The current study presents the development and implementation of a bespoke experimental technique to generate and characterise mode II crack initiation and propagation in arterial tissue. The cur rent study begins with a demonstration that lap -shear testing of arterial tissue results in mixed mode fracture, rather than mode II. We perform a detailed computational design of a bespoke experimental method (which we refer to as a shear fracture ring te st (SFRT)) to robustly and repeatably generate mode II crack initiation and propagation in arteries. This method is based on generating a localised region of high shear adjacent to a cylindrical loading bar. Placement of a radial notch in this region of hi gh shear stress is pred icted to result in a kinking of the crack during a mode II initiation and propagation of the crack over a long distance in the circumferential (c)-direction along the circumferential -axial ( c-a) plane. Fabrication and experimental im plementation of the S FRT on excised ovine aorta specimens confirms that the bespoke test method results in pure mode II initiation and propagation. We demonstrate that the mode II fracture strength along the c-a plane is eight times higher than the corresp onding mode I strength determined from a standard peel test. We also calibrate the mode II fracture energy based on our measurement of crack propagation rates. The mechanisms of fracture uncovered in the current study , along with our quantification of mode II fracture properties have significant implications for current understanding of the biomechanical conditions underlying aortic dissection. Keywords: Soft tissue fracture; Arterial dissection ; Soft tissue anisotrop y; Finite element; Experimental fractur e mechanics. 1 Introduction Aortic dissection is a lethal disease which can lead to stroke, spinal cord infarction, renal failure, cardiac tamponade, aortic rupture, and death [1–3]. Mo rtality rates can be as high as 1% per hour after onset of dissection [1,4] . However, to date the exact pathophysiological processes and related biomechanisms underlying aortic dissection have not been uncovered. Previous investigations of the damage and fracture properties of arterial tissue have primarily relied on peel tests [5–9]. Such a test methodology results in a mode I fracture and allow s for the calibration of a mode I fracture strength (reported to be in the region of 140 kPa [7]). However , given the anisotropic mechanical behaviour of arterial tissue due to its complex microstructure, which consists of families of aligned collagen and elastin fibres, the mode and mechanism of crack propagation indu ced by a standard peel test is of limited relevance to the physiological process of artery dissection and rupture. Given that the high blood pressure in the lumen is the primary source of loading of arterial tissue in vivo, we argue that aortic dissection results from mode II fracture initiation, rather than mode I. Lumen pressure loading induces a high compressive radial component of stress throughout the artery wall, such that a mode I initiation of a dissection in the orthogonal circumferential -axial pla ne is highly improbable. Iatrogenic dissec tions may follow from mixed mode fracture initiation [10], but such dissections primarily result from catheterisation, percutaneous tra nsluminal coronary angioplasty, or cardiac surgery [11]. In any event, iatrogenic dissection s are rep orted to occur in the radial -axial plane [12], which, given the highly anisotropic arterial microstructure, is not at all reflective of the mode I fract ure in the circumferential -axial plane ind uced by a standard peel test. Every aortic dissection involves circumferential and axial crack propagation, whereas not all are associated with a radial crack propagation [1,11,13] . A recent experimental study qualitatively observe d a similar trend of dissection in the c ircumf erential -axial plane of excised aortas subjected to hypertensive levels of lumen fluid pressure [14]. Such di ssection propagation in the circumferential -axial plane is also reported for canine thoracic aortas, where it is qualitatively suggested that bond strength between lamellae are weaker than the lam ellae material strength [15]. 2 To gain further insight into the clinical observation of arterial dissection in the circumferential -axial plane it is essential to accurately characterise the fracture properties of arterial tissue under such mode II load ing condition s. Classical mode II lap -shear fracture mechanics experiments have been carried out by Witzenburg et al . [16] on porcine ascending aorta and by Sommer et al. [17] on excised human aneurysmatic and d issected aorta. However, we present a preliminary finite element analysis of lap -shear testing of arterial tissue in Se ction 1.1 (below) in which we demonstrate that the high levels of material deformation at the point of fractu re initiation consequently results in a mixed mode initia tion and propagation, rather tha n the intended mode II fracture. The generation of a mode II crack in highly deformable , high toughness fibrous soft tissue under controlled experimental conditions remains a significant challenge. This challenge cannot be a ddressed by standard test methodologies developed for traditional engineering materials, such as lap shear tests. A further challenge is that fracture testing of naturally occurring biological materi als such as arteries does not facilitate the engineering of standardised fracture specimen geometries. Therefore, implementation and interpretation of fracture testing subject to the limitation of non - standard geometries , such a s arteries , require in -depth mechanistic analysis. Cohesive zone models (CZMs) have been extensively used to describe an interface undergoing separation [18–21]. The calibration of CZM param eters requires experimen tal testing data [22–24]. The primary outputs of th ese tests are interface strength and fracture energy. It is extremely difficult to measure the stiffness of interfaces in highly elastic materials. The characteristic length ( 𝛿) or the modulus ( 𝐾) of the initial elastic region of some CZMs often dictates the fracture energy ( 𝐺) and vice -versa. This is common in exponential CZM formulations as seen in [19,20,25] . Therefore, the choice of fracture energy may anomalously influence the stiffness and characteristic length of the i nterface. Piecewise CZMs allow for the specification of 𝐾 (and by extension 𝛿) and G independently [26,27] . In the present study a piecewise CZM is developed to simulate the proposed experiment. In the current study we design, develop, and validate a bespoke test methodology for the initiation and propagation of mode II dissection in notched arterial ring spe cimens in the circumferential -axial plane. We demonstrate that this pattern of mode II dissection is highly repeatable. Such mode II fracture always preferentially occurs in our experiments, in contrast to mode I fracture in the radial -axial plane, which n ever occurs due to the significant toughening mechanism of collagen alignment at the crack -tip. Using our bespoke experimental methodology, in parallel with cohesive zone modelling of our experiments, we determine the pure mode II fracture strength and fracture energy of ovine aortas. We demonstrate that the mode II fracture strength is approximately eight times higher than the mode I fracture strength measured using a traditional peel test. Finally, we discuss the implications of our results in terms of ad vancing the current understanding of the biomechanics of artery dissection. 1.1 Lap-shear testing of arterial tissue results in a mixed mode crack propagation Due to the high strength and toughness of arterial tissue, significant material deformation occurs in lap-shear tests prior to damage initiation and crack propagation. This is clearly illustrated by the experiments of Sommer et al . [17] (referred to hereafter as constrained lap -shear testing ) and Witzenburg et al. [16] (referred to hereafter as unconstrained lap -shear testing ). In the constrained lap -shear tests the top and bott om surfaces of the artery sample are bonded to rigid horizontal plates. In the unconstrained lap -shear tests the top and bottom surfaces of the artery sample are unconstrained. In Figure 1 we present finite element cohesive zone an alysis of the constrained experiments of Sommer et al . [17] and the unconstrained experiments of Witzenburg e t al. [16] in which crack initiation and propagation is simulated using a cohesive zone formulation. The fracture s trength is calibrated so that predicted crack initiation occurs at an identical level of applied shear def ormation as reported experimentally. Mode mixity is defined as 𝜑𝑀𝑀=tan−1(𝑇𝑡/𝑇𝑛 ) and is presented as a function of normalised crack tip positio n, where 𝑇𝑡 and 𝑇𝑛 are the tangential (shear) and normal tractions, respectively, at the crack -tip. F ull details of the artery anisotropic hyperelastic material law [28] are presented in Appendix A: Artery material model, and the calibration of the material law to arterial tissue i s presented in Section 3.1. The cohesive zone fracture model developed in this study is presented in Appendix B: Cohesive zone fracture model description and provides an adva nce on previous coupled mixed mode formulations by facilitating indepen dent specification of mode - dependent fracture strength and fracture energy independent of intrinsic interface stiffness, while ensuring positive instantaneous incremental dissipation [29]. 3 As shown in Figure 1(I, III ), simulation of a constrained lap -shear test , predict s a mode mixity of ~ 0.2107 at fracture initiation, with a mean mode mixity of 𝜑𝑀𝑀=~0.1157 rad (~6.63°) during subsequent crack propagation. Remarkably, this represents a mode mixity that is closer to mode I, rather than the intended mode II fracture. Figure 1(III(c) ) illustrates the high level of material deformation at the crack -tip throughout, resulting in a localised mode of loading that resembles a peel test rather than a mode II fracture t est. Also shown in Figure 1, simulation of an unconstrained lap -shear test predicts a mode mixity of ~ 1.229 at fracture initiation, with a mean mode mixity of 𝜑𝑀𝑀=~1.236 rad (~70.85°) during subsequent crack propagation. Again, this is quite different from the intended mode II fracture ( 𝜑𝑀𝑀=π2⁄ rad). Figure 1(II(f)) demonstrates the high levels of material deformation at ini tiation and during propagation, resulting in normal tractions at the crack -tip that are comparable in magnitude to correspondi ng tangential tractions. Figure 1. (I) Mode mixity ( 𝝓𝑴𝑴) as a function of normalised crack tip po sition for a constrained in -plane lap shear test and an unconstrained lap -shear test ; (II) Simulation of an unconstrained lap -shear test where normal and tangential traction is shown . Cohesive zone parameters used were ( 𝝈𝒎𝒂𝒙=𝟐𝟎𝟎 𝒌𝑷𝒂,𝑮𝒏𝟎= 𝟎.𝟎𝟎𝟑𝑵𝒎𝒎⁄,𝝉𝒎𝒂𝒙=𝟒𝟎𝟎 𝒌𝑷𝒂,𝑮𝒕𝟎=𝟎.𝟎𝟎𝟑 𝑵/𝒎𝒎); (III) Simulation of a constrained lap -shear test. The traction at the crack tip is predominantly a mode I traction. Cohesive zone parameters used were ( 𝝈𝒎𝒂𝒙= 𝟐𝟎𝟎 𝒌𝑷𝒂,𝑮𝒏𝟎=𝟎.𝟎𝟎𝟑𝑵𝒎𝒎⁄,𝝉𝒎𝒂𝒙=𝟒𝟎𝟎 𝒌𝑷𝒂,𝑮𝒕𝟎=𝟎.𝟎𝟎𝟑 𝑵/𝒎𝒎). Full details of the artery anisotropic hyperelastic material law and th e cohesive zone fracture model are presented in Appendix A: Artery material model and Appendix B: Cohesive zone fracture model description , respect ively. As neither unconstrained nor constrained lap -shear test protocols result in mode II fracture, in the current study we aim to develop a bespoke test methodology to induce pure mode II fracture initiation and cra ck propagation in arterial tissue. We argue that the quantitative analysis of pure mode II fracture resistance of arteria l tissue is of critical importance . Lumen blood pressure is the primary source of mechanical loading on an artery wall in vivo, resulti ng in a compressive radial component of stress. The presence of an intramural haematoma could potentially result in a mix ed mode or mode I fracture initiation in the c -a plane. However, the high compressive radial component of stress in the vessel wall du e to lumen pressure will promote mode II initiation, particularly in the absence of an intramural hematoma. 4 Therefore, w e suggest that (in the absence of other conditions such as intramural haematoma) mode I initiation of fracture is not possible under in vivo loading and that arterial dissection results from mode II fracture of arterial tissue. 2 Design and computational val idation of a n experimental mode II fracture test (shear fracture ring test (SFRT)) In this section we outline the analysis and design o f a bespoke experimental methodology that results in pure mode II crack propagation in aortic tissue. Motivated by an ana lytical solution developed by Parry and McGarry [30] for the stress state of a bi -layered composite arch, we not e that interface tractions vary from mode I at the top of the arch to mode II at the side of the arch. Attachment of a straight bi -layered composite strut to the bi -layered arch (representing a typical stent design) further increases the shear stress at th e base of the arch [30]. Extending this analysis of the transition of interface from normal stress at the top of an arch to a shear stress at the side of the arch, we propose a test method in which a section (“ring”) of excised artery is mounted o n two cylindrical loading bars. The loading bars are then moved apart, imposing a deformation on the artery ring such t hat that two curved “arch - type” sections (top and bottom) are developed, connected by straight “strut -type” sections, as shown in Figure 2(a). This generates a localised region of high shear stress at the sides of the cylindrical loading bars (analogous to the maximum shear stress at the base of the stent arch uncovered in the study by McGarry and Parry [30]). Computational design of experiment A schematic of the computational design of experiment modelling procedure is outl ined in Figure 2(a). A ring of excised aortic tissue with radius 𝑅𝑠 is placed onto two steel bars of radius 𝑟𝑏. The bottom bar is fixed throughout the analysis while the top bar is displaced at a constant strain rate such that at any time point the distance travelled by the top bar is equal to 𝑢. We define a deformed interface coordinate 𝑆 (shown in Figure 2(a)) which describes the position along a quarter circumference of the deformed configurat ion. 𝑆 begins at the top of the bar and describes the position that is distance 𝑑 through the thickness 𝑡 and ends at 𝑆=𝐿. We also define a radial coordinate 𝑟𝑐𝑡 which describes the radial position of the crack tip through the wall. The original circumference of the ring is described as 𝐶0=2(𝐿0)+2𝜋(𝑟𝑏+𝑡/2) where 𝐿0 is the di stance from the midpoint of one loading bar to the other . The change in circumference is described as follows: Δ𝐶=Δ𝐿 where Δ𝐿 is the displacement applied to the upper loading bar. Also shown in Figure 2(a) is a local/material coordinate system for the artery ring; 𝑟,𝑐,𝑎 indicating axes corresponding to the physiological radial, circumferential, axial axes, respectively. Collag en fibres are assumed to lie in the 𝑐-𝑎 plane [31,32] . A measure of nominal st rain in the c-direction (the physiological circumferential direction) is then given as the 𝐸𝑐𝑐=Δ𝐶/𝐶0 where 𝐶0 is the undeformed circumference of the ring. Hereafter, for brevity, 𝐸𝑐𝑐 is referred to as the nominal circumferential strain imposed on the artery ring, and the bespoke test methodology is referred to as SFRT. A finite element analy sis of the SFRT is presented in Figure 2(b). Figure 2(b) shows a contour plot of circumferential stress 𝜎𝑐𝑐 (normalised by the initial shear mo dulus 𝜇) for the case of a loading bar of radius 𝑟𝑏𝑅𝑠⁄= 0.18. The circumferential stress is the dominant stress component in the artery ring; it is five times higher than the corresponding peak maximum shear stress 𝜏 (when 𝑟𝑏𝑅𝑠⁄=0.18) which occurs a t the side of the arch region, as shown in Figure 2(c). The magnitude of 𝜏 at the side of the bar is strongly influenced by the loading bar radius. Three bar radii are shown in Figure 2(c) (𝑟𝑏𝑅𝑠⁄=0.18,0.36,1.0). The peak value of 𝜏 is increased when the radius of the bar is decreased. In Figure 2(e) the distribution shear traction 𝑇𝑡 along the c-a plane mid -way through the arterial section (r=t/2) is examined. Similar to the distribution of 𝜏 in the ar terial material, 𝑇𝑡 along a c-a plane is highest at the side of the loading bar ( 𝑆=𝜋(𝑟𝑏+𝑑)/2 ) and increases with decreasing value of 𝑟𝑏. In order to determine the radial position with the highest levels of shear traction , the effect of the radial coordinate of a c-a plane throu gh the thickness was also examined, placing the plane a quarter -way through the thickness, such that the plane lies on 𝑆 when 𝑑=𝑡/4, results in a 7.3% increase in 𝑇𝑡 compared to 𝑑=𝑡/2. However, placing th e plane three quarters through the thickness, such that the plane lies on 𝑆 when 𝑑=3𝑡/4, results in a 44% decrease in 𝑇𝑡 compared to 𝑑=𝑡/2. The plane lies on S when 𝑑=𝑡/2 in Figure 2(d-f). 5 Figure 2(f) shows the normal traction s (i.e. in the r-direction actin g on a c-a plane) are negative (i.e. compressive) throughout the arch region and are negligible in the straight section of the specimen . In terms of experimental design of a mode II fracture test, the analyses presented in Figure 2 provide the following insights: • Maximum shear stress and shear traction on a c-a plane occur at the side of the loading bars ( 𝑆=𝜋(𝑟𝑏+ 𝑑)/2). • A smaller loading bar generates highest magnitude of shear stress and shear traction on a c-a plane. • Crack propagation along a c-a plane (as reported clinically) is expected to be pure mode II, based on computed compressive normal tra ctions throughout the arterial specimen. We therefore select a small loading bar radi us of 1.5 mm (corresponding to 𝑟𝑏/𝑅𝑠=0.18 for a n aorta specimen radius of 8.33 mm ). 6 Figure 2. (a) Schematic of the SFRT . An exci sed aortic ring specimen is shown mounted on two bars in the reference configuration. 𝑹𝒔 is the radius of the specimen and 𝒓𝒃 is the radius of the bar. The bottom bar is fixed and u is the displacement of the top bar. 𝑺 is the deformed interf ace coordinate which is a distance 𝒅 offset from the bar. A schematic of the local material coordinate system is shown with local circumferential, axial, and radial directions ( 𝒄-𝒂-𝒓); (b) Local circumferential stress is presented for a 𝒓𝒃𝑹𝒔⁄=𝟎.𝟏𝟖 configuration (𝚫𝑪/𝑪𝟎=𝟏.𝟎) positive values indicate tensile in the local circumferential direction ; (c) Maximum local shear stress ( 𝝉/𝝁) contour is plot for three bar radiuses (𝒓𝒃𝑹𝒔⁄=𝟎.𝟏𝟖,𝟎.𝟑𝟔,𝟏.𝟎); (d) Normalised inte rface shear traction ( 𝑻𝒕/𝝁) contour for the same radiuses as shown in ( c); Tangential traction (e) and normal traction ( f) as a function function of the normalised interface coordinate (𝑺/𝑳) for each of the bar radiuses analysed. Full details of th e artery anisot ropic hyperelastic material law used are presented in Appendix A: Artery material model. 7 In Figure 3, using a co hesive zone formulation (described in Appendix B: Cohesive zone fracture model description ), we examine the mode mixit y during crack initiation and propagation in our SFRT test. We next hypothesise that the insertion of a notch in the radial direction at the location of maximum shear stress (S=π(𝑟𝑏+d)/2), as illustrated in the schematic in Figure 3(a), will result in a kinking of the notch such that m ode II initiation and propagation will occur along the c -a plane through the notch -tip, as shown in the illustrations of Figure 3(a) (we define kinking as the growth of the radial notch directly in the circumferential direction wit h no prior radial crack growth). A cohesive surface is plac ed along the c-a plane through the notch -tip. Simulat ed crack initiation and propagation is presented in Figure 3. The evolution of 𝑇𝑡 along the c-a plane through the notch tip (r=t/2) is shown in Figure 3(b), in addition to the computed specimen deformation. Following initiation at the side of the loading bar (𝑆=𝜋(𝑟𝑏+𝑑)/2), at an applied circumfere ntial strain of Δ𝐶/𝐶0=0.64, the crack front propagates a long distance along the c-a plane. The delaminated portion of artery (i.e. the evolving crack flank) remains flat on the c-a plane, suggesting that the crack growth is pure mode II throughout. T his is confirmed by plotting the mode mixity 𝜑𝑀𝑀 at the crack -tip as a function of crack -tip position during propagation , as shown in Figure 3(c). Corresponding plots for lap-shear test simulations are shown once -again for co mparison, highlighting the significant improvement of the SFRT in generating mode II initiation and propagation in arterial ti ssue. The normal interface calibrated from standard peel tests (see Supplementary Material I: Peel tests) is 𝜎𝑚𝑎𝑥=202 𝑘𝑃𝑎. Figure 3. (a) Schematic of the proposed radial notch and the evolution throughout testing; ( b) Section views of the finite element simulation of the SFRT carried out in the present study. Sections are taken in th e middle of the aortic ring as the crack propagates . Crack tip is shown by the black arrow ; (c) Mode mixity (𝝓𝑴𝑴) as a function of normalised crack tip position for each of the shear experiments analysed in this study. Cohesive zone parameters are as follows: the normal interface strength is calibrated from peel tests, 𝝈𝒎𝒂𝒙=𝟐𝟎𝟐 𝒌𝑷𝒂 (see Supplementary Material I: Peel tests); the shear interface strength is assumed to be approximately twice the no rmal interface strength, 𝝉𝒎𝒂𝒙=𝟒𝟎𝟎 kPa; the mode I and mode II fracture energies are assumed to be 𝑮𝒏𝟎=𝟎.𝟎𝟎𝟑𝑵𝒎𝒎⁄ and 𝑮𝒕𝟎=𝟎.𝟎𝟎𝟑𝑵𝒎𝒎⁄ . Full details of the cohesive zone model are presented in Appendix B: Cohesive zone fracture model description . 8 Apparatus Following from the key computational design and analysis presented in Figures Figure 2Figure 3, a proposed experimental test rig is shown in Figure 4(c). Cantilevered stainless steel loading bars are att ached to the base -plate and the cr oss-head fixture of a mechanical test machine (Zwick/Roell Z005 ). Selection of loadin g bar radius is a key design consideration. While, as shown above, a smaller radius enhances the level of shear stress, bars must be larg e enough to support the forces imposed by the deforming arteri al specimen without exceeding the elastic limit of stainle ss steel . The force imposed by the deforming specimen is, of course, dependent on the chosen specimen width, 𝑤. Specimens are prepared such that 𝑤𝑡⁄≈4.14, so that specimens are sufficiently wide to approximate generalised plane strain conditions, but not so wide that significant anatomical variations are introduced in the a - direction (i.e. specimens are approximately cylindircal). For a typical specimen thickness of t= 2.35 mm , a width 𝑤=9.73 mm is chosen. Specimens are placed as close as possible to the cantilevered ends of the loading bars (while contacting only the loading bars). Preliminary testing (based on the simplified cantilev ered beam subject uniformly distibuted load shown in Figure 4(a)) suggests that a specimen exerts a uniformly distributed load of ~ 0.97 N/mm on the loading bars at ~ Δ𝐶𝐶0⁄≈0.6 prior to rupture. Figure 4(b) shows the stress in stainless steel loading bars, normalised by the yield stress of stainless steel (200 MPa). Clearly, a bar radius in the range 𝑟𝑏/𝑅𝑠≤0.135 (corresponding to 𝑟𝑏≤1.125 𝑚𝑚 based on a mean artery radius of 8.33 mm) doe s not provide an adequate factor of safety for the structural integrity of the loading bar. On the otherhand, large bar radii 𝑟𝑏/𝑅𝑠≥0.25 result in lower magnitudes of shear stress and, consequently, a reduced probability of mode II fracture. Ther efore a bar of radius 𝑟𝑏/𝑅𝑠=0.18 (i.e. 𝑟𝑏=1.5 𝑚𝑚) is chose n to provide a high level of shear stress while ensuring that the yield stress of the loading bar is not exceeded. This bar size also facilitates straight -forward positioning of the radial notch at the position of maximum shear to the side of the bar (𝑆=𝜋(𝑟𝑏+𝑑)/2). A high-resolution camera ( 60 fps, 1080p ) and digital microscope (10 fps, 1080p) are positioned in front of the sample to record specimen deformation and crack prop agation throughout each test. A photograph of the final test set -up is shown in Figure 4(d). 9 Figure 4. (a) Schematic of structural bending of loading bar during stretching of artery SFRT specimen; (b) Identification of optimal experimental design based on required loading bar flexural strength and generation of high shear stress in arterial tissue. The blue shaded region indicates a high loading -bar factor of safety (FOS) and low shear traction. The red region indicates a low loading -bar FOS and a high shear traction. The green region indicates the optimal experimental design whereby high shear tractions are generated in the artery while a sufficiently high loading -bar FOS is achieved; (c) Schematic of t he computationally designed experimental test rig. The section view (Section A -A) shows the front view of the specimen ; (d) Finalised test - rig design with mounted arterial SFRT specimen . Sample preparation The ascending aorta and aortic arch were excise d from 6 sheep sourced from a local abattoir (Brady’s Athenry, Galway, Ireland). Excess connective tissue was carefully removed from the external (adventitial) surface of the vessel. Notched and un -notched specimens were prepared through cutting cylindrica l sections out of the aorta to form circumferentially intact rings. Sample dimensions were taken using a digital Vernier callipers and a digital thickness gauge. Tissue samples were stored at 0 -4℃ in phosphate buffer solution prior to testing to ensure ade quate tissue hydration and preservation , all samples were tested within 72 hours of harvest . Unnotched intact ring extension tests (n=6) and notched ring extension tests (n=1 1) were carried out. Samples were mounted on a rigid bar to allow for constrained, precise notching. Two notches were introduced using a No. 10 scalpel blade at opposite ends of the sample. Notches were approximately a half of the tissue thickness in depth ( 𝑟𝑐𝑡≈𝑡/2). The boundary of the notch was marked with blue waterproof all -surface ink. 10 Protocol To determine the anisotropic hyperelastic properties and anisotropy of the material, three experiments were carried out: axial extension, circumferential extension, and unno tched intact ring extension. Uniaxial testing in two directions was carried out rather than biaxial testing due to the non -uniform strain states in biaxial samples [33]. Strips were excised from the ascending aorta and cut to uniform rectangular shapes in the axial and circumferent ial directions maintaining a mean aspect ratio of 𝐿:𝑤=3.8:1 (all mean specimen dimensions are presented in Table 1). The samples were gripped using Zwick pneumatic spe cimen jaws. Axial (n=10) and circumferential (n=6) extension tests were carried out individually with a crosshead speed of 10 mm/min. Each of the samples were tested to failure. The notched samples were rotated such that the notches were aligned with 𝑆=𝜋(𝑟𝑏+𝑑)/2. Table 1. Mean specimen dim ensions ± SD for each of the experiments. Reported values for SFRT length and unnotched intact ring extension length refer to the diameter of the specimen pre -test. Circumferential extension length is the initial grip to grip separation. *Peel test experim ent is prese nted in Supplementary Material I: Peel tests. Experiment Length or Radius (mm) Width (mm) Thickness (mm) Peel Tests* (n=7) 37.96 ± 6.59 12.77 ± 1.07 1.95 ± 0.5 Circumferential Extension (n=6) 19.06 ± 2.28 4.60 ± 1.17 1.39 ± 0.1 6 Axial Extension (n=10) 27.68 ± 4.35 7.89 ± 0.92 2.06 ± 0.33 Unnotched Intact Ring Extension (n=7) 8.85 ± 0.23 (𝑅𝑠) 5.15 ± 0.49 2.05 ± 0.14 SFRT (n=11) 9.09 ± 0.60 (𝑅𝑠) 9.73 ± 2.32 2.35 ± 0.16 3 Results 3.1 Determination of hyperelastic propertie s Prior to simulation of fracture experiments, the anisotropic hyperelastic behaviour of the arterial tissue must first be calibrated using uniaxial tension testing of the tissue in the circumferential and axial directions. Additional validation of the cal ibrated material model is performed by experimental and computational analysis of the response of unnotched artery sections to our SFRT . Experimental tensile testing reveals significant anisotropy and material non-linearity, as shown in Figure 5𝑎. The material exhibits a higher initial stiffness in the circumferential direction. Furthermore, in the circumferential direction the material transitions to a high stiffness regime at a strain of ~0.4. Strain stiffening is less pronounced in the axial direction. The model provides an accurate representation of the material anisotropy and strain stiffening both in the circumferential direction ( root mean square d error (𝑅𝑀𝑆𝐸)𝑐𝑖𝑟𝑐=0.0059) and in the axial direction (𝑅𝑀𝑆𝐸𝑎𝑥𝑖𝑎𝑙=0.0144) for the unique material parameter set presented in Table 2. Simulation of the s tretching of an unnotched intact ring using the calibrated material model parameters, is shown in Figure 5(b). R esults are in strong agreement with corresponding experim ental measurements for the entire range of applied deformation (𝑅𝑀𝑆𝐸𝑟𝑖𝑛𝑔=0.0081). Table 2. Table of best fit material parameters for the arterial material model. 𝑫𝟏𝒎 𝑫𝟐𝒎 𝑬𝟏𝒎 (MPa) 𝑬𝟐𝒎 (MPa) K (MPa) 𝑫𝟏𝒇 𝑫𝟐𝒇 𝑬𝟏𝒇 (MPa) 𝑬𝟐𝒇 (MPa) 𝛼(∘) 𝛽 0.15 0.9 0.075 0.5 1.52 0.4 0.5 0.16 4.5 28.5 15.0 11 Figure 5. (a) Nominal stress (MPa) as a funct ion of nominal strain for the uniaxial extension of the axial and circumferential datasets. Experimental data and numerical data are shown. The root mean squared error (RMSE) of the axial and cir cumferential data sets are 𝟎.𝟎𝟏𝟒𝟒 and 𝟎.𝟎𝟎𝟓𝟗 respec tively ; (b) Nominal stress (MPa) as a function of the circumferential strain for the unnotched intact ring extension tests. RMSE for the unnotched intact ring extension test is 𝟎.𝟎𝟎𝟖𝟏 . Final material parameters are presented in Table 2. Full details of the artery anisotropic hyperelastic material law are presented in Appendix A: Artery material model. 3.2 Fracture mechanics results Firstly , we introduce the results of our SFRT fracture mechanics investigation by presenting the de tails of a representative aorta test sample , Figure 6 shows the measured force ( 𝐹/𝑤𝑡) versus applied nominal circumferential strain (Δ𝐶/𝐶0). The measured force continues to increase beyond the point of crack initiation at Δ𝐶𝐶0⁄=0.72. This is due to the continued load bearing capacity of the outer section of the ring ( 𝑟>𝑟𝑐𝑡) up to the point of ultim ate rupture. Fracture initiation (shown in Figure 6(a)) appears to be a pure mode II initiation with no evidence of normal separation of the fracture surfaces at the cr ack-tip; i.e. fracture ini tiation entails an immediate kinking of the radial notch so that a mode II initiation along the c-a plane occurs . Following initiation, s ignificant crack propagation is observed in the c-direction through the c-a plane over a long distance (Figure 6(a-d)). This crack propagation appears to be close to pure mode II with no visible normal separation at the crack -tip. A relatively smoot h fracture surface is observed along the c-a plane and the crack flank (delaminated material) is obse rved to remain flat to the c-a plane throughout the test. As shown in Figure 6(b), at Δ𝐶𝐶0⁄=0.86, propagation appears to be a pure mode II in nature. This pattern of mode II propagation continues through Δ𝐶𝐶0⁄=0.93 (Figure 6(c)) up to, and including, the point of ultimate rupture ( Δ𝐶𝐶0⁄=0.97; Figure 6(d)). This observed pattern of mode II initiation and propagation in the c-direction along the c-a plane is remark ably similar to that predicted by experimental design calculations ( Figure 2). This exact pattern of mode II initiation and propagation was observed in all samples ( n=11) subjected to our bespoke SFRT test. In no case was mode I cr acking in the r-direction observed, despite the high levels of circumferential nominal strain applied to the specimen during our SFRT tests. This highlights the benefit of detailed computational analysis to uncover competing mechanism of material damage wh en designing biomechanical tests for biological materials. 12 Figure 6. The gr aph shows force normalised by the cross -sectional area , 𝑭/𝒘𝒕 (MPa) as a function of the circumferential strain ( 𝚫𝑪/𝑪𝟎). The point of crack ini tiation is marked with an X. Experimental results of the SFRT for a representative sample. (a -d) Progression of the crack tip from the point of initiation to ultimate rupture of specimen (S3). The original notch is seen in blue. Figure 7(a) shows the measured force ( 𝐹/𝑤𝑡) versus applied nominal circumferential strain ( Δ𝐶/𝐶0) for all S FRT samples ( n=11). Points of crack initiation are indicated for all cases. As stated above, mode II crack propagation in the 𝑐 direction through th e 𝑐-𝑎 plane is observed in a ll cases. Generally, fracture initiation is observed to occur in the transition between the initial low stiffness regime and the high stiffness regime . The range of nominal strain at initiation is observed as 0.53≤Δ𝐶𝑖/𝐶0≤0.82; mean±SD=0.64±0.23. This illustrates the quantitative repeatability of the SFRT methodology, despite high levels of inter -sample variability typically reported for mechanical be haviour of aortic tissue. Finally, in all samples ( n=11), crack initiation does not eliminate the lo ad bearing capacity of the specimen due to the intact outer layer of each specimen ( 𝑟>𝑟𝑐𝑡). However, superposition of the corresponding mean curve for the intact unnotched specimen s subjected to the same loading regime (Figure 7(a)) confirms that the presence of the notch , and consequent mode II crack propa gation, leads to a reduction in measured force throughout each test. Figure 7(b) shows measurements of the cracked plane as a function of the interface length (S). 𝑡̃ is the measurement of deviation from the original crack plane , it is defined as 𝑡(𝑆)/𝑡(𝑆𝑖) 13 where 𝑡(𝑆) is the thickness of the remaining intact wall at the position S, and 𝑡(𝑆𝑖) is the thickness of the remaining wall at the position where the cr ack initiated. Mean measured values and standard deviatio n is shown by the boxes and error bars, respectively. The dashed line indicates no deviation from the original crack plane (𝑡̃=1). A t-test reveals no statistically significant difference between any of the measured values (p=2.39e -9). Figure 7. (a) Force normalised by the cross -sectional area, 𝑭/𝒘𝒕 (MPa) as a function of the circumferential strain (𝚫𝑪/𝑪𝟎) for each of the radial notch test specimens. The points of crack initiation are marked with the symbol shown in t he legend and are denoted by 𝑺𝒏𝒄 where 𝒏 is the specimen number. The mean stress strain data of all the samples for the unnotched intact ring extension test is shown as the thick dotted black line. The notched samples exhibit a lower stiffness in th e high strain regime compared to the unnotched intact ring. (b) Measurements of the crack plane as a function of the interface length (S). 𝒕̃ is the deviation of the crack plane from the original crack plane. Figure 8(a) shows the deformed crack length as a function of applied deformation beyond the point of initiation ((Δ𝐶−𝐶𝑖)𝐶𝑖⁄). The mean final deformed crack length was 22.65 ±9.68 mm. The typical fracture pattern is characterised by a regime of slow crack growth post-initiation followed by a regime of rapid growth. One likely cause for the initial slow growth followed by the rapid growth is the experimentally observed fibrillation during early stages of crack growth . Extension and pull -out of fibres between fract ure surfaces will provide partial resistance to crack propagation. In 4 samples, a third regime of further slow crack growth subsequent to the fast growth regime is observed. 14 Figure 8. Crack length as a function of circumferenti al stretch past initiation. (𝑪−𝑪𝒊)/𝑪𝒊=𝟏 indicates the point of crack initiation and the final point on each curve is the point at which the specimen underwent final rupture. Images of the fracture surfaces are shown for eight samples in Figure 9. The original notch is marked with blue ink in each of the images. Images confirm that c rack propagation is restricted to the same c-a plane (through the original notch -tip) for the entirety of the test, up to the point of fina l rupture, again confirming the mechanism of mode II fracture predicted in our design of experiment calculations ( Figure 2 and Figure 11). Fibrillation is observed at the crack -tip at the end of SFRTs, as s hown Figure 9(h), in addition to the early stages of fracture propagation and initiation. 15 Figure 9. Fracture surfaces of the specimens. Original notches are shown with blue ink. Fibrils formed during fib rillation are shown in (h). 3.2.1 Computational analysis of SFRT experiments to determine mode II fracture properties Deformed finite element meshes during crack initiation and propagation are shown in Figure 10(a). In all simulations the mixed -mode CZM described in Appendix A: Artery material model is implemented. The value of the param eter 𝜎𝑚𝑎𝑥 of 202 𝑘𝑃𝑎 (the mode I fracture strength along the c-a plane) is determined from the implementation of standard peel test experime nts, described in Appendix B: Cohesive zone fracture model description . The sensitivity of SFRT crack initiation to the mode II fracture strength, 𝜏𝑚𝑎𝑥, is presented in Figure 10(b). 𝜏𝑚𝑎𝑥 values of 1.4 MPa and 1.8 MPa predict fracture initiation within the experimentally observed standard deviations (in terms of circumferential strain and measured force), and a value of 𝜏𝑚𝑎𝑥=1.6 MPa leads to the prediction at Δ𝐶𝐶0⁄=0.63 and 𝐹/𝑤𝑡=0.84 𝑀𝑃𝑎 , which is within 0.86% of th e experimentally measured mean values of Δ𝐶𝐶0⁄=0.63 and 𝐹/𝑤𝑡=0.84 𝑀𝑃𝑎 , respectively. This suggests that the mode II fracture strength for initiation along the c-a plane is eight times higher than the corresponding mode I strength along the c-a plane. It should be emphasised that this is not reflective of the mode I strength for fracture along the r-a plane, as demonstrated by the J-integral analysis presented below in Figure 11, which we expect to be con siderably higher than 𝜏𝑚𝑎𝑥 or 𝜎𝑚𝑎𝑥 along the c-a plane. Similar to experimental measurements, the computed force ( 𝐹/𝑤𝑡) 16 increases following fracture initia tion. Additionally, similar to experiments, computed forces for notched SFRT s are lower than corresponding curves without the presence of a notch/crack propagation. Figure 10. (a) Contour plot of max principal stress showing the progression of a mode II crack propagation , black arrows indicate the area of the crack tip ; (b) Sensitivi ty of the crack initiation to the mode II interface strength. Mean force normalised by the cross -sectional area as a function of circumferential strain (𝚫𝑪/𝑪𝟎) for the SFRT. Points of crack initiation are shown with an ‘X’. In each simulation the mode I interface strength was 𝝈=𝟐𝟎𝟐 𝒌𝑷𝒂 , the mode II fracture energy was 𝑮𝒕𝟎=𝟎.𝟎𝟎𝟏 𝑵/𝒎𝒎, and the mode I fracture energy was 𝑮𝒏𝟎=𝟎.𝟎𝟎𝟏 𝑵/𝒎𝒎. The mean circumferential strain (𝚫𝑪/𝑪𝟎) at initiation is shown by the vertical dotted black line and the standard error is the shown by the horizontal error bar. Full details of the artery anisotropic hyperelastic material law an d cohesive zone model are presented in Appendix A: Artery material model and Appendix B: Cohesive zone fracture model description , respectively. Figure 11 shows the evolution of the J1 and J2 components of the J integral at the c rack tip in addition to the fibre alignment evolution during the simulation. The mean circumferential strain at fracture is shown by the dotted line and the standard deviation is shown in the highlighted region. As s een in Figure 11, J1 is ≈8 times higher than J2 at the mean point of initiation. F racture initiation occurs in the circumferential direction only when fibres are almost fully aligned in the c-a plane. This suggests that the 𝐽𝑐𝑟𝑖𝑡 is at least 8 ti mes higher for crack propagation in the radial direction than the circumferential direction . The experiment generates a sufficient ly high J2 component of th e J- integral at the crack tip to result in circumferential propagation , while fibre alignment result s in a significant toughening mechanism against radial crack initiation so that 𝐽1<𝐽1𝑐𝑟𝑖𝑡 and propagation does not occur in the radial direction . In arterial tissue, collagen fibres lie along c-a planes, as structural inv estigations indicate this depends on age, pathology, and type of artery [34–36]. Therefore, collagen alignment due to increased circumferential strain acts as a toughening mechanism only for fractu re in the radial direction (through the aligned fibres). Crack propagation in the circumferential direction does not benefit from the toughening mecha nism of fibre alignment (essentially cracks propagate between the collagen). 17 Figure 11. (left y -axis): J -Integral calculations for fracture paths in the radial and circumferential directions as a function of circumferential strain in a SFRT. (right y -axis): Fibre angle as a function of circumferential strain. Fibres are almost fully aligned in the c-direction at a circumferential strain of 0.6. A non -dimensional fracture resistance is presented as 𝑱/(𝝓̅𝒂𝒗𝒈𝑹𝒄) where 𝝓̅𝒂𝒗𝒈 is the volume averaged strain energy density in the arterial tissue within a radius 𝑹𝒄 from the notch -tip. Note that 𝝓̅𝒂𝒗𝒈𝑹𝒄 is identical for both crack paths so the figure presents a direct c omparison of the J -integrals. Full details of the artery anisotropic hyperelastic material law used are presented in Appendix A: Artery material model. The effect of notch depth on the J integral is further examined in Table 3. Table 3. Influence of the depth of the radial notch on the J -integral at a circumferential strain of 𝚫𝑪 𝑪𝟎=𝟎.𝟖 Notch Depth J1 (N/mm) J2 (N/mm) 3t/4 0.34 0.0296 t/2 0.25 0.0242 t/4 0.18 0.00789 While t he point of fracture initiation is highly sensitive to the parameter 𝜏𝑚𝑎𝑥, it is not found to be sensitive to the mode II fracture ener gy (𝐺𝑡0). The mode II fracture energy is defined as the area under the tangential traction - separation curve. However, as shown in Figure 12, simulations reveal that the rate of crack propagation exhibits sensitivity to 𝐺𝑡0. Simulations reveal that a value of 𝐺𝑡0=0.25 N/mm results in a rate of crack propagation that is slower than that observed experimentally. On th e other end of the spect rum, values of 𝐺𝑡0≥0.0005 N/mm result in excessively high rates of propa gation compared to experimental measurements. A value of 𝐺𝑡0=0.005 N/mm 18 is predicted to provide a reasonable approximation of experimental measurement, both during early and later stages of crack initiation. Figure 12. Simulated crack length as a function of circumferential stretch past initiation presented alongside the experimental data (shown in light grey). The crack length and velocity are shown to depend on the mode II fracture energy. Cohesive zone parameters are 𝝈𝒎𝒂𝒙=𝟐𝟎𝟐 𝒌𝑷𝒂,𝝉𝒎𝒂𝒙=𝟏.𝟔 𝑴𝑷𝒂, the mode I fracture energy is scaled with the mode II fracture energy according to 𝑮𝒏𝟎=(𝝈𝒎𝒂𝒙𝝉𝒎𝒂𝒙⁄)𝑮𝒕𝟎. Full details of the artery anisotropic hyperelastic material law and cohe sive zone model are p resented in Appendix A: Artery material model and Appendix B: Cohesive zone fracture model description , respectively. 4 Discussion The current study presents the development and implementation o f a bespoke experimental technique (SFRT) to generate and characterise mode II crack initiation and propagation in arterial tissue. A mechanistic consideration of a large body of clinical studies suggest that artery dissection results from mode II initiati on and crack propagation along a c-a plane in the artery wall, rather than mode I propagation. However, the majority of artery fracture experiments rely on mode I peel test. Recently the studies of Sommer et al. [17] and Witzenburg et al. [16] adopt standard lap -shear test techniques to measure mode II fracture properties. The current study begins with a demonstration that lap -shear testing of arter ial tissue results in mixed mode fracture, rather than mode II due to the high levels of tissue deformation at the crack -tip at the point of initiation. The combination of high toughness and high deformability of arterial tissue presents a considerable challenge in generating mode II fracture. A further challenge, albeit an obvious one, is that fracture testing of naturally occurring biological materials such as arteries does not facilitate the engineering of standardised fracture specimen geometries. Therefore, implementation and interpretation of fracture testing subject to the limitation of non -standard geometries, such as arteries, require in - depth mechanistic analysis. In the current study we perform a detailed computational design of a bespoke experimental method (which we refer to as a shear fracture ring test (SFRT)) to robustly and repeatably generate mode II crack initiation and propagation in the c-a plane of arteries. This method is based on generating a localised region of high s hear adjacent to a cylindrical loading bar. P lacement of a radial notch in this region of high shear stress is predicted to result in a kinking of the crack during a mode II initiation and propagation of the crack over a long distance in the c-direction al ong the c-a plane. A J -integral analysis sugg ests radial mode I crack propagat ion will not occur due to significant fracture toughening as a result of collagen alignment in the c-direction behind the crack -tip. The J -Integral was first proposed by Rice [37]. A geometrically nonlinear generalised form of the J - integral was proposed by Maugin et al. [38]. The J -Integral is calculated according to the virtual crack extension/domain integral method s as proposed by [39,40] . Geometric nonlinearity is accounted for using large 19 strain formulations in the commercial software Abaqus. The J -Integral has been traditionally used in engineered materials, however, recently it has also been used in hyperelastic soft tissues [41]. Fabrication and experimental implementation of the SFRT on excised ovine aorta specimens confirms that the test method results in pure mode II initiation and propagation, as predicted by our computational design of experiment. Using a cohesive zone formulation, we simulate our mode II experimental tests, and we demonstrate that the mode II fracture strength along the c-a plane is eight times hig her than the cor responding mode I strength determined from a standard peel test. We also calibrate the mode II fracture energy based on our measurement of crack propagation rates. The mechanisms of fracture uncovered in the current study, along with our qu antification of mode II fracture properties have significant implications current understanding of the biomechanical conditions underlying in vivo aortic dissection. The present study provides an explanation as to why every dissection is associated with c ircumferential a nd axial crack propagation, but not always radial (i.e. the majority of aortic dissections do not rupture [11,13] ). Aortic rupture requires ra dial crack propagation , and collagen fibre alignment provides high levels of toughening against radial fracture. The results of this study suggest radial crack propagation is only likely to occur between collagen fibres or in areas where there is less fibr e toughenin g. The majority of aortic dissection -induced ruptures occur in the ascending aorta leading to cardiac tamponade and death [11,42] . This may be due to the discontinuous and irregular collagen structure observed in the wall of ascending aortic dis sections [36,43] . Such irregular and discontinuous structure reduces the radial toughening mechanism in the aortic wall . This is also a possible mechanistic explanation for the high incidence of aortic dissection amongst patients with connective tissue diseases such as Marfan syndrome and Ehler -Danlos syndrome [44–47]. Connect ive tissue disorders significantly alter the intramural and inter -lamellar aortic microstructure and strength through abnormal fibrillin (Marfan), and collagen deterioration (Ehler -Danlos). This puts patients with connective tissue disorders at higher risk of aortic dissection [48,49] . Previous experimental studies have observed mo de II crack propagation in arterial tissue under uniaxial tension [50] and under radial expansion [14,51] . Figure 13(a,b) shows images (adapted from Helfenstein -Didier et al. [50]) of uniaxial tensile testing of arterial tissue and spo radic, uncontrolled mode II cracks initiating and propagating in the circumferential direction. Figure 13(b) shows a histological section of a sample that underwent spo radic, uncontrolled mode II propagation in the c ircumferential direction. The results of these experiments highlight the toughening mechanism against radial crack propagation due to the alignment of collagen fibres in the circumferential direction. Figure 13(c) (adapted from Has lach et al. [51]) shows a histological image of a radially notched arterial media subjected to radial expansion. Kinking of the radial notch is observed, and pure mode II crack growth is observed in the circumferential direction. 20 Figure 13. (a-b) Experimental images adapted from (Helfenstein -Didier et al. [50]). (a) Segments of arterial media subjected to uniaxial tension result in sporadic uncontrolled mode II crack propagation . (b) Histological image showing mode II fracture of the arterial media in the circumferential direction. (c) Histological image (adapted from Haslach et al. [51]) showing kinking of a radial crack to the circumferential direction . The propagation of a pure mode II crack following insertion of a notch is suggestive that the intimal tear or initiation tear acts in a similar fashion in vivo to the notch [2]. The notch acts as a concentrator of stress at the notch tip and causes shear stress localisations [52]. The notch also significantly reduces the strain in the inner layer of the ar tery relative to the outer layer which results in a shear concentration. The results of the present experiment se rve as supporting evidence to the theory that aortic dissection is far more likely to occur in the presence of an intimal tear [53,54] . This is highlighted further by the additional testing carried out using c ircumferential notches (see Appendix C: Circumferential notch tests). Only 3 of the 8 samples tested with circumferential notches resulted in mode II crack propagation, indicating crack growth in the aorta is much more likely to occur in the presence of a radi al tear or notch. Although initial propagation is predicted to be pure mode II, it is likely that in the presence of a propagating patent false lumen the crack advances in a mixed mode given significantly lower mode I strength in the c -a plane. The cohesiv e zone model used in the present study allows for the accurate calibration of a mode II interface strength and mode II fracture energy. However , it is not capable of capturing the permanent nature of the fracture, this is reflected in the predicted crack l engths which are linear and do not sufficiently capture the two crack growth regimes (fibrillation and rapid crack growth). Future studies coul d consider the development of microstructural models to examine the role of interstitial fluid and species transp ort on fracture initiation. The cohesive zone parameters calculated in the present study should be applied to patient -specific geometries to as sess the risk of dissection. They should also be used to examine a range of biomechanical factors which may influ ence arterial dissection. A companion study to the present study is currently underway examining the application of these CZM parameters to an MRI derived subject -specific aortic geometry to analyse the risk of aortic dissection. Simulations in the present study ignore the viscoelastic behaviour of arterial tissue. The influence of the viscoelastic behaviour is explored in Appendix D: Viscoelasticity . We demonstrate that the inclusion of viscoelasticity does not significantly alter the results of the fracture analyses presented in the current study. 21 The test methodology ( SFRT ) developed in the present study could readily be applied to larg e veins such as the pulmonary veins or other large arteries (left subclavian artery, left common carotid artery, brachiocephalic artery, iliac arteries, renal arteries). The apparatus could be scaled to work on coronary arteries, femoral arteries, carotid arteries, and other arteries of a similar size). Fu rthermore, it could potentially be used with any other tubular soft tissue given a similar arrangement of fibres behind the crack tip resulting in toughing in the mode I (radial) direction . Acknowledgemen ts: The authors acknowledge funding from grant 16/US -C2C/3291 . The authors thank Behrooz Fereidoonezhad and Jamie Concannon for assistance and helpful discussions. Appendix A: Artery material model In order to simulate the anisotropic non -linear hyperela stic behaviour of arterial tissue we use a rece ntly proposed formulation by Fereidoonnezhad and McGarry [28]. Full details are presented in Fereidoonnezhad et al. [28], including extensive demonstr ation that this new formulation robustly avoids the computation of non -physiological auxetic behaviour during large deformation of anisotropic soft tissue, in contrast to established formulations. Here we brie fly present the key features of the model and w e demonstrate that it accurately captures the mechanical behaviour of the ovine arterial specimens used in this study. The anisotropic strain energy density function due to the presence of two fibre families i s given as Ψ𝑎𝑛𝑖𝑠𝑜=∑Ψ𝑓𝑖 , 𝑖=1,2 where Ψ𝑓𝑖= { 𝐸1𝑓(2 3𝜆𝑓𝑖3−𝜆𝑓𝑖2+1 3), 𝜆𝑓𝑖−1≤𝐷1𝑓 2 3𝜆𝑓𝑖3(𝑞−2𝑝)+𝑝 2𝜆𝑓𝑖4+𝜆𝑓𝑖2(𝑝−𝑞+𝑟)+𝜓01, 𝐷1𝑓<𝜆𝑓𝑖−1<𝐷2𝑓 2𝐸2𝑓 3𝜆𝑓𝑖3+𝜆𝑓𝑖2(𝑝𝐷2𝑓2+𝑞𝐷2𝑓+𝑟−𝐸2𝑓−𝐸2𝑓𝐷2𝑓)+𝜓02 ,𝜆𝑓𝑖−1≥𝐷2𝑓 (A1) 𝜆𝑓𝑖 is the fibre stretch of the 𝑖𝑡ℎ fibre family, 𝐷1𝑓 and 𝐷2𝑓 are material parameters, and 𝑝, 𝑞 and 𝑟 are given as: 𝑝=𝐸1𝑓−𝐸2𝑓 2(𝐷1𝑓−𝐷2𝑓),𝑞=𝐸1𝑓−2𝐷1𝑓𝑝,𝑟=(𝐸1𝑓−𝑞)𝐷1𝑓−𝑝𝐷1𝑓2, (A2) in which 𝐸1𝑓 and 𝐸2𝑓 are material parameters. The isochoric strain energy density function is given as 𝜓𝑖𝑠𝑜(𝜆̅1,𝜆̅2,𝜆̅3)=∑𝜓̅(𝜆̅𝑖)3 𝑖=1, 𝜓̅(𝜆̅𝑖)= { 𝐸1(𝜆̅𝑖−ln𝜆̅𝑖−1) |𝜆̅𝑖−1|≤𝐷1 𝑝(𝜆̅𝑖2 2−2𝜆̅𝑖+ln𝜆̅𝑖)+𝑞(𝜆̅𝑖−ln𝜆̅𝑖)+𝑟ln𝜆̅𝑖+𝜓01𝐷1<|𝜆̅𝑖−1|<𝐷2 𝐸2(𝜆̅𝑖−(1+𝐷2)ln𝜆̅𝑖)+(𝑝𝐷22+𝑞𝐷2+𝑟)ln𝜆̅𝑖+𝜓02|𝜆̅𝑖−1|≥𝐷2 (A3) where 𝐷1,𝐷2,𝐸1,and 𝐸2 are material parameters, 𝜆̅𝑖 (𝑖=1,2,3) are the isochoric principal stretches, 𝐽 =𝜆1𝜆2𝜆3 is the jacobian , and 𝜓01 and 𝜓02 are two constants which ensure the continuity of strain energy. Moreover 𝑝,𝑞, and 𝑟 are not independent parameters; in order to maintain C0 and C1 continuity the following relations must be enforced: 22 𝑝=𝐸1−𝐸2 2(𝐷1−𝐷2), 𝑞=𝐸1−2𝐷1𝑝, 𝑟=(𝐸1−𝑞)𝐷1−𝑝𝐷12 (A4) The volumetric strain energy densit y function is given as Ψ𝑣𝑜𝑙=𝐾 2{𝐽−1}2 (A5) A value of 𝜅=1.52 MPa is used resulting in slight material compressibility is based o n the experiments of Nolan and McGarry [55]. For further informati on regarding the material model the reader is directed to Fereidoonnezhad et al. [28]. A user defined material subroutine (UMAT) has been developed for this formulation so that it can be implemented in t he Abaqus finite element solver. Appendix B: Cohesive zone fracture model description To model the delaminatio n at the interface we implement a fully coupled mixed -mode cohesive zone model. For a given interface displacement vector 𝚫 with normal and tangential (shear) components Δ𝑛 and Δ𝑡, respectively, the corresponding displacement mag nitude is given as Δ𝑚=(Δ𝑛2+Δ𝑡2)1/2 and the mode angle is given as 𝜑= tan−1(Δ𝑡Δ𝑛⁄). The magnitude of the interface traction is expressed as a function of Δ𝑚 and 𝜑 by the following formulation: 𝑇𝑚(𝛥𝑚,𝜑)={𝐾𝑚Δ𝑚 ,Δ𝑚<T𝑚𝑚𝑎𝑥(𝜑)/𝐾𝑚 𝐾𝑚T𝑚𝑚𝑎𝑥(𝜑)/𝐾𝑚Ψ(𝜑) ,Δ𝑚≥T𝑚𝑚𝑎𝑥(𝜑)/𝐾𝑚 (B1) We refer to Ψ as the integrity of the interface (i.e. Ψ=(1−𝐷), where D is referred to as the interface damage). Ψ monotonically decreases from 1 to 0 with increasing interface separation, such that Ψ(𝜑)=exp(−𝛥𝑚𝑚𝑎𝑥−T𝑚𝑚𝑎𝑥(𝜑)/𝐾𝑚 𝛿𝑚∗(𝜑))Δ𝑚 Δ𝑚𝑚𝑎𝑥 (B2) where 𝐾𝑚 is the intrinsic elastic stiffness of the interface. 𝐾𝑚 is assumed to be mode -independent. T𝑚𝑚𝑎𝑥(𝜑) is the specified mode -dependent interface strength; the corresponding displacement (𝛿𝑚𝑒𝑙(𝜑)=T𝑚𝑚𝑎𝑥(𝜑)/𝐾𝑚) represents the elastic limit of interface separation at the point of dama ge initiation. The mode -dependent parameter 𝛿𝑚∗(𝜑) governs the rate of softening in the damage region ( Δ𝑚≥T𝑚𝑚𝑎𝑥(𝜑)/𝐾𝑚). The interface strength is defined as a function of the mode as follows: T𝑚𝑚𝑎𝑥(𝜑)=𝜏𝑚𝑎𝑥− ( 𝜏𝑚𝑎𝑥−𝜎𝑚𝑎𝑥 1−exp(−𝜋 2 Ω𝑇) ) (1−exp(−𝜑 Ω𝑇)) (B3) 23 where 𝜏𝑚𝑎𝑥 is the mode II interface strength, Ω𝑇 sets the non -linearity of the transition from mode II to mode I , and 𝜎𝑚𝑎𝑥 is the mode I interface strength. The mode mixity of the initial interface damage parameter 𝛿𝑚∗(𝜑) is obtained from 𝛿𝑚∗(𝜑)=𝐺𝑚(𝜑) T𝑚𝑚𝑎𝑥(𝜑)−T𝑚𝑚𝑎𝑥(𝜑) 2𝐾𝑚 (B4) where 𝐺𝑚(𝜑) is the mode -dependent fracture energy 𝐺𝑚(𝜑)=1 2𝐾𝑚𝛿𝑚𝑒𝑙(𝜑)2+∫ T𝑚𝑚𝑎𝑥(𝜑)exp(−𝛥𝑚𝑚𝑎𝑥−T𝑚𝑚𝑎𝑥(𝜑)/𝐾𝑚 𝛿𝑚∗ )𝑑Δ𝑚∞ T𝑚𝑚𝑎𝑥(𝜑)/𝐾𝑚 (B5) We m ay specify the mode -dependence of 𝐺𝑚(𝜑) using the following function: 𝐺𝑚(𝜑)=𝐺𝑡0− ( 𝐺𝑡0−𝐺𝑛0 1−exp(−𝜋 2 Ω𝐺) ) (1−exp(−𝜑 Ω𝐺)) (B6) where 𝐺𝑡0 is the mode II fracture energy and 𝐺𝑛0 is the mode I fracture energy, Ω𝐺 sets the non -linearity of the transitio n from mode II to mode I . Finally, we complete the description of the cohesive zone formulation by decomposing 𝑇𝑚 into the normal and tangential components, 𝑇𝑛 and 𝑇𝑡, respecti vely, such that 𝑇𝑛={𝐾𝑛𝑜𝑐Δ𝑛,Δ𝑛<0 𝑇𝑚sin(𝜑),Δ𝑛≥0 (B7) 𝑇𝑡=𝑇𝑚cos (𝜑) (B8) where 𝐾𝑛𝑜𝑐 is the overclosure penalty stiffness. Figure B1. (a) Traction -separation r esponse of the CZM to a mode II s eparation , 𝜹𝒕𝒆𝒍 is the maximum mode 24 II separation allowable prior to onset of damage, 𝑮𝒕𝟎 is the area under the traction separation curve ; (b) Illustration of the cohesive zone model mixed mode response. Mixed mode response of fracture energ y (𝑮), and critical interf ace strengths (𝝉,𝝈), and intrinsic elastic energy (𝟏𝟐⁄𝑲𝒎𝜹𝒎𝒆𝒍𝟐) is portrayed. This formulation represents a significant improvement on previous cohesive zone models in which intrinsic stiffness is dependent on specified fracture energy [56]. Additionally, the requirement of positive instantaneous incremental dissipation [29] is satisfied for all proportional and non -proportional loading paths. Appendix C: Circumferential notch tests Circumferen tial notch tests (n=8) were also performed. Notches ( ≈2.1𝑚𝑚) were introduced through puncturing the aortic media through the entire specimen width. Notches were then grown manually to ensure a sharp crack tip was present. Notched samples were mounted ont o the loading bars as described previously in Figure 4. The extent of the notch was marked with ink prior to testing to allow for accurate measurement of crack growth. Crack growth was also measured after the experiment. Figure C1. (a) Image of the circumferential notch pre -testing . Circumferential notch is shown measuring approximately ~2 mm ; (b) Image of the specimen after testing and prior to ultimate failur e. Significant crack growth is observed ; (c) Image showing the extent of the circumferential crack growth. Dissected flap is peeled back to reveal the extent of fracture ; (d) Finite element simulation of SFRT with a circumferential notch. Extensive crack g rowth is observed in the c-direction in the c-a plane. The black arrow indicates the position of the crack tip. Crack growt h was observed in 3 specimens out of the 8 tested. Average crack length was 11.79mm +/− 4.1mm. Aortic dissection is most commonly ass ociated with an intimal tear or notch [53,57 –59], however it can and does occur (although less commonly) without a visible entry tear in certain cases [60–63]. This is reflected in the results of our study, all samples tested with a radial notch exhibited extensive mode II crack growth whereas only 3 out of 8 samples with a circumferential notch exhibited mode II crack growth. The mode mi xity for the circumferential notch test is also explored in Figure C2. It shows the same as Figure 3(c) with the addition of the circumferential notch test. The mean mode mixity for the circumferential notch test is 89.18∘. 25 Figure C2. Mode mixity 𝝋𝑴𝑴 as a function of normalised crack ti p position for the SFRT, circumferential notch test, unconstrained lap -shear, and constrained lap -shear. The mean mode mixity in the circumferential notch test is 𝟖𝟗.𝟏𝟖∘. 26 Appendix D: Viscoelasticity In this appendix viscoelastic material behaviour is considered, and the influence of including viscoelasticity on the calibration of the material model to the experimental F/wt curves is also explored. A two term Prony seri es is calculated (Figure D1 (a)) according to prev iously published stress -relaxation data from Craiem et al. [64]. Figure D1(a) shows the Prony series fit to the stress -relaxation data for human arterial tissue. Figure D1(b) shows the force over area curves as a function of the applied circumferential strain for three different strain rates, the mean experimental data of the SFRT is also presented. The experimental strain rate (ΔĊ 𝐶0=6e-3/s) is lower than the phys iological strain rate of the aorta. The experimental circumferential strain rate was chosen in order to allow for adequate frame rate capture during recording of the experiment as higher circumferential strain rates cause significant difficulty in prec isely identifying the point of fracture initiation . Lumen area vs. time data from dual - VENC MRI of the aorta reveal a physiological circumferential strain rate of ΔĊ 𝐶0≈760e -3/s [65], in Figure D1 (b) we present a similar physiol ogical strain rate of ΔĊ 𝐶0=800e -3/s. The value of F/wt for the chosen experimental strain rate (ΔĊ 𝐶0=6e-3) is only 2% lower than that of th e higher physiological strain rate (ΔĊ 𝐶0 =800e -3) indicating that the effect is not significant. Figure D1. (a) Stress relaxation data and computational model prediction of a two term Prony series to data. The expression for the normalized rela xation shear modulus (𝑮𝑹(𝒕) 𝑮𝟎) is also presented. (b) influence of physiological circumferential strain rate (𝚫𝐂̇ 𝑪𝟎=800e -3/s), experimental circumferential strain rate (𝚫𝐂̇ 𝑪𝟎=6e- 3/s), and slow circumferential strain rate (𝚫𝐂̇ 𝑪𝟎=0.6e-3/s) on the force/area. The mean experimental curve is also shown. Prony series parameters are as follows: 𝒈𝟏= 0.0256106, 𝝉𝟏= 0.27891 , 𝒈𝟐= 0.12103, 𝝉𝟐= 318.35 Appendix E: The influence of friction on shear tracti on In this appendix the influence of the coefficient of friction between the loading bar and the artery is explored. Figure E1 shows the influence of the friction coefficient on the tangential traction at the crack tip. The influence of the friction coeffi cient on the tangential traction is shown to be minimal with only a ~5% increase in traction for a large friction coefficient of 𝜇=0.25 compared to frictionless contact. 27 Figure E1. Tangential traction at the crack tip as a function of the applied circumf erential strain. 5 friction coefficients are shown. Supplementary Material I: Peel tests In this supplementary material we pr esent the methods and results of our standard peel tests carried out on ovine aortic tissue. Methods To characterise the mode I strength of the tissue, peel tests (n=7) were performed. Tubular sections of aorta were cut along the axial length of the artery resulting in planar sheet specimens. Using a scalpel with a No.10 blade, a notch was introduced at one of the boundary faces [6,9] such that the notch was in the circumferential plane. The notch was then manually grown approxima tely 2mm to ensure a sharp crack tip. This created two layers (~8 -10mm) to be gripped in sandpaper and inserted into the grips of the test machine. Due to the well documented nature of the relatively low levels of anisotropy associated with peel tests and large standard deviation [6,9,66] peel tests (n=7) were conducted in the circumferential direction. The tongues of prepared samples were gr ipped on either side with sandpaper and placed into the grips of the test machine. A controlled displacement rate of 10.0mm/min was applied to the sample and the resultant reaction force was recorded. The zero level of the force was defined as the force af ter total separation had occurred. Results Experimental curve 28 Figure S1a shows force/width (mN/mm) as a function of the applied displacement (mm). The mean curve is the thick black curve and each of the individual samples are shown in light grey. The mean force/width is 51.4945 mN/mm, it is shown in dashed red in the figure. Figure S1a. Force/Width (mN/mm) as a function of the displacement (mm) for the peel test specimens. The mean Force/Width is 51.4945 mN/mm and is denoted by the red dotted line. Computational fit All the processes of the experiment are simulated to calibrate our mode I critical interface strength to the peel test data. The sensitivity of the system to variations in the critical interface strength ( 𝜎𝑚𝑎𝑥) is investigated in Figure S1b. No sensitivity of the Force/width to the mode II fracture energy ( 𝐺𝑛0) is found. This finding is in agreement with Ferrara and Pandolfi (2010) , who found no sensitivity to mode I fracture en ergy in a numerical study of mode I peel tests of arterial tissue. The sensitivity of t he force/width to the critical interface strength is explored in Figure S1b where peel tests of various interface strengths are computed. A median filter is applied to t he computed curves to eliminate noise caused by mesh. The mode I critical interface str ength of ovine ascending aorta is calculated as 𝜎𝑚𝑎𝑥=202𝑘𝑃𝑎. 29 Figure S1b. Computed Force/width curves for several Mode I critical interface strengths. The mean force/width of the ascending ovine aorta undergoing a mode I peel is 51.49 mN/mm The results of the peel test agree well with published literature. For example, the mode l presented in the current study with an interface strength of 140kPa calculates a nearly identical mean force/width as seen in previous numerical studies of peel tests (29.03 mN/mm (present study) vs. 28.8 mN/mm [6,7] ). The differences in the mean experimental force/width may be partly explained by the difference in species, and anatomical location of the specimens. The mean force/width is in the region of other peel tests carried out on animals (51.49 mN/mm (present study) vs. 67.4 mN/mm [9,16] . Similarly to Ferrara et al. [7], the mode I interface strength is found to be the primary determinant of the mean force/width and no significant influence of the fracture energy is found. Similar behaviour was also observed in the mesh convergence analysis ( presented in Supplementary Material II : Mesh convergence ) where higher force/width and oscillations were observed with the coarser mesh and a reduction in force/width and oscillations was observed in the intermediate and fi ner meshes. Supplementary Material II : Mesh convergence In this supplementary material we present the mesh convergence study carried out on the mesh used to calibrate the mode II interface strength , the mode II fracture energy , and the mode I interface st rength . In Figure S2(a) the circumferential strain at initiation is shown as a function o f the number of elements in the mesh. The only model parameter changed in each simulation was the mesh density. Mesh 3 is considered sufficiently accurate and was used for the calibration of the mode II interface strength and the mode II fracture energy. Figure S2(b) shows the mean force per width as a function of the number of elements used in the mesh. The intermediate mesh (consisting of 1542 elements) was considered sufficiently accurate to calibrate the mode I interface strength. 30 Figure S2. (a) Circu mferential strain at initiation as a function of the number of elements in the mesh. Mesh 3 is the converged mesh ; (b) M ean force per width as a function of the number of elements used in the mesh. Supplementary Material III: Experimental Images Figure S3: Images of mode II crack propagation in 4 samples undergoing the SFRT . Red arrows approximately indicate the position of the crack tip. Fibrils formed during fibril lation are observed in the top left pane of images. 31 5