look at the geometrical constraints to understand the behaviour for lim ′ →0 F P ′ , , . By geometrical construction, we have that ≤ ′ . This can also be obtained starting from the trigonometrical relations expressed in Eq. (5), which allow us to switch between F P ′ , and F P ( , α) (or any other pair of angles between , ′ , , , describing the detached trait of the tape). We thus obtain that, for ′ → 0 , → 0 . To compute the limit, Eq. (4), should therefore be used: The effect of the geometrical relations can be better understood by looking at Fig. 3, where the limit scenarios are represented. Notice that in these conditions, does not depend on and ′ . If we recall the double peeling equation, we see that if = α , Eq. (7) is the equation for a symmetrical double peeling case (Fig. 4). If instead ′ → 2 , for geometrical considerations it can be seen that ≡ and thus we find again Eq. (4).

Numerical model
We now introduce a numerical model to validate analytical results and to study the evolution of the 3D generalized structure during delamination. In this model, which was introduced in [28,33], we discretize a tape of negligible width using a set of nodes connected by truss elements of length dL . These one-dimensional elements can have an arbitrary constitutive law, allowing e.g. the simulation of hyperelastic materials. No rotational springs are present in the model; thus, the bending resistance is neglected, since we only consider thin wires/tapes. The adhesion between the tape and the substrate is represented by a 3D fibrillar traction-separation law which is introduced for every node of the tape. The law used in this where T is the exerted traction, is the separation, is the adhesive energy per node and is the critical length. We assume a monotonically increasing load, leading to crack opening only. A force-control arc-length method is used to simulate the applied force F at an angle ′ . The displacements and internal forces at equilibrium in the structure are calculated using the Newton-Raphson algorithm shown in [28]. During loading, the two branches undergo deformation and delamination, and the angles characterizing the structure change accordingly.
In the following, the chosen geometrical and mechanical characteristics are where L det and L att are respectively the length of the detached and the attached portions of one branch (these values are chosen arbitrarily, since the peeling force does not depend on them). 800 truss elements are used to describe the 3D structure. We initially study the behaviour of an elastic material, allowing us to compare the numerical results with the theoretical equations. An example of the evolution of the applied force F versus the displacement response u is shown in Fig. 6. The evolution of the two angles and is also shown, together with three dimensional plots which illustrate the behaviour of the structure for different loadings. As shown in Eq. (5), if two of the four angles , , , ′ are given then the system is properly defined and the other two angles can be calculated.
As illustrated in [22], if we suppose that is infinitesimal, the evolution of the delamination observed in the numerical simulations can be theoretically described using the peeling equations derived from the Griffith energy balance. The tension at which delamination occurs is T P , given by Eq. (3). Before reaching T P , the considered tape deforms, and the tension depends exclusively on its elastic properties. We thus indicate this tension as T el : , where F P ≡ F P = 10, = 0; , � , with , as fixed parameters and , ′ as variables. The values which satisfy the geometrical relations are highlighted. The green curve F P , ; , � , , � | = ∕2 describes the single peeling problem (Eq. 1) while the red curve F P , , � = 0; describes the behaviour of the double peeling problem (Eq. 2). The value of is fixed by Eq. (5). Notice that the maximal pull-off forces are always reached for � = 0 , as predicted by tape peeling theory. Two limit scenarios are represented: (I) for = � both angles represent the single tape peeling angle, while (II) for � = 2 , a symmetrical double peeling is created where both and are the governing angles. (Color figure  online) where 0 is the initial value of . We now look at the external load F el during the deformation phase for the 3D generalized structure. F el must consider the superposition of the two peeling processes as already done for Eq. (4): where θ′ is assumed to be constant during peeling. Notice that this is true only for ′ ≠ 0 . For � = 0 , the expression for F el coincides with the double peeling one provided in [22] for an angle . We now indicate the applied load at which delamination occurs as F P ≡ F P , ′ , , with F P , ′ , given by Eq. (6).
When the applied load F ≥ F P , delamination begins and the equilibrium path of the structure follows Eq. (6). Notice that an evolution of at fixed ′ and ′ leads to an evolution of and consequentially and the misalignment . The angles and the load continue to evolve until the delamination takes place over the whole finite length of the tapes or when an equilibrium state is reached.
If the initial hypothesis of an infinitesimal critical length does not hold, the applied forces measured numerically do not follow the theoretical values previously mentioned. The traction-separation law leads to the creation of a cohesive process zone, an area of the substrate where the stresses due to peeling are non-negligible and a curvature of the tape is observed. The dimensions of this area are ruled by the critical length . If ≫ dL , the process zone deforms for F < F P , and thus the equilibrium path becomes smoother, eventually leading to a simultaneous deformation-delamination when the strain distribution at the interface propagates over the entire tape during the elastic phase and thus the expected value of F P is never reached (Fig. 7).
A difference smaller than 0.01% in the maximal pull-off force between the expected results and the numerical simulation is observed during delamination when all three degrees of freedom are considered (Fig. 8). Further data can be found in the Supplementary Information.

Experimental measurements
Quantitative peeling experiments were performed to validate the theoretical and numerical predictions discussed in the previous sections. A transparent elastic tape Tesa ® 4204 was used to perform the symmetric double-peeling experiments. The tape has a mean thickness of t = 52.5 ± 0.105 μm and a mean width of w = 25.39 ± 0.16 mm.

Tensile tests
The mechanical properties of the tape were obtained by 7 preliminary tensile test experiments. These experiments were carried out imposing a constant speed of 0.1 mm/s at one end of the tape with a loading machine (Midi 10, from Messphysik), as shown in Fig. 9a. During the tests, the load was acquired with a DCSRC-1000 N load cell and the displacement with a displacement transducer mounted in the loading frame. The tests were stopped at a strain of about 35%, before the tape failure. An example of a stress-strain curve, representative of the behaviour of all tested samples, is reported in Fig. 9b. The tape's non-linear mechanical behaviour was modelled via a two-segment piecewise linear model, being E 1 and E 2 the slopes of the interpolating lines, representing the two Young's moduli of the tape. The mean values are equal to E 1 = 1722.61 ± 165.02 MPa and E 2 = 60.96 ± 3.77 MPa. The coordinates of the intersection point between the two linear interpolating lines (point A in Fig. 9b) are equal to 1 = 0.0246 ± 0.003 and 1 = 41.40 ± 0.99 MPa.

Single-peeling tests
The adhesion energy γ was determined performing 10 single-peeling tests using the experimental setup shown in Fig. 9c. In each experiment, 380 mm of tape was peeled from a PMMA rigid substrate (the same used in the double peeling experiments) with a constant speed of 1 mm/s. The experimental setup was specifically designed to align the load cell with the tape axis in the limit peeling condition. This experimental expedient is fundamental so that the load cell works under ideal conditions. Therefore, the load measured by the load cell is exactly the applied force F indicated in Fig. 9c responsible for the delamination of the tape. Load and displacement were acquired with a DBBSM-10 kg load cell and with a displacement transducer mounted in the loading frame, respectively. The experiments were also recorded with a camera (Sony PXW-FS5), positioned laterally to the testing platform, and orthogonal to the tape axis. The camera was synchronized with the data acquisition system. Before each test, the PMMA surface was cleaned with a silk cloth dipped in ethanol and the tape was pressed on the PMMA surface with a metal roller coated with a thin (thickness 2 mm) rubber layer to guarantee the same initial condition. Notice that the evolution of and is correlated since ′ is fixed and Eq. (4) must hold. c 3-D plots of the structure in four highlighted points: (I) initial deformation and creation of the process zone, (II) begin-ning of the peeling phase, (III) peak of the force value, (IV) maximal deformation and total delamination. In force-control, after III one should assume the shown curve as the theoretical path, while a sudden failure (i.e. complete delamination) would be observed in an empirical environment. Likewise, in displacement-control the same behaviour should be assumed after IV Since the test is done in a single peeling configuration, we assume that the force measured during detachment equals the predicted critical pull-off force F SP , expressed in Eq. (1). For every experiment, the mean critical pull-off force F SP exp required to peel the tape from the PMMA substrate was assumed to be equal to the horizontal fitting line (black/dashed line in the figure) evaluated between 25 and 100 mm (white range in the figure), after the limit peeling condition ( ′ = ′ lim ) is reached. By averaging such forces F SP exp , estimated for each of the 10 tests, we determined a mean critical pull-off force F SP exp = 54.30 ± 0.95 N. The experimental results obtained from the 10 tests, namely the applied force F as a function of the vertical displacement, are summarized in Fig. 9d. In the figure, the magenta/continuous thick line and the magenta vertical thin lines represent, respectively, the average applied force and the standard deviation computed at all times considering all the experiments. Knowing the critical peeling force F SP exp required to peel the tape from the substrate, we calculated the corresponding adhesion energy through Eq. (1), obtaining where θ′ lim is the limit peeling angle. This angle was estimated equal to 0.319 ± 0.007 rad exploiting the results of the 10 experiments. In Eq. (10), the second term on the right side considers the extensibility of the tape. Since the applied force is small, it is reasonable to consider in such expression the Young Modulus equal to E 1 . A mean value of = 133.42 ± 2.77 J/ m 2 was estimated for the adhesion energy.

3D peeling tests
An ad hoc experimental setup, shown in Fig. 10a, was conceived and realized to perform the 3D peeling experiments. The experiments were conducted by peeling the adhesive tape from a rigid PMMA substrate for two different values of the contact angle ′ . In one case, we imposed � = ∕9 , in the other � = 2 ∕9 . For each angle we performed 16 experiments. At the beginning of each test, the same protocol used for the single peeling experiments was used, again to guarantee the same initial conditions

Fig. 7
Normalized applied force F∕F SP � versus displacement for different values of the process zone critical length (in mm) and for an attached length L att = 4 mm and a discretization step of dL = 0.01 mm . For = 1 mm, the deformation and delamination phases are indiscernible. This happens because the strain field reaches the end of the tape before the process zone reaches its regime. Thus, F < F P during the whole deformation , where F P ≡ F P � = 4 , = 10; , , with ′ , as fixed parameters and , as variables. Two numerical solutions which differ for the initial value 0 are superimposed onto the analytical result. There is a very good agreement between analytical and numerical results during the delamination phase, with a maximal difference smaller than the 0.01% in every test. The rigid PMMA substrate of length 600 mm and width 400 mm was engraved with a 45° conical tool mounted on a milling machine (EGX-600 from Roland, resolution 0.01 mm), as highlighted in Fig. 10a. This machining is fundamental to allow the exact alignment of the tape at the beginning of each experiment for the two investigated cases. Two different PMMA self-aligning mechanisms (one for � = ∕9 , in the other � = 2 ∕9 ) were specifically designed to couple the tape to a miniaturized load cell and to the loading frame machine. The detail of this mechanism is shown in the inset of Fig. 10a.
Four snapshots taken during the execution of one test showing the working principle of the selfaligning mechanism are reported in Fig. 10b. In each experiment, 400 mm of tape was peeled from the PMMA rigid substrate by imposing a constant vertical speed of 1 mm/s to the self-aligning mechanism with a loading frame machine (MIDI 10). During the experiment, the applied force F was measured with the miniaturized load cell (XFTC-300, R.C. 200 N) mounted on the pendular mechanism and the vertical displacement with a displacement transducer mounted on the loading frame machine. To capture the variation of the pull-off angle ′ and the angle between the tape and the substrate , the experiments were also recorded with two cameras (Sony PXW-FS5 and PXW-FS7), positioned orthogonally to the testing platform, one frontally and the other laterally, as sketched in Fig. 10c. Figure 10d shows the superposition of two snapshots taken at the same time with the two synchronized cameras. By exploiting the principles of orthogonal projections, it is possible to determine the pull-off angle ′ directly and the angle by a simple trigonometric relation. The two cameras were triggered with a NI CompactRIO system interfaced with LabView 2020 (National Instruments) to be synchronized with the loading frame and the load cell acquisition.

Comparison between theory, simulations and experiments
The experimental values are compared to numerical results obtained with the described numerical model. The constitutive law for the tape material used in the model is where E 1 , E 2 and 1 are those obtained experimentally. To correctly simulate the experimental setup, the displacement is imposed at the tip of a rigid tape length of 64.0 mm with E load = 10 5 E 1 , representing the load cell. Two segments with the same Young's modulus are connected to the load cell to simulate the PMMA auto-aligning mechanisms. Simulations were carried out using the experimentally derived adhesive energy value of = 133.42 J/m 2 and a cohesive critical length of = 2 mm. To convert the adhesive energy to the adhesive energy per node , we associate with every node an area dL ⋅ w , where dL is the length of each element of the numerical model and w is the width of the tape, measured experimentally. The comparison between theory, numerical and experimental results are shown in Fig. 11. The evolution over time of the applied force per unit width of the tape are reported in Fig. 11a for a contact angle � = ∕9 and, in Fig. 11b for � = 2 ∕9 . The yellow triangles represent the peeling force predicted theoretically, the blue dots represent the numerical results  Fig. 11c, d. The numerical results are obtained using the numerical model described in Sect 4. In the model, we have considered the same geometry of the setup and the same mechanical properties of the tape to mimic the real experiments. Starting from � = 0 , the auto-aligning mechanism attached to the load cell (inset of Fig. 10a and b) rotates until deformation begins ( t ≈ 20 s) causing a rapid increase of the applied force until the mechanism reaches an almost steady inclination. From this moment on, the applied force continues to grow, but with a lower slope, up to the complete detachment of the tape from the substrate ( t ≈ 140 s for � = ∕9 and t ≈ 130 � = 2 ∕9 ). By studying the strain evolution in the tape, it can be seen that its value is constantly lower than the critical value 1 (see Supplementary Information, Figure S.IV). The figures clearly show the very good agreement between the theory and both numerical and experimental results. Figure 11c and d show the evolution over time of the angles ′ and for the two cases � = ∕9 and � = 2 ∕9 , respectively, and their comparison with those predicted by the numerical model. In these figures, the magenta/disks and their relative bars represent, respectively, the mean value and the standard deviation of experimental values of the angles ′ and , while the blue/dashed lines refer to  the numerical results. To estimate the values of such angles, a specific algorithm was developed in Mathematica to perform an image-correlation analysis on the frames recorded by the two cameras (see Fig. 11) at 10-s interval. During the evolution of the experiments, the angle between the tape and the substrate remains almost constant, as correctly predicted by the numerical model. On the contrary, the pull-off angle ′ shows, accordingly with the results reported in Fig. 11a and b, a rapid increase until deformation begins ( t ≈ 20 s), and then tends to decrease. Observing the figures, it is evident that the numerical model captures the behaviour observed in the experiments very well, and can therefore correctly reproduce the different phases of delamination.

Conclusions
We have studied the adhesive properties of a doubly branched 3D generalized structure made of tape-like attachments and shown how the different angles forming the geometry influence the peeling force. We have shown that the 3D structure admits as limiting cases single tape peeling and symmetrical double peeling, and we have derived the corresponding behaviour, proving that both can be described by a single general equation. By using a numerical method based on a finite element analysis, we have confirmed the validity of the analytical solution, and also described the evolution of dendritic two-branched structures during delamination, which give rise to a richer phenomenology than that of single tape peeling or a symmetrical double peeling, since two additional degrees of freedom are introduced. Finally, experimental tests were performed to validate the theoretical/numerical results. The proposed formalism could help biologists better understand the adhesive behaviour of natural structures, generalizing basic geometries such as single or double tape contacts, and to further study the role of geometry in determining tunability of the adhesive force achieved by animals and plants.

Declarations
Conflict of interest The authors declare that they have no conflict of interest.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.