Several medical papers have reported delamination of the coating from the stent -substrate following intravascular deployment leading to adverse outcomes for patients. However, the mechanisms of delamination of such polymer coatings from the surface of a stent due to large deformations during device deployment has not been studied. In this paper, a novel and in- depth investigation of the mechanisms and paramet ers that govern stent -coating delamination is performed, using a cohesive zone formulation to simulation the evolution of traction at the stent -coating interface. The study firstly analyses the behaviour of elastic coatings on idealised elastic stent subst rates. Simulations reveal that the mode mixity of delamination is strongly dependent on the level of stent deployment at initiation . In general, p eak normal tractions exceed peak shear tractions at low levels of stent deployment whereas the reverse trend i s computed at high levels of stent deployment . Interface tractions increase with both increasing stent thickness and coating thickness suggesting that thinner stents and thinner coatings should be utilised for the delivery of antiproliferative drugs in order to reduce the risk of coating delamination . Next, the influence of stent plasticity on interface tractions and coating delamination is investigated. Even a t low levels of deployment, plastic yielding occur s in the stent hin ge region and the patterns of normal and shear tractions are found to be significantly more complex than those computed for elastic stents , with both tensile and compressive regions of normal traction occurring in the stent arch . At a high level of stent d eployment shear tractions at the stent -coating interface are computed to increase with decreasing strain hardening modulus. The findings of this paper provide a new insight into the stress -state at the stent - coating interface as a function of the stent des ign parameters and large deformation elasticity and plasticity during deployment , allowing for a more reliable assessment of the limits relating to safe implantation of coated stents. Keywords: cohesive zone, stent, delamination, elastic -plastic, coating, geometric, material properties 1 Introduction Over 20,000 published articles can be found on finite element simulations of stents. However, to the authors knowledge, fewer than ten previous papers have included stent coating delamination in finite element a nalysis of stent performance (two further papers include a coating in a finite element model, but simplistically assume a rigid perfect bond between the coating and stent (Lee et al., 2014; Schiavone et al., 2014) ). FDA regulatory guidelines for non - clinical analysis of stent performance set a requirement for finite el ement analysis of stents, no guidelines or recommendations have been issued in relation to the finite element analysis of the mechanical behaviour and damage of stent coatings (FDA, 2010) . The lack of mechanistic analysis of coated stents is remarkable, given that implantation of polymer coated drug eluting stents is the dominant treatment strategy for patients with symptomatic coronary artery disease (Torii et al., 2020) . Moreove r, the current global market value of drug eluting stents is $5.63 2 billion and growing. The absence of a robust body of literature on the finite element design and analysis of polymer coated stents is more surprising and alarming given that several experim ental benchtop tests of drug eluting stents using scanning electron microscopy have reported delamination of polymer coatings from the stent surface during deployment (Chen et al., 2018; Fujimoto et al., 2 018; Kollum et al., 2005; Levy et al., 2009) . Such delamination is of critical importance from a clinical viewpoint, potentially leading to thrombosis, inflammation, micro -embolism formation and impaired drug delivery (Hoffmann et al., 2002; Kollum et al., 2005; Levy et al., 2009) . Indeed, it is reasonable to suggest that delamination and fracture of coatings may result in migration of debonded fragments of drug eluting to non - diseased sections of the artery, with subsequent drug elution potentially causing significant damage to healthy arterial tissue. The potential delamination risk of a polymer co ating from a metallic stent surface should not be surprising, given that plastic strains of 30 -50% develop in the hinge region of a stent during balloon expansion/deployment. Indeed the entire design principle upon which stents are based is the generation of high plastic strains so that the stent retains its expanded configuration, and thus acts as a permanent structural support to maintain a large lumen and sufficient blood flow through the previously blocked section of artery . However, only seven papers t o date have presented structural analysis of coating integrity during such large plastic deformation of the metallic stent substrate (four studies by the current authors (Hopkins et al., 2016, 2013; Máirtín et al., 2014; Parry and McGarry, 2012) , and three subsequent studies by Migliavacca and co -workers (Chen et al., 2018; Wu et al., 2015, 2014) . These studies largely provide demonstrations of coating delamination for select number of limited commercially available coated stent designs. However, no study to date has provided a systematic analysis of the role of plastic deformation on traction distri butions and delamination patterns along the stent -coating interface. The study of Parry and McGarry, (2012) presents an analytical solution for the traction distribution of an elastic coating on an elastic stent substrate. This analysis provides fundamental insight into the transition from norma l tractions at the top of the stent arch region to shear tractions at the arch -strut boundary. However, the applicability of this study to the design or real stents is limited by the infinitesimal strain and linear elastic materials required for a close fo rm analytical solution. The extension of this work to include large deformation plastic deformation of the stent substrate is an essential step to advance the current state-of-the-art of stent design. The development of a robust mechanistic approach to ste nt coating design is of particular importance, given the current industry goals of reducing strut thickness so that stents can be deployed in highly stenosed and calcified vessels (Lee and Hernandez, 2018) , and the emerging technology of coated biodegradable polymer stents (Cassese et al., 2018; Kalra et al., 2017; Misra et al., 2017; Rebagay and Bangalore, 2019) . While mechanistic analysis of stent -coating interface tractions and delamination is largely absent from the stent literature, the study of bi -layer systems is a long -established branch of mechanics (Freund and Suresh, 2004) . Such studies largely focus on co atings on flat semi -infinite substrates. For example, it is well known that compressed layers are susceptible to buckling induced delamination especially when the residual compression is large and the interface has low delamination toughness (Faou et al., 2012; Hutchinson and Suo, 1991) . Finite element analyses of thermal barrier coatings have been presented by (Sait et al., 2020; Zhu et al., 2015) . Chen et al., (2015) calculated the fracture energy release rate for brittle elastic coatings on ductile elasto -plastic substrates. The case of thin elastic films delaminating from cylindrical substrates has been investigated (Hutchinson, 2001) , and the effect of subsequent ridge cracking has been studied (Faulhaber et al., 2006) . This kind of damage occurs for thin films on very thick substrates and when very large compressive stresses are present in the film due t o deposition methods, differences in thermal expansion coefficients or when the coating is an oxide scale developing on a metallic substrate. In contrast, stents are structural elements 3 with a film to substrate thickness ratio ranging from 0.01 to 0.5 and a small residual stress. The interface damage is rather linked to the large displacement imposed to the structure during deployment, resulting in plasticity in the substrate and heterogeneous strain distributions in both coating and substrates. The overall aim of the study is to uncover the complex distributions of traction at the stent -coating interface due to large elastic and plastic deformation during stent deployment. The insights provided by the systematic analysis may be used to motivate further mech anistic analysis and design and regulatory guidelines for next -generation polymer coated drug -eluting stents. Firstly, in order to gain an in -depth understanding of the mechanics of coating delamination due to finite deformation stent deformation, hyperela stic coatings on hyperelastic stent substrates are simulated. Specifically, the influence of stent geometry ( strut length, strut thickness and coating thickness ) on interface tractions and coating delamination is analysed. Such analysis of elastic stent su bstrates may be relevant to next -generation biodegradable polymer stent substrates (Lee and Hernandez, 2018) . The influence of stent plasticity on interface tractions and coating delamination is then investigated. We demonstrate that large plastic deformation of the stent -substrate (the very basis of balloon expandable metallic stent designs) profoundly affect the distribution of traction at the stent -coating interface. Analyses suggest that re liable design of coated stent systems cannot be performed without finite deformation computational plasticity simulation. 2 Model Construction Parameterisation of stent geometry for finite element investigation Stent geometries consists of a series of inter connected repeating units, as shown in Figure 1 (a), with each repeating unit consisting of an arch connected to straight struts (Parry & McGarry, 2012). The reference configuration of the stent and coating geometry utilised in the present study is depicted in Figure 1(b). A 2D unit cell idealisation is utilised and stent and coating dimensions are chosen based on commercially available stent designs (Regar et al. , 2001) . A displacement boundary condition is applied on the bottom edge of the strut to simulate stent deployment. A polar coordinate system (r,θ) is utilised with the stent -coating interface located at r=R. The coating material and the stent materials are identified as layers 1 and 2, of thickness ℎ1 and ℎ2 respectively. The arch an d strut sections are also clearly indicated in Figure 1(b) along with the strut length, L. The distance along the stent -coating interface, x, originating at the top of the arch (x⁄(R=0)) is also indicated. A custom python code was developed to automaticall y generate stent meshes for parametric analysis. Both stent and coating are meshed using generalised plane strain elements. A mesh sensitivity study reveals that ten elements through the thickness of the coating provides a converged solution in terms of pe ak interface tangential traction at 𝑥𝑅=𝜋2⁄ ⁄ . Coatings are modelled as Neo -Hookean hyperelastic materials. In section 3.1 the stent material is modelled as neo -Hookean elastic materials, representative of next -generation polymer stents. In Section 3.2 the stent material is modelled using J2 plasticity materials with isotropic hardening, representative of metallic balloon expandable stents. (a) 4 Figure 1: (a) An example of a cardiovascular stent architecture consisting of a series of repeating interconnected unit cells (highlighted), where each unit cell consists of an arch connected to straight struts (Parry and McGarry, 2012). (b) Geometry of th e stent and stent coating in the initial configuration. Boundary conditions at the base of the stent struts and applied displacements, 𝑼, are highlighted at the strut ends. The radial distance to the stent -coating interface, 𝑹, and the distance along the stent -coating interface from the top of the arch, 𝒙, is also indicated. The arch and strut sections are also highlighted. The coating thickness, 𝒉𝟏, stent thickness, 𝒉𝟐 and strut length, 𝑳, are also shown. (c) Geometry of a deployed stent configurat ion with the stent and coating materials highlighted. The arch deployment (𝚺𝒂) and circumferential strain (𝛆𝒄𝒊𝒓𝒄𝒖𝒎𝒇 ) are also defined, with 𝛆𝒐and 𝐃𝒐 defined in ( b). As the arch deformation is the critical determinant of the interface tractions (P arry & McGarry, 2012), non -dimensional arch deformation is defined as; Σ𝑎=(𝐷−𝐷0)𝐷0⁄ (1.01) where 𝐷0and𝐷 are the i nitial and final arch deployment s respectively, as depicted in Figure 1(b,c) . Σ𝑎 is referred to as the deployment level for the remainder of this paper.The circumferential strain (𝜀𝑐𝑖𝑟𝑐𝑢𝑚𝑓 ) is also defined in Figure 1 (c) as; 𝜀𝑐𝑖𝑟𝑐𝑢𝑚𝑓 =(𝜀−𝜀0)𝜀0⁄ (1.02) Cohesive Zone Formulation The interface between the stent (substrate) and the coating is modelled using mixed mode cohesive zone formulations previously developed by (McGarry et al., 2014) . Mixed mode interface separation (∆𝑛>0) is modelled using the following expression for normal traction 𝑇𝑛and tangential traction 𝑇𝑡 𝑇𝑛(∆𝑛,∆𝑡)=𝜎𝑚𝑎𝑥𝑒𝑥𝑝(1)(∆𝑛 𝛿𝑛) 𝑒𝑥𝑝 (−√∆𝑛2 𝛿𝑛2+∆𝑡2 𝛿𝑡2) (1.03) 𝑇𝑡(∆𝑛,∆𝑡)=𝜏𝑚𝑎𝑥𝑒𝑥𝑝(1)(∆𝑡 𝛿𝑡) 𝑒𝑥𝑝 (−√∆𝑛2 𝛿𝑛2+∆𝑡2 𝛿𝑡2) (1.04) where ∆𝑛and ∆𝑡are the normal and tangential components of the interface separation vector, ∆;the normal and tangential interface characteristic len gths are 𝛿𝑛 and 𝛿𝑡; 𝜎𝑚𝑎𝑥 is the mode I interface strength and 𝜏𝑚𝑎𝑥 is the mode II interface strength. This model provides identical behaviour in mode I and mode II separation when 𝜎𝑚𝑎𝑥 =𝜏𝑚𝑎𝑥 and 𝛿𝑛=𝛿𝑡. Additionally, for this specific case the traction magnitude √𝑇𝑛2+𝑇𝑡2 is independent of the mode mixity, depending only on the separation magnitude . Furthermore, it can easily be demonstrated that (b) (c) 5 𝜕𝑇𝑛𝜕∆𝑡 ⁄ =𝜕𝑇𝑡𝜕∆𝑛 ⁄ , i.e. the traction separation equations can be derived from a potential function 𝜙(∆𝑛,∆𝑡)=𝜙𝑜+𝜎𝑚𝑎𝑥𝑒𝑥𝑝(1)(1+√∆𝑛2 𝛿𝑛2+∆𝑡2 𝛿𝑡2)𝑒𝑥𝑝 (−√∆𝑛2 𝛿𝑛2+∆𝑡2 𝛿𝑡2) (1.05) when 𝜎𝑚𝑎𝑥 =𝜏𝑚𝑎𝑥 and 𝛿𝑛=𝛿𝑡. While equations (1.03) and (1.04) provide physically realistic coupling between normal and tangential behaviour during mixed mode separation (∆𝑛>0), correctly penalization of mixed mode over -closure in the event of compressive (negative) normal interface tr actions is not obtained . Therefore, when Δn< 0 the following tractions -separation equations are implemented (McGarry et al., 2014): 𝑇𝑛(∆𝑛,∆𝑡)=𝜎𝑚𝑎𝑥exp (1)(∆𝑛 𝛿𝑛)exp (−∆𝑛 𝛿𝑛)exp (−𝛼√∆𝑡2 𝛿𝑡2) (1.06) 𝑇𝑡(∆𝑛,∆𝑡)=𝜏𝑚𝑎𝑥exp (1)(∆𝑡 𝛿𝑡)exp (−√∆𝑡2 𝛿𝑡2)exp (−𝛽∆𝑛 𝛿𝑛) (1.07) All symbols have the same meaning as equations (1 .03) and ( 1.04) above, while α and β specify the weighting of the mixed mode coupling terms. Here, 𝛼=𝛽=√2−1. In the current study it is assumed that n = t = 1 µm and 𝜎𝑚𝑎𝑥 =τ𝑚𝑎𝑥, giving identical behaviour in mode I and mode II separation unless otherwise stated . Values of interface strength 𝜎𝑚𝑎𝑥 are selected based on the range identified in the experimental study of (Hopkins et al., 201 3). The cohesive zone model is implemented in the Abaqus Standard (implicit) solver via a user -defined interface subroutine (UINTER). 3 Parametric Analysis 3.1 Elastic Stent In this section, elastic material properties are utilised to describe both the co ating and stent materials. The stent stiffness, 𝐸𝑠, and the coating stiffness, 𝐸𝑐, are chosen so that 𝐸𝑠𝐸𝑐⁄ =2.5, unless otherwise stated. This coating to stent stiffness ratio is based on a polymer coating on a polymer stent. The choice of a relativ ely stiff coating is supported by the fact that diamond like carbon coatings (Kim et al., 2007) and aluminium coatings (Kollum et al., 2005) have frequently been coated on stents. In this section the influence of interface strength ( 𝜎𝑚𝑎𝑥 𝐸𝑐⁄ = 𝜏𝑚𝑎𝑥 𝐸𝑐⁄), strut length (𝐿), coating thickness (ℎ1) and stent thickness (ℎ2) on computed interface tractions is considered. Unless otherwise stated, cohesive zone parameters are chosen so that peak mode I traction and peak mode II tractions are equal ( 𝜎𝑚𝑎𝑥 𝐸𝑐⁄ =𝜏𝑚𝑎𝑥 𝐸𝑐⁄ =2) with the peak occurring at t he same effective separation in mode I and mode II separation (|𝑇|𝑚𝑎𝑥𝜃=0=|𝑇|𝑚𝑎𝑥𝜃=𝜋2⁄; |∆|𝑚𝑎𝑥𝜃=0=|∆|𝑚𝑎𝑥𝜃=𝜋2⁄). A characteristic distance of 𝛿𝑛=𝛿𝑡=0.04𝜇𝑚 was chosen based on the parameter sensitivity and mesh sensitivity analyses . A Poisso n’s ratio of 0.3 is assumed for both the stent and coating materials. 6 Influence of Prescribed Interface Strength on the Initial Debonding Mechanism The interface strength dramatically influences the initial delamination mechanism at the stent - coating interface. Figure 2 shows the initial coating delamination patterns for a strut length of 𝐿𝑅=2 ⁄ when differing interaction strengths are chosen. When a low interface strength of 𝜎𝑚𝑎𝑥 𝐸𝑐⁄ =𝜏𝑚𝑎𝑥 𝐸𝑐⁄ =0.0019 is prescribed ( Figure 2a(i) ), mode I coating delamination initiates at the top of the arch (𝑥/𝑅=0) at a moderate stent deployment (Σ𝑎=0.025 ). Following delamination, a significant reduction in coa ting stress is computed, whereas an increase in tensile stress is computed on the stent surface in the delamination region ( Figure 2a(ii)). A markedly different pattern of coating delamination is observed if a stronger interface strength (𝜎𝑚𝑎𝑥 𝐸𝑐⁄ =𝜏𝑚𝑎𝑥 𝐸𝑐⁄ =0.026 ) is prescribed. As the interface strength is high, delamination does not occur until a high level of stent deployment is reached (Σ𝑎=0.47)(Figure 2b(i)). As demonstrated in Figure 2b(i), prior to the initiation of debonding (Σ𝑎=0.43) tangent ial tractions at the bottom of the arch become greater in magnitude than normal tractions at the top of the arch when high stent deployment levels are imposed. Again, it should be noted that this effect is particularly dominant for stents with shorter stru ts. Hence, predominantly mode II coating delamination initiates in the region of the bottom of the arch (𝑥/𝑅=/2). Following initiation of predominantly mode II debonding, a reduction of both tangential and normal tractions is observed at the bottom of t he arch in Figure 2b(ii), demonstrating the importance of correct coupling in the cohesive zone formulations. The predominantly tangential separation of the interface at the bottom of the arch (𝑥/𝑅=/2) is shown in Figure 2c(ii). Following mode II initia tion, mixed -mode delamination propagates from the bottom of the arch towards the top of the arch, as shown in Figure 2c(iii). The coating remains attached to the stent at the top of the arch as mode I delamination is not achieved until further deployment. Figure 2: (a) Stent and coating von Mises stress (𝝈) distribution plotted on the deformed geometry for an arch deployment, 𝚺𝒂, of (i) 𝟎.𝟎𝟐𝟑 and (ii) 𝟎.𝟎𝟐𝟓. Mode I coating delamination initiates at the top of the arch (𝒙𝑹⁄=𝟎) as shown in (ii) . 𝑬𝒄=𝟐𝟎𝟎𝐆𝐏𝐚 , 𝑬𝒔=𝟓𝟎𝟎𝐆𝐏𝐚 , 𝝈𝒎𝒂𝒙 𝑬𝒄⁄ =𝝉𝒎𝒂𝒙 𝑬𝒄⁄ =𝟎.𝟎𝟎𝟏𝟗 , 𝒉𝟏𝑹=𝟎.𝟐 ⁄ , 𝒉𝟐𝑹=𝟏 ⁄ , 𝐋𝑹=𝟐 ⁄ . Deformation factor =3 in (ii). (b) Computed interface traction (𝑻𝑬𝒄⁄) as a function of the distance along the stent -coating in terface (𝒙𝑹⁄) for an arch deployment, 𝚺𝒂, of (i) 𝟎.𝟒𝟑 and (ii) 𝟎.𝟒𝟕. Predominantly mode II coating delamination initiates at the bottom of the arch (𝒙𝑹⁄=𝝅𝟐⁄) as indicated by the reduction in traction in (ii). 𝑬𝒄=𝟐𝟎𝟎𝐆𝐏𝐚 , 𝑬𝒔=𝟓𝟎𝟎𝐆𝐏𝐚 , 𝝈𝒎𝒂𝒙 𝑬𝒄⁄ =𝝉𝒎𝒂𝒙 𝑬𝒄⁄ = 𝟎.𝟎𝟐𝟔, 𝒉𝟏𝑹=𝟎.𝟐 ⁄ , 𝒉𝟐𝑹=𝟏 ⁄ , 𝐋𝑹=𝟐 ⁄ . (c) Stent and coating von Mises stress (𝝈) distribution plotted on the deformed geometry for an arch deployment, 𝚺𝒂, of (i) -(ii) 𝟎.𝟒𝟕 and (iii) 𝟎.𝟒𝟕𝟓. Predominantly mode II coating delamination initiates at the bottom of the arch (𝒙𝑹⁄=𝝅𝟐⁄) as shown in (ii). Mixed -mode 7 coating delamination computed at higher stent deployment is shown in (iii). 𝑬𝒄=𝟐𝟎𝟎𝐆𝐏𝐚 , 𝑬𝒔=𝟓𝟎𝟎𝐆𝐏𝐚 , 𝝈𝒎𝒂𝒙 𝑬𝒄⁄ =𝝉𝒎𝒂𝒙 𝑬𝒄⁄ =𝟎.𝟎𝟐𝟔, 𝒉𝟏𝑹=𝟎.𝟐 ⁄ , 𝒉𝟐𝑹=𝟏 ⁄ , 𝐋𝑹=𝟐 ⁄ . Deformation factor =3 for (iii). Effect of Strut Length In this section the effect of strut length on stent -coating interface tractions is investigated, assuming a constant stent thickness, coating thickness and stent -coating stiffness ratio for all simulations. Assuming parameters of ℎ1𝑅=0.2 ⁄ , ℎ2𝑅=1 ⁄ and 𝐸𝑠𝐸𝑐=2.5 ⁄ , three strut length are considered: 𝐿/𝑅=0.5, 𝐿/𝑅=2, and 𝐿/𝑅=10, representing stent designs with very short, typical and very long struts respectively. Figure 3 shows the computed normal (𝑇𝑛) and tangential (𝑇𝑡) tractions at the stent -coating interface for a range of strut lengths. A high interface strength is assumed for all simulations presented in Figure 3 so that the distribution of tangential and normal tractions can be examined in the absence of interface d ebonding. Additionally, a high interface stiffness is assumed so that the infinitesimal strain analytical elastic solution of Parry & McGarry (2012) is reproduced when the strut is removed from the geometry . Results are presented at three stent deployment levels: Low deployment (Σ𝑎=0.024 ) results are shown in Figure 3(a), where the solution is in the infinitesimal strain regime; Medium deployment (Σ𝑎=0.134 ) is shown in Figure 3(b), where the solution is in the finite strain regime; High deployment (Σ𝑎=0.608) is shown in Figure 3(c), again with the solution in the finite strain regime. An examination of the computed tractions for the shortest strut length (𝐿/𝑅=0.5) at a low deployment (Σ𝑎=0.024 ) in Figure 3(a) reveals that the peak normal traction at the top of the arch (𝑥/𝑅=0) is greater than the peak tangential traction at the bottom of the arch (𝑥/𝑅=/2).A sinusoidal distribution of normal and shear traction can be observed at the interface along the arch (0≤𝑥/𝑅≤/2), whereby the tangential traction red uces to 0 at the top of the arch (as required by symmetry) and the normal traction reduces to a non -zero value at the bottom of the arch (𝑥/𝑅=/2). It is worth noting that the range of tangential and normal tractions is approximately equal over the arch interface (𝑇𝑛(𝑥/𝑅=0)-𝑇𝑛(𝑥/𝑅=/2)𝑇𝑡(𝑥/ 𝑅=/2)). These observations are closely aligned with the traction distribution determined by the analytical approach of Parry & McGarry (2012) , where the incorporation of a strut simply effects an increased rotation of the base of the arch , reducing the tangential traction and increasing the normal traction. 8 Figure 3: Computed normal (𝑻𝒏)and tangential (𝑻𝒕) tractions along the stent -coating interface for a range of strut lengths for (a) low deployment (𝚺𝒂=𝟎.𝟎𝟐𝟒 ) (b) medium deployment (𝚺𝒂=𝟎.𝟏𝟑𝟒 ) and (c) high deployment (𝚺𝒂=𝟎.𝟔𝟎𝟖 ). At low deployment, normal tractions increase from the bottom of the arch (𝒙𝑹⁄=𝝅𝟐⁄) to the top of the arch (𝒙𝑹⁄=𝟎)while tangential tractions increase from the top of the arch (𝒙𝑹⁄=𝟎) to the bottom of the arch (𝒙𝑹⁄=𝝅𝟐⁄). 𝑬𝒄=𝟐𝟎𝟎𝐆𝐏𝐚 , 𝑬𝒔=𝟓𝟎𝟎𝐆𝐏𝐚 , 𝝈𝒎𝒂𝒙 𝑬𝒄⁄ = 𝝉𝒎𝒂𝒙 𝑬𝒄⁄ =𝟐, 𝒉𝟏𝑹=𝟎.𝟐 ⁄ , 𝒉𝟐𝑹=𝟏 ⁄ . The key observations for low deployment (infinitesimal strain) are as follows: (i) Peak normal traction occurs at the top of the arch; (ii) Peak tangential traction oc curs at the bottom of the arch and is lower than the peak normal traction at the top of the arch for the range of strut length s considered (𝐿/𝑅≤10); (iii) Sinusoidal distributions are computed in the arch only for a very short strut length (𝐿/𝑅≤ 0.5); (iv) Tractions in the strut are lower than those in the arch, with normal tractions reducing to zero and tangential tractions reducing to a constant non - zero value that depends on strut length. The patterns of computed traction for a medium deployment leve l ((Σ𝑎=0.134), Figure 3(b)) are reasonably similar to those presented for a low deployment level. However, for the highest deployment level (Σ𝑎=0.608 ), Figure 3(c)) significantly different trends can be observed. The peak tangential traction at the bottom of the arch is greater in magnitude than the peak normal traction for all strut lengths. Shorter struts lead to the greatest increase in tangential traction. Additionally, a further increase in tangential traction (and reduction in normal traction) is computed when the strut is omitted (𝐿𝑅⁄=0). The normal traction is constant for all points on the interface where 𝑥/𝑅<0.5 for all strut lengths. Additionally , it can be observed that compressive normal tractions are computed along the strut, highlight ing the importance of the cohesive zone behaviour under conditions of mixed -mode over-closure . It should be noted that f or each specified value of Σ𝑎, the radial expansion of the stent will depend also on the strut length. This is clearly evident from Figure 4 where stress distributions and deformed geometries for the three deployment levels (Σ𝑎=0.024 ,0.134, 0.608) are shown for a strut length of 𝐿/𝑅=2 in Figure 4(a-c) and for a strut length of 𝐿/𝑅= 10 in Figure 4(d-f). For example, for the high deplo yment level (Σ𝑎=0.608 ) a strut length of 𝐿/𝑅=2 provides an 87% increase in stent radius, representing a moderate radial expansion 9 (as shown in Figure 4(c)) whereas a strut length of 𝐿/𝑅=10 provides a 466% increase in stent radius, representing a significant radial overexpansion (as shown in Figure 4(f)). Finally, it should be noted that rotational constraints are not imposed on the strut ends for all simulations presented in this paper, as this is representative of the majority of commercially available stent designs (McGarry et al. , 2004; Hopkins et al. , 2010) . However, certai n stent designs may result in a rotational constraint at the bottom of the struts, e.g. a repeating diamond arrangement of struts. This constraint essentially results in a significant increase in tangential traction for all deployment levels and also resul ts in a normal compression at the stent –coating interface at the bottom of the arch. Figure 4: von Mises stress (𝝈)distribution for deformed stent geometries at low (𝚺𝒂=𝟎.𝟎𝟐𝟒 ), medium (𝚺𝒂=𝟎.𝟏𝟑𝟒 )and high (𝚺𝒂=𝟎.𝟔𝟎𝟖 ) deployment for (a) -(c) 𝑳𝑹=𝟐 ⁄ and (d)-(f)𝑳𝑹=𝟏𝟎 ⁄ . The black line at base of the arch is included to highlight the increase in arch rotation with increasing stent deployment. Effect of Coating Thickness In this section the effect of coating thickness on stent -coating interface tractions is investigated, assuming a constant stent thickness, strut length and stent -coating stiffness ratio for all simulations. Figure 5 shows the computed normal (𝑇𝑛) and tange ntial (𝑇𝑡) tractions at the stent -coating interface for a range of coating thicknesses. Again, normal and tangential tractions ar e assessed in the absence of interface debonding. Interface tractions increase with coating thickness with higher normal and tan gential interface tractions being computed for thicker coating at all stent deployment levels. This suggests that the use of a thinner coating could prevent coating debonding. At a low level of stent deployment (Σ𝑎=0.0063 ) (Figure 5 (a)), the peak normal tr action at the top of the arch (x/R=0) is greater than the peak tangential traction at the bottom of the arch ( x/R=/2) for the thicker coatings (ℎ1𝑅⁄=0.2;ℎ1𝑅⁄=0.5). However, the peak tangential traction at the bottom of the arch is similar in magnitude to th e peak normal traction at the top of the arch for the thinner coating (ℎ1𝑅⁄=0.05) with |𝑇𝐸𝑐⁄|≈1.25 x10−4.At medium stent deployment (Σ𝑎= 0.236)(Figure 5(b)), the peak normal traction computed at the top of the arch (x/R=0) again exceeds the peak tangential tra ction at the bottom of the arch (x/R=/2) except for the case of the thinner coating (ℎ1𝑅⁄=0.05), where the computed peak tangential tractions are marginally greater. At high stent deployment (Σ𝑎=0.737 )(Figure 5(c)), the peak tangential traction computed at the bottom of the arch exceeds the peak normal traction at the top of the 10 arch for all coating thicknesses. At this level of stent deployment, instead of normal traction reducing monotonically from the top of the arch towards the bottom of the arch along the arch interface (0≤x/R≤/2), as seen in Figure 5, an increase in normal traction is computed along the interface from the top of the arch (x/R=0) to x/R=1.05 and x/R=0.82 for coating thicknesses of ℎ1𝑅⁄=0.05,and 0.2 respectively. A marginal increase in normal traction is also computed for the thickest coating ( ℎ1𝑅⁄=0.5) from the top of the arch (x/R= 0) to x/R=0.28 before reducing towards the bottom of the arch. Therefore, at a high level of stent deployment, the location of maximum normal t raction is computed further from the top of the arch as coating thickness reduces. Additionally, tangential traction along the strut also increases with coating thickness. A small degree of coating compression is computed for coating thicknesses of ℎ1𝑅=0.05 ⁄ and ℎ1𝑅=0.2 ⁄ at a low level of stent deployment ( Figure 5(a)) along the stent strut (𝑥/𝑅>/2) with no compression computed for the thickest coating. Minor coating compression is computed along the stent strut for all coating thicknesses at medium sten t deployment ( Figure 5(b)). At a high level of deployment, in the case of the thinnest coating (ℎ1𝑅⁄=0.05), maximum normal compression (𝑇𝐸𝑐⁄ ≈−4.8 𝐸−4) is computed in the region of the bottom of the arch at 𝑥/𝑅=1.9. Maximum normal compression is computed further along the stent strut, at 𝑥/𝑅=2.2 and 2.5 for the thicker coatings of ℎ1𝑅⁄= 0.2 and 0.5 respectively. Again, it is worth noting that the cohe sive zone formulation correctly penalises mixed -mode coating over -closure along the stent struts. Figure 5: Computed normal (𝑻𝒏)and tangential (𝑻𝒕) tractions along the stent -coating interface for a range of coating thicknesses for (a) low deployment (𝚺𝒂=𝟎.𝟎𝟎𝟔𝟑 ) (b) medium deployment (𝚺𝒂=𝟎.𝟐𝟑𝟔 ) and (c) high deployment (𝚺𝒂=𝟎.𝟕𝟑𝟕 ). The insert in (c) shows increasing normal tractions away from the top of the arch (𝒙𝑹⁄=𝟎) for 𝒉𝟏𝑹=𝟎.𝟎𝟓 𝐚𝐧𝐝 𝟎.𝟐 ⁄ . 𝑬𝒄=𝟐𝟎𝟎𝐆𝐏𝐚 , 𝑬𝒔=𝟓𝟎𝟎𝐆𝐏𝐚 , 𝝈𝒎𝒂𝒙 𝑬𝒄⁄ =𝝉𝒎𝒂𝒙 𝑬𝒄⁄ =𝟐, 𝒉𝟐𝑹=𝟏 ⁄ , 𝐋𝑹=𝟐 ⁄ . 11 Effect of Stent T hickness In this section the effect of stent thickness on stent -coating interface tractions is investigated, assuming a constant coating thickness, strut length and stent -coating stiffness ratio for all simulations. Analyses presented in previous sections assume a stent thickness of ℎ2/𝑅=1, based on typical dimensions of commercially available stents. In Figure 6, three additional stent thicknesses are considered: ℎ2/𝑅=5 (significantly thicker than typical commercially available stent designs); ℎ2/𝑅=0.5 (significantly thinner than typical commercially available stent designs; ℎ2/𝑅=0.1 (unconventionally thin stent: In this case, the coating is twice as thick as the stent). The investigation of stent thickness on stent -coating interface traction is particularly relevant given recent clinical studies that suggest that thinner stent struts lead to reduced rates of restenosis (Kastrati et al., 2001; Turco et al., 2008) . Figure 6 shows the computed normal (𝑇𝑛) and tangential (𝑇𝑡) tractions at the stent -coating interface for the three aforementioned stent thicknesses. At a low level of stent deployment (Σ𝑎=0.017 ) (Figure 6(a)) interface tractions generally increase with stent thickness. However, a 10 -fold increase in stent thickness from ℎ2/𝑅=0.5 to ℎ2/𝑅=5 results in only a ~2 fold increase in peak normal traction at the top of the arch. Peak normal tractions at the top of the arch (𝑥/𝑅=0) are greater than the peak tangential tractions at the bottom of the arch (𝑥/𝑅=/2) for all stent thicknesses. Most notably, tangential tractions are negligible for the thinnest stent geometry (ℎ2/𝑅=0.1) at a low level of deployment. Similar trends can also be observed at a medium level of stent deployment (Σ𝑎=0.327), as shown in Figure 6(b). In contrast to lower stent deployment ( Figure 6(a) and (b)), at a high level of stent deployment (Σ𝑎=0.831 ) (Figure 6(c)) peak tangential tractions are greater than peak normal tractions for all strut thicknesses, in particular for thinner struts (ℎ2/𝑅=0.1,0.5). Normal (𝑇𝑛𝐸𝑐⁄) and tangential (𝑇𝑡𝐸𝑐⁄) tractions are depicted in two separate figures in part (c) to enhance the visualisation of the tractions. Interestingly, for the two thinnest stents, the normal tractions decrease from the values compu ted for the lower deployment level considered in Figure 6(b). In fact, negligible normal tractions are computed for the thinnest stent design (ℎ2/𝑅=0.1). Additionally, coating compression is computed in the region of maximum tangential traction (the bottom of the arch) for the thinnest stent. Tangential tractions are significantly higher than normal tractions at the high deployment level for all stent thicknesses. From a design perspective , very thin stent struts will result in dominant normal tractions at l ow deployment levels , and dominant tangential tractions at high deployment levels. 12 Figure 6: Computed normal (𝑻𝒏)and tangential (𝑻𝒕) tractions along the stent -coating interface for a range of strut thicknesses for (a) low deployment (𝚺𝒂=𝟎.𝟎𝟏𝟕 ) (b) med ium deployment (𝚺𝒂=𝟎.𝟑𝟐𝟕 ) and (c) high deployment (𝚺𝒂=𝟎.𝟖𝟑𝟏 ). Computed normal (𝑻𝒏𝑬𝒄⁄) and tangential (𝑻𝒕𝑬𝒄⁄) tractions are shown separately in (c) to enhance the visualisation . 𝑬𝒄=𝟐𝟎𝟎𝐆𝐏𝐚 , 𝑬𝒔=𝟓𝟎𝟎𝐆𝐏𝐚 , 𝝈𝒎𝒂𝒙 𝑬𝒄⁄ =𝝉𝒎𝒂𝒙 𝑬𝒄⁄ =𝟐, 𝒉𝟏𝑹=𝟎.𝟐 ⁄ , 𝐋𝑹=𝟐 ⁄ . Further results on the influence of (i) stent -coating stiffness ratio, (ii) Mode I to Mode II interface strength ratio on coating delamination, and (iii) geometrical parameters on peak tangential to peak normal interface traction r atio can be found in the Supplementary Material Sections (A -C) respectively. 3.2 Elastic -Plastic Stent In this section, the influence of stent plasticity on computed interface tractions is investigated. Plastic deformation is required for the correct functionality of a metallic balloon expandable stent. Plastic strains of 30 -50% are developed in the region of the arch while the diameter of the stent increases by a factor of 3 -5 during balloon deployment ( McGarry et al., 2004). Typically, elastic recoil results in a 2 -5% reduction in deployed diameter following balloon removal. Analysis of Interface Tractions Without Coating Debonding A high interface strength is assumed for all simulations presented in this section so that the distribution of tangential and normal tractions can be examined in the absence of interface debonding. In this section, computed interface tractions are presented for a range of both yield stresses (𝜎𝑦=80MPa ,200 MPaand 600 MPa ) and stent strain hardening moduli (𝐸ℎ= 2MPa ,20 MPa and 200 MPa ). Coatings are modelled as elastic materials with a Young’s modulus (𝐸𝑐= 20MPa ), based on the stiffness range of commercially available polymer coatings (Yung & Cooper, 1998; Antony et al. , 2003) . In all cases, coating thickness (ℎ1𝑅=0.2 ⁄ ) and strut length (L𝑅=2 ⁄ ) are based on the reference stent geometry. A 13 Young’s modulus (𝐸𝑠) of 200GPa, yield stress (𝜎𝑦) of 200MPa and Poisson’s ratio (𝜈) of 0.3 is chosen for the elastic -plastic stent material (representative of 316L stainless steel) unless otherwise stated. Influence of Yield Stress Figure 7 reveals the influence of yield stress on the computed normal (𝑇𝑛) and tangential (𝑇𝑡) tractions at the stent -coating interface. It is shown that yield stress (𝜎𝑦) dramatically influences the computed interface tractions, particularly at high levels of sten t deployment. Three yield stresses are considered, 𝜎𝑦=80MPa ,200 MPa ,and 600 MPa , representative of the yield stress of biodegradable magnesium, 316L stainless steel, and cobalt chromium, respectively. At a low level of stent deployment (Σ𝑎=0.056 ), peak ta ngential traction exceeds peak normal traction for each yield stress considered ( Figure 7(a)). Even at this low deployment level, yielding occurs in all three cases and the patterns of normal and tangential tractions differ significantly to those reported for elastic stents in previous sections of this paper . Peak tangential tractions are computed close to the top of the arch at 𝑥/𝑅=0.4 with greatest tangential traction computed for 𝜎𝑦=200𝑀𝑃𝑎 . A minimum in normal traction is computed in all cases at 𝑥/𝑅= 0.36 with coating compression (characterised by negative normal traction) computed for 𝜎𝑦= 200𝑀𝑃𝑎 . In all cases two local maxima are computed for normal traction, one at the top of the arch (𝑥/𝑅=0) and one at 𝑥/𝑅0.6. At a medium level of ste nt deployment (Σ𝑎=0.303 ), the largest normal and tangential tractions are computed at the top of the arch (𝑥/𝑅=0) and at 𝑥/𝑅=0.375 respectively for 𝜎𝑦=600𝑀𝑃𝑎 (Figure 7(b)). Furthermore, significant coating compression is computed for 𝜎𝑦=600𝑀𝑃𝑎 , from 0.26≤𝑥/𝑅≤0.43. In the case of the lower yield stresses ( 𝜎𝑦= 80𝑀𝑃𝑎 and 200 𝑀𝑃𝑎 ) the peak normal traction is not computed at the top of the arch and does not fluctua te very significantly along the arch interface. Yield stress has a significant influence on the location and magnitude of maximum tangential traction at this deployment level with peak tangential traction computed at 𝑥/𝑅=0.85 and 𝑥/𝑅=0.57 for 𝜎𝑦= 80𝑀𝑃𝑎 and 200 𝑀𝑃𝑎 respectively. At a high stent deployment (Σ𝑎=0.627 ) similar trends in tangential traction are computed ( Figure 7(c)). Significant coating compression is again computed for 𝜎𝑦=600 𝑀𝑃𝑎 from 0.16≤𝑥/𝑅≤0.47. Interestingly, relative to Figure 7(b), a significant decrease in normal traction is computed at the top of the arch (𝑥/𝑅=0), for 𝜎𝑦= 200 𝑀𝑃𝑎 , with negligible normal traction computed (𝑇𝐸𝑠⁄ ~1.7𝑥10−6). Similar to Figure 7(b), largest normal and tange ntial tractions traction are still computed for 𝜎𝑦=600 𝑀𝑃𝑎 at the top of the arch (𝑥/𝑅=0)and at 𝑥/𝑅=0.37 respectively . 14 Figure 7: Computed normal (𝑻𝒏) and tangential (𝑻𝒕) tractions along the stent -coating interface for a range of yield stresses (𝝈𝒚=𝟖𝟎𝑴𝑷𝒂 ,𝟐𝟎𝟎 𝑴𝑷𝒂 ,𝟔𝟎𝟎 𝑴𝑷𝒂 ), for (a) low deployment (𝚺𝒂=𝟎.𝟎𝟓𝟔 ) (b) medium deployment (𝚺𝒂=𝟎.𝟑𝟎𝟑 ) and (c) high deployment (𝚺𝒂=𝟎.𝟔𝟐𝟕 ). In all cases, an elast ic coating (𝑬𝒄= 𝟐𝟎𝟎 𝑴𝑷𝒂 ) is considered . For the elastic -plastic stent, 𝑬𝒔=𝟐𝟎𝟎 𝑮𝑷𝒂 , the strain hardening modulus, 𝑬𝒉= 𝟐𝟎𝐌𝐏𝐚 , and yield stress. 𝑳/𝑹=𝟐, 𝒉𝟏𝑹⁄=𝟎.𝟐, 𝒉𝟐𝑹⁄=𝟏. 𝝈𝒎𝒂𝒙 𝑬𝒔⁄ =𝝉𝒎𝒂𝒙 𝑬𝒔⁄ =𝟐. Influence of Stent Strain Hardening on Interface Tractions The computed normal (𝑇𝑛) and tangential (𝑇𝑡) tractions for a range of stent strain hardening moduli (𝐸ℎ=2MPa ,20 MPa and 200 𝑀𝑃𝑎 )are shown in Figure 8 for a coating stiffness (𝐸𝑐) of 200𝑀𝑃𝑎 . At a low stent deployment, the influence of stent strain hardening modulus on interface traction distributions is not significant ( Figure 8(a)). At medium stent deployment (Σ𝑎=0.255 ), it is evident that the strain hardening modulus significantly affects the computed traction distributio ns (Figure 8(b)).The greatest normal and tangential tractions are computed for 𝐸ℎ=2𝑀𝑃𝑎 at this level of stent deployment. For 𝐸ℎ=2𝑀𝑃𝑎 and 20𝑀𝑃𝑎 , peak tangential tractions are computed in the same location along the stent -coating interface as minimum normal tractions at 𝑥𝑅≈0 ⁄ .4. At a high level of stent deployment (Σ𝑎=0.5), the greatest normal and tangential tractions are again computed for 𝐸ℎ=2𝑀𝑃𝑎 (Figure 8(c)). However, it should be noted that the location of peak normal traction is computed at 𝑥𝑅≈0 ⁄ .9 while a significant reduction in normal traction is computed at the top of the arch for 𝐸ℎ=2𝑀𝑃𝑎 and 20𝑀𝑃𝑎 . This reduction in normal traction can be attr ibuted to the high concentration of plastic deformation in the region of the top of the arch at high stent deployment. Lowest tangential tractions are computed for the stiffer strain hardening modulus (𝐸ℎ=200𝑀𝑃𝑎 ) for 𝑥𝑅<0 ⁄ .7. 15 Figure 8: Computed nor mal (𝑻𝒏) and tangential (𝑻𝒕) tractions along the stent -coating interface for (a) low deployment (𝚺𝒂=𝟎.𝟎𝟑) (b) medium deployment (𝚺𝒂=𝟎.𝟐𝟓𝟓 ) and (c) high deployment (𝚺𝒂=𝟎.𝟓) for an elastic -plastic stent with the substrate thickness, 𝒉𝟐𝑹⁄=𝟏. Three st rain hardening moduli are considered for the strut (𝑬𝒉=𝟐,𝟐𝟎 𝐚𝐧𝐝 𝟐𝟎𝟎 𝑴𝑷𝒂 ). In all cases, an elastic coating (𝑬𝒄=𝟐𝟎𝟎 𝑴𝑷𝒂 ) is considered .𝒉𝟏𝑹⁄=𝟎.𝟐,𝑬𝒔=𝟐𝟎𝟎 𝑮𝑷𝒂 , 𝑳/𝑹=𝟐, 𝝈𝒎𝒂𝒙 𝑬𝒔⁄ =𝝉𝒎𝒂𝒙 𝑬𝒔⁄ =𝟐. The strut yield stress (𝝈𝒚)= 𝟐𝟎𝟎 𝑴𝑷𝒂 . Analysis of Coating Debonding Mechanisms The previous section illustrates that complex distributions of normal and tangential traction are computed along the stent -coating interface when plastic deformation of th e stent is considered. The ratio of normal to tangential traction at any point on the interface is shown to be highly dependent on the deployment level in addition to the yield stress and strain hardening modulus. While delamination is not considered in th e previously presented analyses of plastically deforming stents, with a very strong stiff interface being assumed, the complex traction distributions suggest that if a lower interface strength is considered the location and mode mixity of debonding will be highly dependent on several factors, including deployment level at debonding initiation, ratio of normal to tangential interface strength, in addition to stent geometric and material parameters. In the final section of this paper, two examples of diverse coating debonding patterns during deployment of a plastically deforming stent are presented. The first example entails debonding of a coating from a thin stent geometry (ℎ2𝑅=0.1 ⁄ ) and the second example entails the debonding of a coating from a thick stent g eometry (ℎ2𝑅=2 ⁄ ). These debonding simulations do not represent an exhaustive collection of all possible debonding patterns. They merely serve to illustrate some possible debonding patterns that emerge as a result of the complex distributions of normal and ta ngential tractions at the stent – coating interface in the case of a plastically deforming stent. 16 Coating Debonding Behaviour for a Thin Stent Strut Design (𝒉𝟐𝑹⁄=𝟎.𝟏) For the thin stent geometry (ℎ2𝑅⁄=0.1) it can be seen that tangential tractions are dominant at an arch deployment of Σ𝑎=0.371 for 𝐸𝑐=20𝑀𝑃𝑎 and 𝐸ℎ=200𝑀𝑃𝑎 with peak tangential tractions computed at 𝑥/𝑅=0.75. In the first sample simulation of coating debonding, presented in Figure 9, interface strengths are chosen so that coating debonding initiates at this level of deployment. Furthermore, it is assumed that mode I and mode II interface strengths are equal (𝜎𝑚𝑎𝑥 𝐸𝑠⁄ =𝜏𝑚𝑎𝑥 𝐸𝑠⁄ =2x10−6). The coating node located at 𝑥/𝑅=0.75 is hig hlighted in Figure 9(b). Corresponding interface tractions are depicted in Figure 9(c) and (d). Mixed -mode delamination is initially computed at the interface at 𝑥/𝑅= 0.75as shown in Figure 9(b). As stent deployment increases, a predominantly mode II delamination is computed at the interface with 𝑇𝑡𝜏𝑚𝑎𝑥 ≈0.914 ⁄ (Figure 9(c)). Due to the dominant mode II type delamination, the resistance to normal separation is significantly reduced with computed peak normal traction of 𝑇𝑛𝜎𝑚𝑎𝑥 ≈0.46 ⁄ (Figure 9(d)). No coating compression is computed at the stent -coating interface. Following full coating delamination at 𝑥/𝑅=0.75, mixed -mode delamination progresses along the interface with pure mode I coating delamination eventually occurring at the top of the arch (𝑥/𝑅=0). Further progression of coating delamination at higher stent deploym ent is shown in Figure 9(e). However, at the level of stent deployment shown in Figure 9(e), pure mode I delamination has not yet occurred. Figure 9: (a) Stent and coating von Mises (𝝈) stress distribution plotted on the deformed geometry at an arch deployment of 𝚺𝒂=𝟎.𝟑𝟕𝟏 (b) Mixed -mode coating delamination computed at 𝒙/𝑹=𝟎.𝟕𝟓 (c) Normalised tangential traction ( 𝑻𝒕𝝉𝒎𝒂𝒙⁄ ) as a function of normalised tangential separation (∆𝒕𝜹𝒕⁄) computed at coating node (𝒙/𝑹=𝟎.𝟕𝟓). (d) Normalised normal traction ( 𝑻𝒏𝝈𝒎𝒂𝒙⁄ ) as a function of normalised normal separation ( ∆𝒏𝜹𝒏⁄) computed at coating node (𝒙/𝑹=𝟎.𝟕𝟓). (e) von Mises (𝝈) stress distribution plotted on the deformed st ent geometry at an arch deployment of 𝚺𝒂=𝟎.𝟒𝟐. An elastic coating (𝑬𝒄=𝟐𝟎𝑴𝑷𝒂 ) is considered . Elastic -plastic properties are assigned for the stent strut with 𝑬𝒔= 𝟐𝟎𝟎 𝑮𝑷𝒂 , strain hardening modulus, 𝑬𝒉=𝟐𝟎𝟎𝑴𝑷𝒂 and yield stress, 𝝈𝒚=𝟐𝟎𝟎𝑴𝑷𝒂 . 𝒉𝟏𝑹⁄=𝟎.𝟐,𝒉𝟐𝑹⁄= 𝟎.𝟏,𝑳/𝑹=𝟐, 𝝈𝒎𝒂𝒙 𝑬𝒔⁄ =𝝉𝒎𝒂𝒙 𝑬𝒔⁄ =𝟐𝒙𝟏𝟎−𝟔. 17 Coating Debonding Behaviour for a Thick Stent Strut Design (𝒉𝟐𝑹⁄=𝟐) In the final sample simulation of coating debonding from a plastically deforming stent a thick stent geometry (ℎ2𝑅⁄=2) is considered, with 𝐸𝑐=200 𝑀𝑃𝑎 ,𝐸ℎ=2𝑀𝑃𝑎 . The interface strengths are reduced so that debonding initiates at a deployment level of Σ𝑎=0.101, assuming that the normal and tangential interface strengths are equal (𝜎𝑚𝑎𝑥 𝐸𝑠⁄ =𝜏𝑚𝑎𝑥 𝐸𝑠⁄ = 4.8 x10−5). The peak tangential traction is computed at 𝑥/𝑅≈0.6 at this low level of deployment as indicated by the non -debonding simulation shown in Supplementary Material Figure S10 . Additionally, in this region the coating is compressed normally into the stent surface, hence nodes in this region undergo mixed -mode over -closure. The node at 𝑥/𝑅=0.6 is highlighted in Figure 10(a). The insert shown i n Figure 10(b) illustrates the mixed -mode over-closure of the node. It can be noted that while the coating is compressed into the stent surface, the magnitude of over -closure is very small, as the cohesive zone framework penalises such mixed -mode over -closure (characterised by the computation of a rapid increase in negative normal tractions highlighted by red arrow in Figure 10(e)). During the debonding of the node highlighted in Figure 10, the compression of the node into the stent surface is reversed and the node separates from the stent in a mixed -mode fashion. Resistance to normal separation is significantly reduced due to the earlier initiation of tangential debonding, with peak normal traction of 𝑇𝑛𝜎𝑚𝑎𝑥 ≈0.17 ⁄ being computed ( Figure 10(e)). The insert in Figure 10(d) illustrates the final mixed -mode separation of the node from the stent surface. Further progression of coating delamination at higher stent deployment is shown in Figure 10(f) where coating delamination is computed all along the arch, including pure mode I debonding at the top of the arch (𝑥/𝑅=0). The insert in Figure 10(f) shows that the stent surface at the interface becomes highly non -uniform at finite deformation. Such non -uniform undulations on the surface of a plastically deforming stent surface underlie the complex patterns of interface traction and mixed -mode delamination illustrated in this study. 18 Figure 10: (a) Stent and coating von Mises (𝝈) stress distribution plotted on the deformed geometry at an arch deployment of 𝚺𝒂=𝟎.𝟎𝟗𝟖 (b) Mixed -mode coating compression computed at 𝒙/𝑹=𝟎.𝟔 (c) Normalised tangential traction ( 𝑻𝒕𝝉𝒎𝒂𝒙⁄ ) as a function of normalised tangential separation ( ∆𝒕𝜹𝒕⁄) computed at coating node (𝒙/𝑹=𝟎.𝟔) (d) Mixed -mode coating delamination computed at 𝒙/𝑹=𝟎.𝟔 and 𝚺𝒂=𝟎.𝟏𝟏𝟐 (e) Normalised normal traction ( 𝑻𝒏𝝈𝒎𝒂𝒙⁄ ) as a function of normalised normal separation (∆𝒏𝜹𝒏⁄) computed at coating node (𝒙/𝑹=𝟎.𝟔). (f) von Mises (𝝈) stress distributi on plotted on the deformed stent geometry at an arch deployment, 𝚺𝒂=𝟎.𝟑𝟎𝟓. Insert highlights non -uniform strut surface. An elastic coating (𝑬𝒄=𝟐𝟎𝟎 𝑴𝑷𝒂 ) is considered . Elastic -plastic properties are assigned for thestent strut with 𝑬𝒔=𝟐𝟎𝟎 𝑮𝑷𝒂, strain hardening modulus, 𝑬𝒉=𝟐𝑴𝑷𝒂 and yield stress, 𝝈𝒚=𝟐𝟎𝟎 𝑴𝑷𝒂 . 𝒉𝟏𝑹⁄= 𝟎.𝟐,𝑳/𝑹=𝟐, 𝒉𝟐𝑹⁄=𝟐. 𝝈𝒎𝒂𝒙 𝑬𝒄⁄ =𝝉𝒎𝒂𝒙 𝑬𝒄⁄ =𝟒.𝟖 𝐱𝟏𝟎−𝟓. Deformation factor=3 for (b) and (d). Corresponding results for thin (h_2/R=0.1) and thick (h_2/R=2) stent struts are presented in the Supplementary Material Sections (D) and (E) respectively. 4 Concluding Remarks Several medical papers have reported delamination of the coating from the stent -substrate following intravascular deployment leading to adverse outcomes for patients. However, the mechanisms of delamination of such polymer coatings from the surface of a stent due to large deformations during device deployment had not been investigated before. In this paper, a novel and in-depth investigat ion of the mechanisms and parameters that govern stent -coating delamination is performed, using a cohesive zone formulation to simulate the evolution of traction at the stent -coating interface. The study firstly analyses the behaviour of elastic coatings o n idealised elastic stent substrates. Such analysis of elastic stent substrates may be relevant to next -generation biodegradable polymer stent substrates (Lee and Hernandez, 2018) . Simulations reveal that the mode mixity of delamination is strongly dependent on the level of stent deployment at initiation . In general, p eak normal tractions exceed peak shear tractions at low levels of stent deployment whereas the reverse trend is computed at high levels of stent deployment . Interface tractions increase with both increasing stent thickness and coating thickness suggesting that thinner stents and thinner coatings should be utilised for the delivery of antiproliferative drugs in order to reduce the risk of coating delamination . Next, the influence of stent plasticity on interface tractions and coating delamination is investigated. Even a t low levels of deployment, plastic yielding occur s in the stent hinge region and the patterns of norma l and shear tractions are found to be significantly more complex than those computed for elastic stents , with both tensile and compressive regions of normal traction occurring in the stent arch . At a high level of stent deployment shear tractions at the st ent-coating interface are computed to increase with decreasing strain hardening modulus. This study represents a significant advancement on the study of Hopkins et al. (2010) where interface traction distributions were not examined and only mode I debondi ng initiation at the top of the arch was considered. This study also represents a significant advance on analytical solution for stent -coating interface tractions for infinitesimal linear elastic deformation developed by Parry & McGarry (2012). 19 The finding s of this paper provide a new insight into the stress -state at the stent -coating interface as a function of the stent design parameters and plastic deformation during deployment , allowing for a more reliable assessment of the limits relating to safe implan tation of coated stents. In this study the influence of geometrical parameters (strut length and stent thickness ) and material properties (yield stress and strain hardening modulus ) on the stress state at a stent - coating interface is investigated. The infl uence of finite deformation on interface tractions is examined, significantly advancing on the analytical analyses of Parry & McGarry (2012) which were limited to infinitesimal linear elastic deformation s. Most importantly, the present paper also considers the effect of stent plasticity on interface tractions and provides simulations of coating debonding. The current study demonstrates that the mechanisms of coating debonding are complex and depend on numerous stent , coating and interface design criteria. The full complexity of coating debonding for elastic and elastic -plastic stents is uncovered. Several experimental studies using scanning electron microscopy have reported extensive delamination of polymer coatings from the stent surface during deployment (Basalus and von Birgelen, 2010; Levy et al., 2009; Otsuka et al., 2007; Wiemer et al., 2010) . Numerous studies have suggested that coating damage may play a significant role in late stent thrombosis , following the use of d rug- eluting stents (Balakrishnan et al., 2005; Daemen et al., 2007; Hoffmann et al., 2002; Iakovou et al., 2005; Levy et al., 2009; Lüscher et al., 2007; Otsuka et al., 2007; Wang et al., 2002) . It is reasonable to suggest that some of the other clinical implications associated with coating damage may include micro -embolism formation, impaired drug de livery and obstruction of blood flow. Therefore, computational analyses and reliable design techniques are required in order to determine the limits relating to the safe implantation of coated stents. The analyses presented in this paper provide valuable insight into the stress -state at the stent -coating interface as a function of the stent design parameters. Such analyses allow for a more reliable assessment of the limits relating to safe implantation of coated stents. Future extensions of this work should include simulation of the stent microstructure using crystal plasticity formulations (Liu et al., 2018; McGarry et al., 2007; Shanthraj et al., 2015) and strain gradient plasticity (Gurt in, 2008; Qu et al., 2006) . Future implementations should consider nitinol shape memory alloy stent substrates (Arghavani et al., 2010; Grabe and Bruhns, 2009; Sedlák et al., 2012) . Simulations should also consider coating/stent due to cyclic loading due to cardiac motion and blood pressure (Han et al., 2020; McCarthy et al., 2014a, 2014b) . Non -linear hyper -viscoelastic behaviour should be considered for simulation of commercially used drug eluting polymers. Interaction between stent coatings and t he occluded artery wall should also be simulated, including endothelial cell denudation due to interaction with stent coatings ( Reynolds et al., 2014 ; McGarry and McHugh, 2008) . Analysis should also investigate the fracture of thin struts during deploy ment due to webbing coating defects (Hopkins et al., 2016) . Experimental characterisation should be performed to determine the stent -coating interface mode I, mode II and mixed mode fracture strength. The coating strength implemented in the simu lation of debonding from a thin strut stent design (𝜎𝑚𝑎𝑥 𝐸𝑠⁄ = 2x10−6) is similar to that measured by Hopkins et al., (2013) for a p olyurethane Chronoflex coating bonded to a stainless steel substrate ( 𝜎𝑚𝑎𝑥 𝐸𝑠⁄ =~1.5x10−6. However, the analyses above suggest that such an interface strength is insufficient for stent struts within the range of thicknesses used commercially. Key implications for the improv ed design of coated stents uncovered in this study are summarised below. An elastic stent is first considered in order to develop an initial fundamental understanding of interface traction behaviour. A number of important findings are revealed for an elas tic stent: 20 • Firstly, it is important to note that peak normal tractions exceed peak tangential tractions at low stent deployment whereas peak tangential tractions exceed peak normal tractions at high stent deployment (finite deformation) when interface tractions are assessed for a range of strut lengths, stent thicknesses and coating thicknesses. Since stents typically undergo high deployment levels, mode II coating debonding initiation at the bottom of the arch should be carefully considered. • The implementation of a longer stent strut effects an increased rotation at the base of the stent arch which reduces the tangential tractions and increases the normal tractions along the stent -coating interface. This effect can be counteracted by using a sh orter strut design. • It is shown that interface tractions increase with coating thickness suggesting that antiproliferative drugs should be delivered via a thinner coating in order to reduce the risk of coating debonding. Commercially available stent coati ngs range in thickness from 10–20𝜇𝑚 (Hopkins et al., 2010) . Since interface tractions increase with increasing coating thickness , a higher bond strength must be achieved when thicker coatings are utilised. • The influence of stent thickness on interface t ractions is particularly relevant given that clinical studies suggest that thinner stent struts lead to reduced rates of restenosis (Kastrati et al., 2001; Turco et al., 2008 ). Interface tractions increase with increasing stent thickness suggesting that th inner stents should be utilised to reduce the risk of coating delamination. Increased tangential tractions and negligible normal tractions are computed for a thin stent at high stent deployment due to excessive straightening of the thin stent geometry. In the second section of this study, an elastic -plastic stent is considered: • Yield stress is shown to significantly influence the location and magnitude of peak normal and tangential interface tractions. Additionally, a high yield stress (𝜎𝑦= 600 𝑀𝑃𝑎 )is shown to promote significant coating compression at high deployment. • Interface tractio ns are also presented for a range of stent strain hardening modul i. For a typical stent thickness (ℎ2ℎ1⁄ =5), the strain hardening modul ushas a significant influence on the computed interface tractions at high stent deployment. • For a thin elastic -plastic stent (ℎ2ℎ1⁄ =0.5), it is demonstrated that the prescribed ratio of peak tangential to peak normal traction significantly alters the mechanism and location of initial coating debonding. A highly non -uniform stent surface is computed at the stent -coating interface f or a thick elastic -plastic stent. In this case, mixed -mode coating over -closure is computed prior to mixed -mode coating delamination, highlighting the importance of implementing a cohesive zone formulation which can appropriately penalise mixed -mode compre ssion. 5 Acknowledgements Funding: The authors acknowledge funding from Science Foundation Ireland grant no. 18/ERCD/5481 6 Author Contributions E. O’M: Formal analysis; Methodology ; Writing JC: Writing; Formal analysis GP: Conceptualization; Formal analysis; Methodology; Software PMcG: Conceptualization; Formal analysis; Methodology; Writing; Software; Supervision 21 7