Development of a test method to investigate mode II fracture and dissection of arteries

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 current 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 test (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 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 circumferential (c)- direction along the circumferential-axial ( c-a) plane. Fabrication and experimental implementation of the SFRT 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 corresponding 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.


Introduction
Aortic dissection is a lethal disease which can lead to stroke, spinal cord infarction, renal failure, cardiac tamponade, aortic rupture, and death [1][2][3].Mortality 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][6][7][8][9].Such a test methodology results in a mode I fracture and allows 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 induced 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 plane is highly improbable.Iatrogenic dissections may follow from mixed mode fracture initiation [10], but such dissections primarily result from catheterisation, percutaneous transluminal coronary angioplasty, or cardiac surgery [11].In any event, iatrogenic dissections are reported to occur in the radial-axial plane [12], which, given the highly anisotropic arterial microstructure, is not at all reflective of the mode I fracture in the circumferential-axial plane induced 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 observed a similar trend of dissection in the circumferential-axial plane of excised aortas subjected to hypertensive levels of lumen fluid pressure [14].Such dissection 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 lamellae material strength [15].
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 loading conditions.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 dissected aorta.However, we present a preliminary finite element analysis of lap-shear testing of arterial tissue in Section 1.1 (below) in which we demonstrate that the high levels of material deformation at the point of fracture initiation consequently results in a mixed mode initiation and propagation, rather than 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 addressed 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 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 nonstandard geometries, such as arteries, require in-depth mechanistic analysis.
Cohesive zone models (CZMs) have been extensively used to describe an interface undergoing separation [18][19][20][21].The calibration of CZM parameters requires experimental testing data [22][23][24].The primary outputs of these tests are interface strength and fracture energy.It is extremely difficult to measure the stiffness of interfaces in highly elastic materials.The characteristic length () or the modulus () of the initial elastic region of some CZMs often dictates the fracture energy () 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 interface.Piecewise CZMs allow for the specification of  (and by extension ) 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 specimens 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 never 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 advancing the current understanding of the biomechanics of artery dissection.

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 bottom 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 analysis of the constrained experiments of Sommer et al. [17] and the unconstrained experiments of Witzenburg et al. [16] in which crack initiation and propagation is simulated using a cohesive zone formulation.The fracture strength is calibrated so that predicted crack initiation occurs at an identical level of applied shear deformation as reported experimentally.Mode mixity is defined as   = tan −1 (  /  ) and is presented as a function of normalised crack tip position, where   and   are the tangential (shear) and normal tractions, respectively, at the crack-tip.Full 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 is 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 advance on previous coupled mixed mode formulations by facilitating independent specification of modedependent fracture strength and fracture energy independent of intrinsic interface stiffness, while ensuring positive instantaneous incremental dissipation [29].
As shown in Figure 1(I, III), simulation of a constrained lap-shear test, predicts a mode mixity of ~0.2107 at fracture initiation, with a mean mode mixity of   = ~0.1157rad (~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 test.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   = ~1.236rad (~70.85°)during subsequent crack propagation.Again, this is quite different from the intended mode II fracture (  = π 2 ⁄ rad).Figure 1(II(f)) demonstrates the high levels of material deformation at initiation and during propagation, resulting in normal tractions at the crack-tip that are comparable in magnitude to corresponding tangential tractions.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 crack propagation in arterial tissue.We argue that the quantitative analysis of pure mode II fracture resistance of arterial tissue is of critical importance.Lumen blood pressure is the primary source of mechanical loading on an artery wall in vivo, resulting in a compressive radial component of stress.The presence of an intramural haematoma could potentially result in a mixed mode or mode I fracture initiation in the c-a plane.However, the high compressive radial component of stress in the vessel wall due to lumen pressure will promote mode II initiation, particularly in the absence of an intramural hematoma.
Therefore, we 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.

Design and computational validation of an experimental mode II fracture test (shear fracture ring test (SFRT))
In this section we outline the analysis and design of a bespoke experimental methodology that results in pure mode II crack propagation in aortic tissue.Motivated by an analytical solution developed by Parry and McGarry [30] for the stress state of a bi-layered composite arch, we note 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 the 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 on two cylindrical loading bars.The loading bars are then moved apart, imposing a deformation on the artery ring such that that two curved "archtype" 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 outlined in Figure 2(a).A ring of excised aortic tissue with radius   is placed onto two steel bars of radius   .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 .We define a deformed interface coordinate  (shown in Figure 2(a)) which describes the position along a quarter circumference of the deformed configuration. begins at the top of the bar and describes the position that is distance  through the thickness  and ends at  = .We also define a radial coordinate   which describes the radial position of the crack tip through the wall.The original circumference of the ring is described as  0 = 2( 0 ) + 2(  + /2) where  0 is the distance from the midpoint of one loading bar to the other.The change in circumference is described as follows: Δ = Δ where Δ 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; , ,  indicating axes corresponding to the physiological radial, circumferential, axial axes, respectively.Collagen fibres are assumed to lie in the - plane [31,32].A measure of nominal strain in the c-direction (the physiological circumferential direction) is then given as the   = Δ/ 0 where  0 is the undeformed circumference of the ring.Hereafter, for brevity,   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 analysis of the SFRT is presented in Figure 2(b).Figure 2(b) shows a contour plot of circumferential stress   (normalised by the initial shear modulus ) for the case of a loading bar of radius     ⁄ = 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  (when     ⁄ = 0.18) which occurs at the side of the arch region, as shown in Figure 2(c).The magnitude of  at the side of the bar is strongly influenced by the loading bar radius.Three bar radii are shown in Figure 2(c) (    ⁄ = 0.18,0.36,1.0).The peak value of  is increased when the radius of the bar is decreased.In Figure 2(e) the distribution shear traction   along the c-a plane mid-way through the arterial section (r=t/2) is examined.Similar to the distribution of  in the arterial material,   along a c-a plane is highest at the side of the loading bar ( = (  + )/2 ) and increases with decreasing value of   .
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 through the thickness was also examined, placing the plane a quarter-way through the thickness, such that the plane lies on  when  = /4, results in a 7.3% increase in   compared to  = /2.However, placing the plane three quarters through the thickness, such that the plane lies on  when  = 3/4, results in a 44% decrease in   compared to  = /2.The plane lies on S when  = /2 in Figure 2(d-f).
Figure 2(f) shows the normal tractions (i.e. in the r-direction acting 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 ( = (  + )/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 tractions throughout the arterial specimen.
We therefore select a small loading bar radius of 1.5 mm (corresponding to   /  = 0.18 for an aorta specimen radius of 8.33 mm).In Figure 3, using a cohesive zone formulation (described in Appendix B: Cohesive zone fracture model description), we examine the mode mixity 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(  +d)/2), as illustrated in the schematic in Figure 3(a), will result in a kinking of the notch such that mode 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 with no prior radial crack growth).A cohesive surface is placed along the c-a plane through the notch-tip.Simulated crack initiation and propagation is presented in Figure 3.The evolution of   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 ( = (  + )/2), at an applied circumferential strain of Δ/ 0 = 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.This is confirmed by plotting the mode mixity   at the crack-tip as a function of crack-tip position during propagation, as shown in Figure 3

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 attached to the base-plate and the cross-head fixture of a mechanical test machine (Zwick/Roell Z005).Selection of loading bar radius is a key design consideration.While, as shown above, a smaller radius enhances the level of shear stress, bars must be large enough to support the forces imposed by the deforming arterial specimen without exceeding the elastic limit of stainless steel.The force imposed by the deforming specimen is, of course, dependent on the chosen specimen width, .Specimens are prepared such that   ⁄ ≈ 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 adirection (i.e.specimens are approximately cylindircal).For a typical specimen thickness of t=2.35 mm, a width =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 cantilevered 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 ~Δ  0 ⁄ ≈ 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   /  ≤ 0.135 (corresponding to   ≤ 1.125  based on a mean artery radius of 8.33 mm) does not provide an adequate factor of safety for the structural integrity of the loading bar.On the otherhand, large bar radii   /  ≥ 0.25 result in lower magnitudes of shear stress and, consequently, a reduced probability of mode II fracture.Therefore a bar of radius   /  = 0.18 (i.e.  = 1.5 ) is chosen 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 ( = (  + )/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 propagation throughout each test.A photograph of the final test set-up is shown in Figure 4(d).

Sample preparation
The ascending aorta and aortic arch were excised 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 cylindrical 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 adequate 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=11) 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 (  ≈ /2).The boundary of the notch was marked with blue waterproof all-surface ink.

Protocol
To determine the anisotropic hyperelastic properties and anisotropy of the material, three experiments were carried out: axial extension, circumferential extension, and unnotched 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 circumferential directions maintaining a mean aspect ratio of :  = 3.8: 1 (all mean specimen dimensions are presented in Table 1).The samples were gripped using Zwick pneumatic specimen 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  = (  + )/2.3 Results

Determination of hyperelastic properties
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 calibrated 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 squared error ()  = 0.0059) and in the axial direction (  = 0.0144) for the unique material parameter set presented in Table 2. Simulation of the stretching of an unnotched intact ring using the calibrated material model parameters, is shown in Figure 5(b).Results are in strong agreement with corresponding experimental measurements for the entire range of applied deformation (  = 0.0081).

Fracture mechanics results
Firstly, we introduce the results of our SFRT fracture mechanics investigation by presenting the details of a representative aorta test sample, Figure 6 shows the measured force (/) versus applied nominal circumferential strain (Δ/ 0 ).The measured force continues to increase beyond the point of crack initiation at Δ  0 ⁄ = 0.72.This is due to the continued load bearing capacity of the outer section of the ring ( >   ) up to the point of ultimate 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 crack-tip; i.e. fracture initiation entails an immediate kinking of the radial notch so that a mode II initiation along the c-a plane occurs.Following initiation, significant 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 smooth fracture surface is observed along the c-a plane and the crack flank (delaminated material) is observed to remain flat to the c-a plane throughout the test.As shown in Figure 6(b), at Δ  0 ⁄ = 0.86, propagation appears to be a pure mode II in nature.This pattern of mode II propagation continues through Δ  0 ⁄ = 0.93 (Figure 6(c)) up to, and including, the point of ultimate rupture (Δ  0 ⁄ = 0.97; Figure 6(d)).This observed pattern of mode II initiation and propagation in the c-direction along the c-a plane is remarkably 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 cracking 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 when designing biomechanical tests for biological materials.

Figure 6. The graph shows force normalised by the cross-sectional area, 𝑭/𝒘𝒕 (MPa) as a function of the circumferential strain (𝚫𝑪/𝑪 𝟎 ). The point of crack initiation 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 (/) versus applied nominal circumferential strain (Δ/ 0 ) for all SFRT samples (n=11).Points of crack initiation are indicated for all cases.As stated above, mode II crack propagation in the  direction through the - plane is observed in all 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 ≤ Δ  / 0 ≤ 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 behaviour of aortic tissue.Finally, in all samples (n=11), crack initiation does not eliminate the load bearing capacity of the specimen due to the intact outer layer of each specimen ( >   ).However, superposition of the corresponding mean curve for the intact unnotched specimens subjected to the same loading regime (Figure 7(a)) confirms that the presence of the notch, and consequent mode II crack propagation, 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). ̃ is the measurement of deviation from the original crack plane, it is defined as ()/(  ) where () is the thickness of the remaining intact wall at the position S, and (  ) is the thickness of the remaining wall at the position where the crack initiated.Mean measured values and standard deviation is shown by the boxes and error bars, respectively.The dashed line indicates no deviation from the original crack plane ( ̃= 1).A t-test reveals no statistically significant difference between any of the measured values (p=2.39e-9).).The mean final deformed crack length was 22.65±9.68mm.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 fracture 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.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 crack 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 final 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 shown Figure 9(h), in addition to the early stages of fracture propagation and initiation.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 considerably higher than   or   along the c-a plane.Similar to experimental measurements, the computed force (/) increases following fracture initiation.Additionally, similar to experiments, computed forces for notched SFRTs are lower than corresponding curves without the presence of a notch/crack propagation.Figure 11 shows the evolution of the J1 and J2 components of the J integral at the crack 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 seen in Figure 11, J1 is ≈8 times higher than J2 at the mean point of initiation.Fracture initiation occurs in the circumferential direction only when fibres are almost fully aligned in the c-a plane.This suggests that the   is at least 8 times higher for crack propagation in the radial direction than the circumferential direction.The experiment generates a sufficiently high J2 component of the Jintegral at the crack tip to result in circumferential propagation, while fibre alignment results in a significant toughening mechanism against radial crack initiation so that 1 < 1  and propagation does not occur in the radial direction.In arterial tissue, collagen fibres lie along c-a planes, as structural investigations indicate this depends on age, pathology, and type of artery [34][35][36].Therefore, collagen alignment due to increased circumferential strain acts as a toughening mechanism only for fracture in the radial direction (through the aligned fibres).Crack propagation in the circumferential direction does not benefit from the toughening mechanism of fibre alignment (essentially cracks propagate between the collagen).The effect of notch depth on the J integral is further examined in Table 3.While the point of fracture initiation is highly sensitive to the parameter   , it is not found to be sensitive to the mode II fracture energy (  0 ).The mode II fracture energy is defined as the area under the tangential tractionseparation curve.However, as shown in Figure 12, simulations reveal that the rate of crack propagation exhibits sensitivity to   0 .Simulations reveal that a value of   0 = 0.25 N/mm results in a rate of crack propagation that is slower than that observed experimentally.On the other end of the spectrum, values of   0 ≥ 0.0005 N/mm result in excessively high rates of propagation compared to experimental measurements.A value of   0 = 0.005 N/mm is predicted to provide a reasonable approximation of experimental measurement, both during early and later stages of crack initiation.

Discussion
The current study the development and implementation of 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 initiation 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 arterial 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 indepth 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 shear adjacent to a cylindrical loading bar.Placement 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 along the c-a plane.A J-integral analysis suggests radial mode I crack propagation 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 Jintegral was proposed by Maugin et al. [38].The J-Integral is calculated according to the virtual crack extension/domain integral methods as proposed by [39,40].Geometric nonlinearity is accounted for using large 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 higher than the corresponding 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 current of the biomechanical conditions underlying in vivo aortic dissection.
The present study provides an explanation as to why every dissection is associated with circumferential and axial crack propagation, but not always radial (i.e. the majority of aortic dissections do not rupture [11,13]).Aortic rupture requires radial 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 fibre toughening.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 dissections [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][45][46][47].Connective 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 mode 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 sporadic, uncontrolled mode II cracks initiating and propagating in the circumferential direction.Figure 13(b) shows a histological section of a sample that underwent sporadic, uncontrolled mode II propagation in the circumferential 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 Haslach 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.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 artery relative to the outer layer which results in a shear concentration.The results of the present experiment serve 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 circumferential 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 radial 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 cohesive 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 lengths which are linear and do not sufficiently capture the two crack growth regimes (fibrillation and rapid crack growth).Future studies could consider the development of microstructural models to examine the role of interstitial fluid and species transport on fracture initiation.The cohesive zone parameters calculated in the present study should be applied to patient-specific geometries to assess the risk of dissection.They should also be used to examine a range of biomechanical factors which may influence 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.
where   is the mode II interface strength, Ω  sets the non-linearity of the transition from mode II to mode I, and   is the mode I interface strength.We may specify the mode-dependence of   () using the following function: where   0 is the mode II fracture energy and   0 is the mode I fracture energy, Ω  sets the non-linearity of the transition from mode II to mode I. Finally, we complete the description of the cohesive zone formulation by decomposing   into the normal and tangential components,   and   , respectively, such that   = {    Δ  , Δ  < 0   sin() , Δ  ≥ 0 (B7) where    is the overclosure penalty stiffness.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
Circumferential notch tests (n=8) were also performed.Notches (≈ 2.1) 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 onto 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.Crack growth was observed in 3 specimens out of the 8 tested.Average crack length was 11.79mm+/− 4.1mm.Aortic dissection is most commonly associated with an intimal tear or notch [53,[57][58][59], however it can and does occur (although less commonly) without a visible entry tear in certain cases [60][61][62][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 mixity for the circumferential notch test is also explored in Figure C2.It shows the same as

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 (  ) is investigated in Figure S1b.No sensitivity of the Force/width to the mode II fracture energy (  0 ) is found.This finding is in agreement with Ferrara and Pandolfi (2010), who found no sensitivity to mode I fracture energy in a numerical study of mode I peel tests of arterial tissue.The sensitivity of the 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 the computed curves to eliminate noise caused by mesh.The mode I critical interface strength of ovine ascending aorta is calculated as   = 202.The results of the peel test agree well with published literature.For example, the model 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. 28.8mN/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 (51.49mN/mm (present study) vs. 67.4mN/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 finer meshes.

Figure 1 .
Figure 1.(I) Mode mixity (  ) as a function of normalised crack tip position 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 (  =  ,    = .   ⁄ ,   =  ,    = . /); (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 (  =  ,    = .   ⁄ ,   =  ,    = . /).Full details of the artery anisotropic hyperelastic material law and the cohesive zone fracture model are presented in Appendix A: Artery material model and Appendix B: Cohesive zone fracture model description, respectively.

Figure 2 .
Figure 2. (a) Schematic of the SFRT.An excised aortic ring specimen is shown mounted on two bars in the reference configuration.  is the radius of the specimen and   is the radius of the bar.The bottom bar is fixed and u is the displacement of the top bar. is the deformed interface coordinate which is a distance  offset from the bar.A schematic of the local material coordinate system is shown with local circumferential, axial, and radial directions (--); (b) Local circumferential stress is presented for a     ⁄ = . configuration (/  = .) positive values indicate tensile in the local circumferential direction; (c) Maximum local shear stress (/) contour is plot for three bar radiuses (    ⁄ = ., ., .); (d) Normalised interface shear traction (  /) 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 (/) for each of the bar radiuses analysed.Full details of the artery anisotropic hyperelastic material law used are presented in Appendix A: Artery material model.
(c).Corresponding plots for lap-shear test simulations are shown once-again for comparison, highlighting the significant improvement of the SFRT in generating mode II initiation and propagation in arterial tissue.The normal interface calibrated from standard peel tests (see Supplementary Material I: Peel tests) is   = 202 .

Figure 3 .
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 the middle of the aortic ring as the crack propagates.Crack tip is shown by the black arrow; (c) Mode mixity (  ) 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,   =   (see Supplementary Material I: Peel tests); the shear interface strength is assumed to be approximately twice the normal interface strength,   =  kPa; the mode I and mode II fracture energies are assumed to be    = .   ⁄ and    = .   ⁄ .Full details of the cohesive zone model are presented in Appendix B: Cohesive zone fracture model description.

Figure 4 .
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 the computationally designed experimental test rig.The section view (Section A-A) shows the front view of the specimen; (d) Finalised testrig design with mounted arterial SFRT specimen.

Figure 5 .
Figure 5. (a) Nominal stress (MPa) as a function 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 circumferential data sets are . and . respectively; (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 ..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.

Figure 7 .
Figure 7. (a) Force normalised by the cross-sectional area, / (MPa) as a function of the circumferential strain (/  ) for each of the radial notch test specimens.The points of crack initiation are marked with the symbol shown in the legend and are denoted by   where  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 the high strain regime compared to the unnotched intact ring.(b) Measurements of the crack plane as a function of the interface length (S). ̃ is the deviation of the crack plane from the original crack plane.

Figure 8 (
Figure 8(a)  shows the deformed crack length as a function of applied deformation beyond the point of initiation ((Δ −   )   ⁄ ).The mean final deformed crack length was 22.65±9.68mm.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 fracture 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.

Figure 8 .
Figure 8. Crack length as a function of circumferential stretch past initiation.( −   )/  =  indicates the point of crack initiation and the final point on each curve is the point at which the specimen underwent final rupture.

Figure 9 .
Figure 9. Fracture surfaces of the specimens.Original notches are shown with blue ink.Fibrils formed during fibrillation 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 parameter   of 202  (the mode I fracture strength along the c-a plane) is determined from the implementation of standard peel test experiments, described in Appendix B: Cohesive zone fracture model description.The sensitivity of SFRT crack initiation to the mode II fracture strength,   , is presented in Figure 10(b).  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   = 1.6 MPa leads to the prediction at Δ  0 ⁄ = 0.63 and / = 0.84 , which is within 0.86% of the experimentally measured mean values of Δ  0 ⁄ = 0.63 and / = 0.84 , respectively.

Figure 10 .
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) Sensitivity of the crack initiation to the mode II interface strength.Mean force normalised by the cross-sectional area as a function of circumferential strain (/  ) for the SFRT.Points of crack initiation are shown with an 'X'.In each simulation the mode I interface strength was  =  , the mode II fracture energy was    = . /, and the mode I fracture energy was    = . /.The mean circumferential strain (/  ) 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 and cohesive zone model are presented in Appendix A: Artery material model and Appendix B: Cohesive zone fracture model description, respectively.

Figure 11 .
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 /( ̅    ) where  ̅  is the volume averaged strain energy density in the arterial tissue within a radius   from the notch-tip.Note that  ̅    is identical for both crack paths so the figure presents a direct comparison of the J-integrals.Full details of the artery anisotropic hyperelastic material law used are presented in Appendix A: Artery material model.

Figure 12 .
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   =  ,   = . , the mode I fracture energy is scaled with the mode II fracture energy according to    = (    ⁄ )   .Full details of the artery anisotropic hyperelastic material law and cohesive zone model are presented in Appendix A: Artery material model and Appendix B: Cohesive zone fracture model description, respectively.

Figure 13 .
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.

Figure
Figure B1.(a) Traction-separation response of the CZM to a mode II separation,    is the maximum mode

Figure C1 .
Figure C1.(a) Image of the circumferential notch pre-testing.Circumferential notch is shown measuring approximately ~2 mm; (b) of the specimen after testing and prior to ultimate failure.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 growth is observed in the c-direction in the c-a plane.The black arrow indicates the position of the crack tip.

Figure 3 (
c) with the addition of the circumferential notch test.The mean mode mixity for the circumferential notch test is 89.18 ∘ .

Figure C2 .
Figure C2.Mode mixity   as a function of normalised crack tip position for the SFRT, circumferential notch test, unconstrained lap-shear, and constrained lap-shear.The mean mode mixity in the circumferential notch test is . ∘ .

Figure
FigureS1ashows 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 .
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.

Figure S1b .
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 51.49mN/mm

Figure S2 .
Figure S2.(a) Circumferential strain at initiation as a function of the number of elements in the mesh.Mesh 3 is the converged mesh; (b) Mean force per width as a function of the number of elements used in the mesh.

Figure S3 :
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 fibrillation are observed in the top left pane of images.

Table 2 . Table of best fit material parameters for the arterial material model.
The mode mixity of the initial interface damage parameter *