Modelling the non-steady peeling of viscoelastic tapes

We present a model to study the non-steady V-shaped peeling of a viscoelastic thin tape adhering to a rigid flat substrate. Geometry evolution and viscoelastic creep in the tape are the main features involved in the process, which allows to derive specific governing equations in the framework of energy balance. Finally, these are numerically integrated following an iterative scheme to calculate the process evolution assuming different controlling conditions (peeling front velocity, peeling force, tape tip velocity). Results show that the peeling behavior is strongly affected by viscoelasticity. Specifically, for a given applied force, the peeling can either be prevented, start and stop after some while, or endlessly propagate, depending on the original undeformed tape geometry. Viscoelasticity also entails that the interface toughness strongly increases when the tape tip is fast pulled, which agrees to recent experimental observations on tougher adhesion of natural systems under impact loads, such as see waves and wind gusts.

Relaxation time P 0 , θ 0 Critical force and angle for peeling initiation P S , θ S Long-term steady-state peeling force and angle

INTRODUCTION
In modern science, the study of attachment and detachment mechanisms is of practical importance for several applications, such as climbing ability in soft-robotics [1,2], deposition and removal of coating for specialized interfaces [3], pick-and-place processes in manufacturing [4], self-healing heterogeneous materials for construction [5] and wound dressing for medical industries [6].Among the others, when dealing with tapes and membranes, as well as fibrils and thin bristles, detachment through mechanical peeling has recently seen a growing interest, quickly becoming the main mechanism for systems such as electro-adhesive [7] and shear-activated nano-structured [8] grippers for objects manipulation made of compliant membranes, band-aids [9] and tunable skin patch [10] to minimize the removal damage of biological tissues [11,12], spray coatings [13] and transfer printing [14,15] for flexible cir-cuits fabrication (also adopting micro-fibrils adhesives [16]), and highly-stretchable structural adhesive tapes [17].
Although peeling itself is a local phenomenon involving a crack propagation at the interface between a layer and a substrate, the macroscopic detachment response is also affected by the global system properties.Those of main interest for similar tribological problems usually are the system geometry [37][38][39] and the materials rheology [40,41].Indeed, studying peeling geometries other than single peeling (in which case the peeling angle equals the force angle) has been recently urged by biomimetics in the attempt, for instance, to mimic [42] the superior locomotion performance of spiders, insects, and reptiles.This depends on the ability to quickly detach their hierarchical-structured toes by exploiting simultaneous peeling fronts propagation, ranging from the macro-scale (e.g., the leg) to the nano-scale (e.g., the toe spatula) [43].Pugno and coworkers [44,45] suggested that both the hierarchy and V-shape of the peeling geometry of such systems may play a key role in the overall toughness as multiple peeling fronts coexist, and the peeling angle varies during the detachment process.Later, Lepore et Al.[46] showed that the angles assumed by Tokay geckos at the two characteristic sizes of feet and toes are in excellent agreement with Pugno's multiple peeling theory predictions.Similarly, V-peeling geometry has been observed in spiders' webs anchors [47,48] and byssus threads networks of mussels [49,50], both showing superior adhesive performance and the ability to withstand heavy winds and waves.Moreover, since peeling has also been successfully employed in characterizing the adhesive properties of materials and adhesives [51], as well as to assess the toughness of interfaces, specific tests (e.g., ASTM Loop Tack test) have been defined relying on the V-peeling geometry to reduce the possible effect of tape bending [52], compared to standard 90°-180°peel tests.Nonetheless, existing models for V-peeling geometry only focus on elastic tapes [53], whereas biological systems and commercial tapes usually exhibit a certain degree of viscoelasticity.Indeed, the effect of the materials viscoelasticity and, in turn, of energy dissipation during creep deformation, has been mostly addressed with reference to the single peeling configuration.Both physical [54][55][56] and phenomenological [28,29,[57][58][59] models have been developed, showing that tape viscoelasticity makes the peeling toughness increase with peel rate.More in detail, Ceglie et Al.[54] have shown that viscoelasticity and local frictional sliding close to the peeling front may lead to unbounded peeling toughness at very low peeling angles, in agreement with existing experimental results [60].Zhu et Al.[61] focused on the tape visco-hyperelasticity effect on specific zero-degree peeling configuration, showing that for relatively thick tapes the peeling expected peeling force is less sensitive to bulk properties and interfacial adhesion (i.e., surface defects) compared to the case of linear rheology materials.
Viscoelasticity can also be localized in the substrate [62] leading to an "ultra-tough" behavior achievable at specific peel rates.Similar results were also confirmed by Pierro et Al.[63] for a real viscoelastic material with a broader relaxation spectrum and by Zhu at Al. [64] also assuming rate dependent interface adhesion.The substrate rheology is crucial, for instance, when adhesive tapes are removed from human skin, in which case Renvoise at Al. [65] showed that at relatively large peeling velocity the multi-layer nature of human skin can also matter.Surprisingly, less has been done combining rheology effects and V-peeling geometry, although Menga et Al.[66,67] have shown that, for purely elastic conditions, highly compliant substrates can drastically alter the peeling toughness in V-shaped systems due to the elastic interaction between adjacent peeling fronts.
In this study, we present a model for the V-peeling process of viscoelastic tapes backed onto rigid substrates, aiming at fostering the understanding of the mechanisms underlying the superior performance shown by insect toes, mussels attachment structures, and other biological systems relying on V-shaped peeling geometry in the presence of viscoelasticity, as well as to enhance the accuracy of loop tack test analysis to predict the adhesive performance of real interfaces.Since the process under investigation is non-steady, in Section 2 we set the appropriate theoretical energy-based framework and derive the governing equations for the peeling load, angle and front velocity, while the numerical procedure to integrate such equations and predict the process evolution over time is given in Appendix A and B. Results are presented in Section 3, focusing on three possible peeling procedures (constant peeling front velocity, constant peeling load, and constant velocity of the tape tip), each of which leads to qualitatively different results highlighting the interplay between V-shaped geometry and tape viscoelasticity.

FORMULATION
We consider the peeling configuration shown in Fig. 2(a) Double V-shaped peeling scheme of a viscoelastic tape adhering to a rigid substrate.v c is the peeling front propagation velocity, and v P is the pulling velocity (i.e., the velocity of the tape tip).(b) By exploiting the system symmetry, the study only focuses on half of the tape.We show three different configurations: the undeformed tape, the tape at peeling propagation start (subscript 0 ), and a generic time instant with peeling force P (t) and angle θ(t).In the bottom part, we also show qualitative diagrams of the stress σ (blue) and deformation ε (orange) along the tape coordinate λ.figure.caption.6a,where a thin viscoelastic tape of thickness d and width w adhering to a rigid substrate is pulled away by a normal force 2P .Since two peeling fronts propagate in opposite directions (V-shaped double peeling), the whole process is symmetric with respect to the force direction and the study can be limited to half of the system, as shown in Fig. 2(a) Double V-shaped peeling scheme of a viscoelastic tape adhering to a FIG.2: (a) Double V-shaped peeling scheme of a viscoelastic tape adhering to a rigid substrate.v c is the peeling front propagation velocity, and v P is the pulling velocity (i.e., the velocity of the tape tip).(b) By exploiting the system symmetry, the study only focuses on half of the tape.We show three different configurations: the undeformed tape, the tape at peeling propagation start (subscript 0 ), and a generic time instant with peeling force P (t) and angle θ(t).In the bottom part, we also show qualitative diagrams of the stress σ (blue) and deformation ε (orange) along the tape coordinate λ.
rigid substrate.v c is the peeling front propagation velocity, and v P is the pulling velocity (i.e., the velocity of the tape tip).(b) By exploiting the system symmetry, the study only focuses on half of the tape.We show three different configurations: the undeformed tape, the tape at peeling propagation start (subscript 0 ), and a generic time instant with peeling force P (t) and angle θ(t).In the bottom part, we also show qualitative diagrams of the stress σ (blue) and deformation ε (orange) along the tape coordinate λ.figure.caption.6b.
Before the peeling force P being applied, the (undeformed) non-adhering tape length and angle are L i and φ, respectively.Once the force is applied, at the time instant when the peeling starts to propagate, the tape angle is θ 0 and the peeling force is P 0 .According to Fig. 2(a) Double V-shaped peeling scheme of a viscoelastic tape adhering to a rigid substrate.v c is the peeling front propagation velocity, and v P is the pulling velocity (i.e., the velocity of the tape tip).(b) By exploiting the system symmetry, the study only focuses on half of the tape.We show three different configurations: the undeformed tape, the tape at peeling propagation start (subscript 0 ), and a generic time instant with peeling force P (t) and angle θ(t).In the bottom part, we also show qualitative diagrams of the stress σ (blue) and deformation ε (orange) along the tape coordinate λ.figure.caption.6b,at a generic time t, the peeling front coordinate and velocity are λ c (t) and v c (t) = −dλ c /dt, respectively, with λ being the (undeformed) tape-fixed reference frame.Similarly, s d (t) = t 0 v c (t)dt is the detached tape length, and the peeling angle θ(t) is given as where ∆L(t) = λc+s d +L i λc ε(λ, t)dλ is the elongation of the overall non-adhering tape, with ε(λ, t) being the extensional deformation field in the tape.
The instantaneous energy balance governing the peeling process is where W P (t) is the work per unit time done by the peeling force P (t), W in (t) is the work per unit time done by the internal stress field, and W ad (t) is the rate of the surface adhesion energy.In Eq. (2equation.2.2), minor energy contributions ascribable to acoustic emissions and heat transfer are neglected, as well as dynamic and inertial effects which might lead to stick-slip unstable delamination [68][69][70][71][72].Moreover, we assume a fully stuck adhesion between the tape and the rigid substrate in the adhering region, thus no friction energy dissipation occurs due to relative sliding, as instead considered in Refs.[33,54].
The term W in (t) is associated with both the rate of elastic energy stored in the detached tape and the viscoelastic energy loss occurring during the tape relaxation.Large deformations can be reasonably expected for soft polymeric tapes; however, both numerical [20] and experimental [60] studies have clearly shown that real systems exhibiting strains as large as beyond 60% can still be both qualitatively and quantitatively described in linear theory approximation, especially at relatively large peeling angles [73].Similar results are confirmed for visco-hyperelastic tapes [61], where qualitatively different behaviors are expected only beyond approximately 100% strain value.Moreover, we assume purely extensional stress σ(λ, t) and deformation ε(λ, t) fields in the tape, as experiments have shown that bending effects vanish for very thin tapes [74] (i.e., the tape bending stiffness depends on d 3 ).Therefore, we have with ε(λ, t) = σ(λ, t) = 0 for λ < λ c (adhering tape) and σ(λ, t) = σ(t) = P/ (A t sin θ) for λ > λ c (detached tape).In the peeling section (i.e., for λ = λ c ), a step change of the stress occurs [54], so that where H is the Heaviside step function.In the framework of linear viscoelasticity, the deformation field within the tape is given by where J is the viscoelastic creep function which, for a single characteristic creep time τ , is given by where , with E 0 and E ∞ being the low and high frequency viscoelastic moduli, respectively.
The term W P (t) in Eq. (2equation.2.2) is given by where is the pulling velocity (see Fig. 2(a) Double V-shaped peeling scheme of a viscoelastic tape adhering to a rigid substrate.v c is the peeling front propagation velocity, and v P is the pulling velocity (i.e., the velocity of the tape tip).(b) By exploiting the system symmetry, the study only focuses on half of the tape.We show three different configurations: the undeformed tape, the tape at peeling propagation start (subscript 0 ), and a generic time instant with peeling force P (t) and angle θ(t).In the bottom part, we also show qualitative diagrams of the stress σ (blue) and deformation ε (orange) along the tape coordinate λ.figure.caption.6).
Finally, in Eq. (2equation.2.2), W ad (t) represents the energy per unit time associated with the rupture of interfacial bonds between the tape and the rigid substrate; being γ the energy of adhesion (also called Dupre's energy), we have The adhesion energy γ might, in general, depend on the peeling velocity, as reported by several experiments [15,28,29,59].This is usually ascribed to viscoelastic non-conservative (stiffening) effects in the tape close to the peeling front, as recently predicted in Ref. [54].
Here, we precisely model the tape viscoelastic creep, thus the latter effect is intrinsically accounted for.However, as pointed out by Marin & Derail [75] with ad hoc tests on inextensible tapes, velocity-dependent power loss is also localized in the thin adhesive layer between the tape and the substrate, with γ given by the following power-law where γ 0 is the nominal adhesion energy for v c v γ , with v γ being a reference peeling velocity, and n being a constant which depends on the properties of the adhesive (typically in a range of 0.3 − 0.7) [15,59,75].
To set the range of validity of the present model, we observe that the real deformation process occurring across the peeling front is continuous and cannot be formally represented by a step-change in the stress field.Nonetheless, physical arguments suggest that the length of the region undergoing the stress increase from 0 to σ is of the same order of magnitude as the tape thickness d [62,67], which results in a local excitation frequency ω ≈ v c /d and allows to identify three different qualitative behaviors across the peeling section.For v c d/τ (i.e., ω 1/τ ), the tape behaves almost elastically, with elastic modulus approaching the lowfrequency modulus E 0 .Since no viscoelastic dissipation occurs, this case is clearly out of the scope of this study, and the corresponding peeling behavior follows the elastic predictions given in Refs.[44,76].For v c ≈ d/τ (i.e., ω ≈ 1/τ ), the tape response strongly depends on the specific deformation process across the peeling front (small-scale energy dissipation cannot be neglected), and a local ad hoc solid mechanics formulation is required to model the peeling.Finally, the third case is the one of interest for the present model, as for v c d/τ (i.e., ω 1/τ ) the tape behavior is elastic across the peeling section with high frequency elastic modulus E ∞ , and viscoelastic losses are localized in the non-adhering tape (largescale).Daily-life adhesive tapes are commonly very thin, with a corresponding threshold velocity usually being in the range of d/τ ≈ 10 − 100 µm/s, which makes the third case of most relevant practical interest.

RESULTS AND DISCUSSION
In this section, we discuss the peeling behavior resulting from Eqs. (1equation.2.1,2equation.2.2), which can be numerically solved by following the procedure outlined in Appendix Asection*.17.To simplify the analysis of the results, we refer to dimensionless quantities, i.e. t = t/τ , γ = γ/E 0 d, ṽc = v c τ /d, ṽP = v P τ /d, P = P/dwE 0 = σ sin θ/E 0 .In our calculations, we consider a tape of thickness d ≈ 100 µm with initial non-adhering length L i = 100d.The tape material is viscoelastic with low-frequency modulus E 0 = 10 MPa and creep time τ = 1 s.Marin and Derail [75] measured the peeling force P as a function of the peeling velocity v c for real adhesives with inextensible aluminum backing, and Rivlin peeling theory [18] allows to calculate the corresponding effect of v c on the adhesion energy γ.Since no viscoelastic relaxation occurs, the latter effect is only ascribable to non-conservative phenomena localized in the very proximity of the peeling front.According to their results, we set v γ ≈ 10 −3 m/s, γ 0 ≈ 20 J/m 2 , and n = 0.5 in Eq. 10equation.2.10, whose corresponding dimensionless quantities are ṽγ = v γ τ /d = 10 and γ0 = γ 0 /E 0 d = 0.02.
Results are presented considering three different controlling parameters, corresponding to specific physical scenarios: (i) peeling propagation occurring at constant peeling front velocity v c ; (ii) the case of a constant peeling force P applied at the tape tip; and (iii) the case of the tape tip pulled at constant velocity v P .

Constant peeling front velocity
We firstly consider the peeling process occurring at constant peeling front velocity v c .This case corresponds to time-varying values of both the peeling force P and pulling velocity v P , thus resulting harder to be straightforwardly associated with common applications.However, since v c also represents the length of undeformed tape that detaches the substrate per unit time and, once deformed, undergoes viscoelastic relaxation, fundamental insight on the interplay between peeling propagation and tape viscoelasticity.
With reference to Fig. 2(a) Double V-shaped peeling scheme of a viscoelastic tape adhering to a rigid substrate.v c is the peeling front propagation velocity, and v P is the pulling velocity (i.e., the velocity of the tape tip).(b) By exploiting the system symmetry, the study only focuses on half of the tape.We show three different configurations: the undeformed tape, the tape at peeling propagation start (subscript 0 ), and a generic time instant with peeling force P (t) and angle θ(t).In the bottom part, we also show qualitative diagrams of the stress σ (blue) and deformation ε (orange) along the tape coordinate λ.figure.caption.6b,we assume that the peeling front propagation starts with velocity v c at time t = 0 under the action of the critical force P 0 , which is instantaneously applied.At this time, the tape deformation and angle undergo a step-change, varying from ε = 0 and φ at time t → 0 − to ε = σ 0 /E ∞ and θ 0 at time t → 0 + .As a consequence, no viscoelastic loss occurs in the non-adhering tape at t = 0, and the critical values of σ 0 and θ 0 for peeling initiation are given by Kendall's equation [34,66,67] and, from Eq. (1equation.2.1) with Finally, the critical force is calculated as P 0 = A t σ 0 sin θ 0 .For t > 0, the peeling process evolution follows Eqs.(1equation.2.1,2equation.2.2) and is calculated by exploiting the numerical procedure outlined in Appendix Asection*.17.
Comparing the critical peeling force P 0 with the long-term limit P S allows to differentiate from toughening and weakening overall peeling behaviors.This is done in the top row of Fig. 3Top row: the initial P0 and long-term PS dimensionless peeling force as functions of the dimensionless adhesion energy γ0 (a), the viscoelastic parameter κ = E ∞ /E 0 (b), and the dimensionless peeling front velocity ṽc (c).In figure (c), the rate-dependent dimensionless adhesion energy γ is also shown for comparison (dashed line).Notably, κ = 1 corresponds to elastic tapes.Bottom row: the time-history of the normalized peeling force P / PS (d) and peeling angle θ (e) for different values of the dimensionless peeling front velocity ṽc and initial undeformed configurations.Non-monotonic behavior occurs in the red regions, i.e. for t ≈ τ .Transient diagrams P versus θ are shown for different values of ṽc and for φ = 30°(f) and φ = 45°(g).Blue and red curves represent the starting condition and the steady-state limit, respectively.Black arrows indicate the time evolution of the process.figure.caption.10,where we consider the effect of (a) the dimensionless adhesion energy γ0 , (b) the viscoelastic parameter κ = E ∞ /E 0 , and (c) the dimensionless peeling front velocity ṽc .While in the elastic V-peeling case, the toughest behavior always occurs in FIG. 3: Top row: the initial P0 and long-term PS dimensionless peeling force as functions of the dimensionless adhesion energy γ0 (a), the viscoelastic parameter κ = E ∞ /E 0 (b), and the dimensionless peeling front velocity ṽc (c).In figure (c), the rate-dependent dimensionless adhesion energy γ is also shown for comparison (dashed line).Notably, κ = 1 corresponds to elastic tapes.Bottom row: the time-history of the normalized peeling force P / PS (d) and peeling angle θ (e) for different values of the dimensionless peeling front velocity ṽc and initial undeformed configurations.Non-monotonic behavior occurs in the red regions, i.e. for t ≈ τ .Transient diagrams P versus θ are shown for different values of ṽc and for φ = 30°(f) and φ = 45°(g).Blue and red curves represent the starting condition and the steady-state limit, respectively.Black arrows indicate the time evolution of the process.
the steady-state limit, as clearly shown for κ = 1 in Fig. 3Top row: the initial P0 and longterm PS dimensionless peeling force as functions of the dimensionless adhesion energy γ0 (a), the viscoelastic parameter κ = E ∞ /E 0 (b), and the dimensionless peeling front velocity ṽc (c).In figure (c), the rate-dependent dimensionless adhesion energy γ is also shown for comparison (dashed line).Notably, κ = 1 corresponds to elastic tapes.Bottom row: the time-history of the normalized peeling force P / PS (d) and peeling angle θ (e) for different values of the dimensionless peeling front velocity ṽc and initial undeformed configurations.Non-monotonic behavior occurs in the red regions, i.e. for t ≈ τ .Transient diagrams P versus θ are shown for different values of ṽc and for φ = 30°(f) and φ = 45°(g).Blue and red curves represent the starting condition and the steady-state limit, respectively.Black arrows indicate the time evolution of the process.figure.caption.10bwhere P S > P 0 , the same principle cannot be generalized to the viscoelastic case, where the value of both P 0 and P S depends on the tape relaxation process and, as a consequence, on the specific combination of the parameters γ0 , κ , and ṽc , as shown in Figs.3Top row: the initial P0 and long-term PS dimensionless peeling force as functions of the dimensionless adhesion energy γ0 (a), the viscoelastic parameter κ = E ∞ /E 0 (b), and the dimensionless peeling front velocity ṽc (c).In figure (c), the rate-dependent dimensionless adhesion energy γ is also shown for comparison (dashed line).Notably, κ = 1 corresponds to elastic tapes.Bottom row: the time-history of the normalized peeling force P / PS (d) and peeling angle θ (e) for different values of the dimensionless peeling front velocity ṽc and initial undeformed configurations.Non-monotonic behavior occurs in the red regions, i.e. for t ≈ τ .Transient diagrams P versus θ are shown for different values of ṽc and for φ = 30°(f) and φ = 45°(g).Blue and red curves represent the starting condition and the steady-state limit, respectively.Black arrows indicate the time evolution of the process.figure.caption.10a,b,c.Moreover, according to Eqs. (14equation.3.14-15equation.3.15), the critical starting force P 0 depends on the undeformed tape angle φ, specifically leading to tougher peeling initiation with φ reducing.In agreement with theoretical [34] and experimental results [77], stiffer tapes entail higher peeling forces, as shown in Fig. 3Top row: the initial P0 and long-term PS dimensionless peeling force as functions of the dimensionless adhesion energy γ0 (a), the viscoelastic parameter κ = E ∞ /E 0 (b), and the dimensionless peeling front velocity ṽc (c).In figure (c), the rate-dependent dimensionless adhesion energy γ is also shown for comparison (dashed line).Notably, κ = 1 corresponds to elastic tapes.Bottom row: the time-history of the normalized peeling force P / PS (d) and peeling angle θ (e) for different values of the dimensionless peeling front velocity ṽc and initial undeformed configurations.Non-monotonic behavior occurs in the red regions, i.e. for t ≈ τ .Transient diagrams P versus θ are shown for different values of ṽc and for φ = 30°(f) and φ = 45°(g).Blue and red curves represent the starting condition and the steady-state limit, respectively.Black arrows indicate the time evolution of the process.figure.caption.10b;nonetheless, at very large values of κ both P 0 and P S are almost constant and the Rivlin [18] solution for rigid tapes is asymptotically approached.
The bottom row of Fig. 3Top row: the initial P0 and long-term PS dimensionless peeling force as functions of the dimensionless adhesion energy γ0 (a), the viscoelastic parameter κ = E ∞ /E 0 (b), and the dimensionless peeling front velocity ṽc (c).In figure (c), the rate-dependent dimensionless adhesion energy γ is also shown for comparison (dashed line).Notably, κ = 1 corresponds to elastic tapes.Bottom row: the time-history of the normalized peeling force P / PS (d) and peeling angle θ (e) for different values of the dimensionless peeling front velocity ṽc and initial undeformed configurations.Non-monotonic behavior occurs in the red regions, i.e. for t ≈ τ .Transient diagrams P versus θ are shown for different values of ṽc and for φ = 30°(f) and φ = 45°(g).Blue and red curves represent the starting condition and the steady-state limit, respectively.Black arrows indicate the time evolution of the process.figure.caption.10shows the peeling transient evolution from start to steady-state behavior.The most important feature is that V peeling of viscoelastic tapes may present non-monotonic trends of both θ and P , in contrast with results achieved for the elastic case in Refs.[66,67,76].More in detail, focusing on 3Top row: the initial P0 and long-term PS dimensionless peeling force as functions of the dimensionless adhesion energy γ0 (a), the viscoelastic parameter κ = E ∞ /E 0 (b), and the dimensionless peeling front velocity ṽc (c).In figure (c), the rate-dependent dimensionless adhesion energy γ is also shown for comparison (dashed line).Notably, κ = 1 corresponds to elastic tapes.Bottom row: the time-history of the normalized peeling force P / PS (d) and peeling angle θ (e) for different values of the dimensionless peeling front velocity ṽc and initial undeformed configurations.Non-monotonic behavior occurs in the red regions, i.e. for t ≈ τ .Transient diagrams P versus θ are shown for different values of ṽc and for φ = 30°(f) and φ = 45°(g).Blue and red curves represent the starting condition and the steady-state limit, respectively.Black arrows indicate the time evolution of the process.figure.caption.10e,given the undeformed tape angle φ, the tape angle at peeling start θ 0 increases with ṽc , as expected from Eq. (15equation.3.15) and Fig. 3Top row: the initial P0 and long-term PS dimensionless peeling force as functions of the dimensionless adhesion energy γ0 (a), the viscoelastic parameter κ = E ∞ /E 0 (b), and the dimensionless peeling front velocity ṽc (c).In figure (c), the rate-dependent dimensionless adhesion energy γ is also shown for comparison (dashed line).Notably, κ = 1 corresponds to elastic tapes.Bottom row: the time-history of the normalized peeling force P / PS (d) and peeling angle θ (e) for different values of the dimensionless peeling front velocity ṽc and initial undeformed configurations.Non-monotonic behavior occurs in the red regions, i.e. for t ≈ τ .Transient diagrams P versus θ are shown for different values of ṽc and for φ = 30°(f) and φ = 45°(g).Blue and red curves represent the starting condition and the steady-state limit, respectively.Black arrows indicate the time evolution of the process.figure.caption.10cshowing θ 0 increasing with P0 and P0 increasing with ṽc , respectively.Following Eq. (1equation.2.1), once the peeling starts, the value of θ(t) depends on the interplay between (i) the peeling front propagation, causing a linear increase of s d , and (ii) the detached tape relaxation, increasing the term ∆L.For t 1, a rough estimation of θ can be derived from Eq. (1equation.2.1) as θ ∝ d (∆L) /dt − βv c , with β = β (θ 0 ) being a monotonically increasing function; indeed, in agreement with Fig. 3Top row: the initial P0 and long-term PS dimensionless peeling force as functions of the dimensionless adhesion energy γ0 (a), the viscoelastic parameter κ = E ∞ /E 0 (b), and the dimensionless peeling front velocity ṽc (c).In figure (c), the rate-dependent dimensionless adhesion energy γ is also shown for comparison (dashed line).Notably, κ = 1 corresponds to elastic tapes.Bottom row: the time-history of the normalized peeling force P / PS (d) and peeling angle θ (e) for different values of the dimensionless peeling front velocity ṽc and initial undeformed configurations.Non-monotonic behavior occurs in the red regions, i.e. for t ≈ τ .Transient diagrams P versus θ are shown for different values of ṽc and for φ = 30°(f) and φ = 45°(g).Blue and red curves represent the starting condition and the steady-state limit, respectively.Black arrows indicate the time evolution of the process.figure.caption.10e,f, and g, the peeling angle θ at t 1 can either decrease (at high velocity, i.e. ṽc ≈ 1000) or increase (at low velocity, i.e. ṽc ≈ 10), while in the long-term limit θ( t 1) ≈ θ S eventually leading to non-monotonic behavior, depending on the specific value of θ S .The normalized peeling force P / PS in Fig. 3Top row: the initial P0 and longterm PS dimensionless peeling force as functions of the dimensionless adhesion energy γ0 (a), the viscoelastic parameter κ = E ∞ /E 0 (b), and the dimensionless peeling front velocity ṽc (c).In figure (c), the rate-dependent dimensionless adhesion energy γ is also shown for comparison (dashed line).Notably, κ = 1 corresponds to elastic tapes.Bottom row: the time-history of the normalized peeling force P / PS (d) and peeling angle θ (e) for different values of the dimensionless peeling front velocity ṽc and initial undeformed configurations.Non-monotonic behavior occurs in the red regions, i.e. for t ≈ τ .Transient diagrams P versus θ are shown for different values of ṽc and for φ = 30°(f) and φ = 45°(g).Blue and red curves represent the starting condition and the steady-state limit, respectively.Black arrows indicate the time evolution of the process.figure.caption.10d,f, and g is non-monotonic, as well, since high values of θ lead to low values of P / PS and vice versa, as expected [34,54,76].

Constant peeling force
In this section, we investigate the viscoelastic V peeling behavior under a constant peeling force P , such as under the action of a dead weight.Surprisingly, the results show that the peeling process can either start and indefinitely propagate, start and then stop after some time, or not even start at all, depending on the value of P and initial tape geometry (i.e., the undeformed angle φ and length L i ).The boundaries between these qualitatively different behaviors depend on the peeling front velocity, which is not known a priori in this case; therefore, the critical (minimum) forces for peeling start and steady-state propagation must be sought for both v c d/τ and v c d/τ assuming γ ≈ γ 0 , as for real thin tapes ṽγ ≈ 10d/τ .As discussed at the end of Section 2section*.5, in the former case, critical loads for peeling start P 1 and steady-state propagation P 2 are given by Eqs.(14equation.3.14,15equation.3.15) and Eqs.(11equation.2.11,12equation.2.12), respectively, with γ ≈ γ 0 .On the contrary, in the latter case (i.e., for v c d/τ ), the critical forces P 3 (start) and P 4 (steady-state propagation) are given by the same equations with the high-frequency modulus E ∞ replaced by the low-frequency one E 0 and, again, with γ ≈ γ 0 .
The map in Fig. 4(a) The initial [ṽ c ] 0 and long-term [ṽ c ] S dimensionless peeling front velocity as functions of the dimensionless peeling force P .(b) The time-history of the normalized peeling front velocity ṽc /[ṽ c ] 0 for different values of dimensionless peeling force P .(c) The state map of the possible peeling behavior as a function of the dimensionless applied peeling force P and undeformed tape angle φ.P1 and P2 are the critical (minimum) force for peeling start and steady-state propagation calculated with v c d/τ , while P3 and P4 refer to the same critical conditions when v c d/τ (further know are given in the text).Results refer to κ = 10 and γ0 = 0.02.figure.caption.12cshows the possible peeling behaviors as discussed above as functions of the dimensionless applied peeling force P and undeformed tape angle φ (we assume L i = 100d for all calculations).Specifically, in region I the peeling does not propagate, in region II-III the peeling propagation starts and then stops after some time, and in regions IV-V-VI-VII the peeling propagates indefinitely approaching the steady-state regime (though in IV and V the long-term velocity is lower than d/τ ).The present model predictions are rigorously valid in regions I and VII; nonetheless, qualitative insight can also be inferred for regions III and V, as a steady-state propagation with very low velocity (about d/τ ≈ 10 −4 m/s) qualitatively corresponds to a peeling stop (i.e., as in case III), for region VI, where v c ≈ d/τ when peeling starts and then rapidly increases, and for region II, as the peeling cannot be sustained in steady-state conditions regardless of the starting behavior.Finally, region IV cannot be accounted for in the present framework, as the specific viscoelastic behavior close to the peeling front does really matter throughout the whole process evolution.In most cases, real systems belong to the first scenario, with v c d/τ.The peeling front velocity calculated at the process start [v c ] 0 and in the long-term steadystate [v c ] S limit are shown in Fig. 4 d/τ , while P3 and P4 refer to the same critical conditions when v c d/τ (further know are given in the text).Results refer to κ = 10 and γ0 = 0.02.figure.caption.12b,with respect to the time-history of the normalized peeling front velocity ṽc / [ṽ c ] 0 for different values of P belonging to regions VII and III.In the latter case (blue curve), for P < P4 , the peeling front velocity decreases down to full stop.In the other cases, all belonging to region VII, after the initial non-monotonic behavior, in the long-term a steady-state behavior is approached, with endless propagation occurring at velocity [v c ] S .

Constant pulling velocity
The final case we deal with is with the tape being pulled at a constant velocity ṽP .In this case, the start of peeling propagation does not coincide with the application of the pulling velocity.Indeed, before peeling starts, the stress σ in the non-adhering tape must increase from zero to a certain critical value σ cr .The energy-based procedure to calculate such a critical condition is given in Appendix Bsection*.19.  and c, we show, respectively, the time-history of P , θ and ṽc /ṽ P , for different values of the dimensionless pulling velocity ṽP and undeformed angle φ.Circles indicate the peeling start, which increases with φ reducing, as expected from Eq. (B1equation.B.1).In the long-term limit, steady-state propagation occurs, with [v c ] S /ṽ P = 1/ tan θ S according to Eq. (13equation.2.13).The most interesting result from Fig. 5The time-history of the dimensionless peeling force P (a), the peeling angle θ (c), and the peeling velocity ratio ṽc /ṽ P (c), together with the transient diagram P vs. θ (d) for different values of the dimensionless pulling velocity ṽP .Two undeformed tape angles are considered.The circles indicate the instant when peeling front propagation starts.The red curve in (d) represents the steadystate peeling limit, and black arrows indicate the time evolution of the process.Results refer to κ = 10 and γ0 = 0.02.figure.caption.14a, is that the peeling force P may present a maximum at the early stage, right after the peeling start, at a relatively high pulling velocity ṽP .This is also shown in Fig. 5The time-history of the dimensionless peeling force P (a), the peeling angle θ (c), and the peeling velocity ratio ṽc /ṽ P (c), together with the transient diagram P vs. θ (d) for different values of the dimensionless pulling velocity ṽP .Two undeformed tape angles are considered.The circles indicate the instant when peeling front propagation starts.The red curve in (d) represents the steady-state peeling limit, and black arrows indicate the time evolution of the process.Results refer to κ = 10 and γ0 = 0.02.figure.caption.14d,clearly indicating that this is associated with a temporary reduction of the peeling angle θ which results from a fast increase of v c before viscoelastic relaxation occurs (i.e., for t 1).Such a peculiar feature may partially link to the superior adhesive performance of V-shaped natural systems, such as spider webs [48] and mussels byssus [49], under the action of high-speed (impact) loading conditions.In the latter case, for instance, Cohen et Al.[78] have shown that the single byssus is highly stretchable, due to the heterogeneous filament structure (a system of nonlinear swollen springs); here, we suggest that also the interplay between byssus rheology and V-shaped multiple threads geometry (see Fig. 1Examples of V-peeling configurations in natural systems and practical applications: (a) spider web anchors (from Ref. [66]); (b) gecko upside down climbing (adapted from Wikipedia); (c) mussel byssus threads (from Wikipedia); (d) loop tack test schematic (from Ref. [51]).figure.caption.4c)may contribute to the observed tougher adhesive response under dynamic loads [49].

CONCLUSIONS
In this study, we model the peeling behavior of a viscoelastic thin tape arranged in Vshaped peeling configuration.Specifically, the velocity-dependent condition for peeling front propagation is found in terms of energy balance between the work per unit time done by the internal stress in the tape, the external forces acting on the system, and the surface adhesion forces.An ad hoc numerical procedure is derived to predict the time-evolution of the peeling process, taking into account the time-varying viscoelastic relaxation of the detached tape.We consider three possible physical scenarios for peeling propagation: constant peeling front velocity, constant peeling force, and constant pulling velocity at the tape tip.
In the long-term limit, the peeling propagation asymptotically approaches a steady-state elastic-like behavior, regardless of the specific controlled parameter.However, the initial transient peeling behavior is strongly affected by the tape viscoelasticity and undeformed geometry, and presents non-monotonic time evolution of the peeling force and angle.More in detail, when a constant force is applied, we found that the peeling can either endlessly propagate, start and stop after some time, or not even start.Which of these scenarios occurs seems to depend only on the applied force value and undeformed non-adhering tape geometry (angle and length).
More surprisingly, when the pulling velocity at the tape tip is assigned, as in the case of impact loads acting on an attached object, the peeling propagation is delayed against the instant of force application, and the force required to sustain the peeling propagation (i.e., the peeling toughness) can be temporarily larger than in stationary conditions.This mechanism might be qualitatively related to the high-speed superior adhesive performance observed in several natural systems.
We consider that at time t = 0 the tape tip is pulled at a constant velocity v P .In this case, the deformation ε(t) and stress σ(t) in the non-adhering tape monotonically increase, and peeling front propagation starts at time t * when σ(t * ) = σ cr (t * ).The value σ cr depends on the energy balance where we assumed that v c (t * ) v γ .According to Fig. B.1(a) A schematic of the tape in undeformed condition, and at a generic time t < t * , i.e. before the peeling propagation starts.(b) The dimensionless stress σ in the tape (solid curves) and the dimensionless critical stress σcr required to start the peeling propagation (dashed curves) as functions of the dimensionless time t for different dimensionless tape tip velocity ṽP .Circles indicate the instant of propagation start.figure.caption.20a,before peeling front propagation (i.e., for t < t * ), the peeling angle is given by tan θ = L i sin φ + v P t L i cos φ , (B2) where v P t is the tape tip vertical displacement at the generic time t.Similarly, since L i + ∆L = L i cos φ/ cos θ is the deformed tape length, the tape uniform deformation before peeling propagation is Finally, using the viscoelastic constitutive equation, we can calculate the uniform stress in the detached tape as Once the peeling front propagation starts, the numerical algorithm described in Appendix Asection*.17can be used to calculate the time evolution of the peeling process, with t * corresponding to j = 0.In the present formalism, the peeling front velocity v c (t * ) at the instant of the peeling propagation start cannot be exactly determined; however, in the reasonable assumption for practical applications that v P / tan θ S = [v c ] S d/τ , the system

FIG. 4 :
FIG. 4: (a) The initial [ṽ c ] 0 and long-term [ṽ c ] S dimensionless peeling front velocity as functions of the dimensionless peeling force P .(b) The time-history of the normalized peeling front velocity ṽc /[ṽ c ] 0 for different values of dimensionless peeling force P .(c) The state map of the possible peeling behavior as a function of the dimensionless applied peeling force P and undeformed tape angle φ.P1 and P2 are the critical (minimum) force for peeling start and steady-state propagation calculated with v c d/τ , while P3 and P4 refer to the same critical conditions when v c d/τ (further know are given in the text).Results refer to κ = 10 and γ0 = 0.02.

FIG. 5 :
FIG. 5: The time-history of the dimensionless peeling force P (a), the peeling angle θ (c), and the peeling velocity ratio ṽc /ṽ P (c), together with the transient diagram P vs. θ (d) for different values of the dimensionless pulling velocity ṽP .Two undeformed tape angles are considered.The circles indicate the instant when peeling front propagation starts.The red curve in (d) represents the steady-state peeling limit, and black arrows indicate the time evolution of the process.Results refer to κ = 10 and γ0 = 0.02.
− t ) ε(t )dt , (B4)where R is the stress-relaxation function given byR(t) = E 0 + (E ∞ − E 0 )e t/τr ,(B5) with τ r = τ /(1 + ∆) being the relaxation time, and ∆ = E ∞ /E 0 − 1 being the relaxation strength.The peeling starting condition σ(t * ) = σ cr (t * ) is then obtained by simultaneously solving Eqs.(B1equation.B.1-B4equation.B.4).In Figure B.1(a) A schematic of the tape in undeformed condition, and at a generic time t < t * , i.e. before the peeling propagation starts.(b) The dimensionless stress σ in the tape (solid curves) and the dimensionless critical stress σcr required to start the peeling propagation (dashed curves) as functions of the dimensionless time t for different dimensionless tape tip velocity ṽP .Circles indicate the instant of propagation start.figure.caption.20bwe show the time-history of σ(t) and σcr (t), for different dimensionless pulling velocities ṽP .Increasing ṽP leads to faster stress increase in the viscoelastic tape, thus peeling propagation starts sooner.
Tape width W in , W P , W ad Power of the internal stress, the peeling force, and the adhesive bonds