Bifurcation Analysis of a Cylindrical Tube for a Residually-Stressed Material and its Application to Aneurysms in Arterial Tissue

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(3) Universidad Politécnica de Madrid Escuela Técnica Superior de Ingenieros de Caminos, Canales y Puertos de Madrid. Máster Universitario en Ingenierı́a de Estructuras, Cimentaciones y Materiales. Trabajo Fin de Máster. Bifurcation analysis of a cylindrical tube for a residually-stressed material and its application to aneurysms in arterial tissue.. Author Alejandro Font Rojas Ingeniero Civil. Tutor Profesor José Merodio Doctor Ingeniero Mecánico. Madrid, julio de 2019.

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(5) Abstract An evaluation of the parameters that define a proposed boundary value problem solved with the finite element method as an approach to emulate the bio-mechanical response of arteries at an instability point. Blood vessels are idealized as cylinder tubes with different geometries and boundary conditions that incorporate an invariant based, hyper-elastic and residually stressed material model as a subroutine in a finite element software. A sequential, three-step, mechanical analysis that simultes the in vivo conditioning and physiological actions is applied; the analysis consists on the application of residual stresses, an imposed axial stretch and the application of uniform internal pressure. The models are restricted to obtain instabilities of bending and bulging bifurcation modes, the first is related to mechanical buckling for the mode m=1, while the second one to the propagation of a bulge in the middle section. The average azimuthal and axial stresses are evaluated along the axial direction; uniform distributions are found at the bifurcation point for models in the bending mode, while peaks in the middle section are found for bulging which indicate radial displacements. Stresses in the axial direction tend to increase due to the stretch in that direction, eventually surpassing the ones in the hoop direction. The critical uniform pressure reached at a bifurcation point defines the corresponding onset of bifurcation in terms of the axial stretch. The bending mode is related to low values of axial stretch, but bulging becomes the onset of bifurcation at values above a defined transition value. The transition value increases for more slender cylinder tubes and for lower values of the magnitude of residual stresses. The critical pressure values at their corresponding transition values also tend to decrease for higher magnitudes of residual stresses. A description of the stages of the mechanical analyses are included in terms of the history of configurations for both bifurcation modes. The bulging mode is characterized for having both maximum critical uniform pressure and stresses at the bifurcation point. For certain axial stretches, the bending mode presents the propagation of a secondary lateral bulge at a post-bifurcation stage at a higher critical pressure applied which is related to abdominal aortic aneurysms. i.

(6) Keywords: Residual stress, Bending bifurcation, Bulging bifurcation, Arterial tissue, Aneurysm.. Resumen. Una evaluación de los parametros que definen un problema de contorno resuelto con el método de los elementos finitos como planteamiento para emular la respuesta biomecánica de arterias en un punto de inestabilidad. Vasos sanguı́ineos son idealizados como tubos cilı́ndricos de distintas geometrı́as y condiciones de apoyo que incorporan un modelo material de tipo hiperelástico, tensionado residualmente, basado en invariantes e introducido como una subrutina en un software de elementos finitos. Un análisis secuencial de tres pasos que simula las condiciones in vivo y acciones fisiológicas es aplicado; el análisis consiste en la aplicación de tensiones residuales, un estiramiento axial impuesto y la aplicación de presión uniforme en el interior. Los modelos están restringidos a capturar inestabilidades de modos de tipo flexión y abultamiento, el primero relacionado al pandeo mecánico en el modo m=1, mientras que el segundo al de la propagación de un bulto en la sección media. Las tensiones azimutales y axiales promedio son evaluadas a lo largo de la dirección axial; distribuciones uniformes son halladas en el punto de bifurcación para modelos del modo de flexión, mientras que picos en la sección media son hallados para el modo de abultamiento que indican desplazamientos radiales. Las tensiones en la dirección axial tienden a disminuir debido al estiramiento en dicha dirección, eventualmente superando a aquellas en la dirección angular. La presión crı́tica uniforme alcanzada en el punto de bifurcación define al modo de propagación de la bifurcación en función del estiramiento axial. El modo de flexión está relacionado a valores bajos de estiramiento axial, pero el de abultamiento se convierte en el propagador para valores mayores al que está definido como valor de transición. El valor de transición aumenta cuanto más esbeltos sean los tubos cilı́ndricos y cuanto menores sean los de magnitud de tensión residuales. Los valores de presión crı́tica en su correspondiente valor de transición también tienden a disminuir para mayores magnitudes de tensiones residuales. ii.

(7) Una descripción de las etapas de los análisis mecánicos son incluidos según historias de configuración para ambos modos de bifurcación. El modo de abultamiento es caracterizado por tener valore máximos, tanto de presión crı́tica uniforme como de tensiones en el punto de bifurcación. Para ciertos estiramientos axiales, el modo de flexión presenta la propagación lateral de un bulto secundario en una etapa de post bifurcación a una presión mayor que está relacionado con aneurismas de aorta abdominal. Palabras clave: Tensión residual, Bifurcación de tipo flexión, Bifurcación de tipo abultamiento, Tejido arterial, Aneurisma.. iii.

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(9) Acknowledgements This project is only possible with the guidance and expertise of the supervisor José Merodio and also the guidance of Hamidreza Dehghani for the development of the numerical procedure. The project also represents the culmination of another stage in my professional career; therefore I need to acknowledge the support of the colleagues and friends that have accompanied me during the masters courses at the Universidad Politécnica de Madrid and also at the graduate courses at the Facultad de Ingenierı́a de la Universidad Nacional de Asunción. I am grateful to my parents, Hernán and Kathy, to my brothers Gabriel and Esteban and also to Gabriela, my girlfriend.. v.

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(11) Contents. Abstract. i. Resumen. ii. Acknowledgements. v. 1. Introduction. 1. 1.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2. 1.2. Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2. 1.3. Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 2. Theoretical Framework. 4. 2.1. Arterial histology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 2.2. Geometry of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. 2.3. Continuum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2.3.1. Kinematics and strain . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2.3.2. Motion equations and balance laws . . . . . . . . . . . . . . . . .. 9. vii.

(12) viii. CONTENTS. 2.4. 3. Constitutive equation . . . . . . . . . . . . . . . . . . . . . . . . .. 10. 2.3.4. Residual stress in the reference configuration . . . . . . . . . . . .. 11. Finite element analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12. 2.4.1. The finite element method . . . . . . . . . . . . . . . . . . . . . . .. 13. 2.4.2. The arc length method . . . . . . . . . . . . . . . . . . . . . . . . .. 14. Methodology. 16. 3.1. Material description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16. 3.1.1. Energy-strain equation . . . . . . . . . . . . . . . . . . . . . . . . .. 16. 3.1.2. Residual stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16. 3.1.3. Invariant formulation . . . . . . . . . . . . . . . . . . . . . . . . .. 17. Finite element analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17. 3.2.1. Geometry and elements . . . . . . . . . . . . . . . . . . . . . . . .. 17. 3.2.2. Material model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18. 3.2.3. Analysis description . . . . . . . . . . . . . . . . . . . . . . . . . .. 18. Bifurcation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19. 3.3.1. Bulging mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20. 3.3.2. Bending mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21. 3.3.3. Bifurcation in the arc length method . . . . . . . . . . . . . . . . .. 21. 3.2. 3.3. 4. 2.3.3. Numerical Results. 24.

(13) 4.1. Parameters and conditions . . . . . . . . . . . . . . . . . . . . . . . . . . .. 24. 4.1.1. Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .. 24. 4.1.2. Effects of boundary conditions . . . . . . . . . . . . . . . . . . . .. 25. Bifurcation criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28. 4.2.1. Membrane approximation criteria . . . . . . . . . . . . . . . . . .. 28. 4.2.2. Onset of bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . .. 29. 4.3. Distribution of azimuthal and axial stresses . . . . . . . . . . . . . . . . .. 30. 4.4. Bifurcation in terms of the axial stretch . . . . . . . . . . . . . . . . . . . .. 33. 4.4.1. Transition axial stretch . . . . . . . . . . . . . . . . . . . . . . . . .. 34. 4.4.2. Effects of residual stresses . . . . . . . . . . . . . . . . . . . . . . .. 35. History of configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36. 4.2. 4.5. 5. Conclusions. 42. Bibliography. 44. ix.

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(15) List of Figures. 2.1. Structural model of the composing layers for an artery in healthy conditions [13]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2. 5. Stress-strain curve for loading-unloading cycles of an elastic artery. Ultimately a non-linear elastic behavior is observed after the initial cycles. [13]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.3. 6. Mapping effect. Finite lines in the reference and current configurations [19]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8. 2.4. Stress t and normal unit direction n in surface S [19]. . . . . . . . . . . .. 9. 2.5. Outline of a boundary value problem. In a incompressible material V 0 = V [28]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13. 2.6. Graphical outline of the arc length method [12] . . . . . . . . . . . . . . .. 15. 3.1. Residual stresses τθθ and τrr in the hoop and radial directions along the thickness in the middle cross-section for a value of β = 2.0. The analytical values of these stresses are stated in equations 3.3 and 3.4 and similar results are obtained when compared with the numerical ones. . . . . . .. 3.2. 19. Acting forces in an infinitesimal section of a membrane. Basis for equilibrium equations in the radial and axial directions [16]. . . . . . . . . . . xi. 21.

(16) xii. LIST OF FIGURES 3.3. Interpretation of forces acting in a infinite length membrane for the bending mode [16]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22. 3.4. Load proportionality factor plot for a bulging motion with λz = 1.0. . . .. 23. 3.5. Load proportionality factor plot for a bending motion with λz = 1.0. . .. 23. 4.1. Values of τθθ along the normalized thickness in the middle cross-section for variable magnitudes of residual stress β. . . . . . . . . . . . . . . . . .. 4.2. 25. Average stresses σθθ and σzz of cross-sections along their position in the axial direction. In continuous lines the analyses are performed with restricted displacements (BC 01) in the radial direction, while in dashed lines the displacements are permitted (BC 02). The values are extracted at the corresponding bending bifurcation point for a model with β = 2.0. 4.3. 27. Values of the criteria M = σθθ − 2 × σzz , as seen in equation 3.15, for an average stresses in the middle cross-section. The analysis considers β = 2.0, for a first model λz = 1.0 and a second one λz = 1.40. Positive values σθθ − 2 × σzz > 0 indicate an instability of bending motion while negative ones where obtained for the bulging one. . . . . . . . . . . . . .. 4.4. 28. Load proportionality factor against arc length plot for a cylinder tube model at β = 2.0 and λz = 1.0. The first equilibrium path showcases an instability of bending motion, that separates itself from the second one, which is the bulging one in its bifurcation point. The noticed bifurcation point for bulging modes are close to the maximum values of LPF. . . . .. 29.

(17) LIST OF FIGURES 4.5. xiii. Normalized pressure development against hoop strain λθ for bulging mode analyses at λz = 1.30. The hoop strains considered are in two distinct points along the axial direction, one inside the bulge (in) located in the middle cross-section and another one outside of the bulge (out) in an intermediate cross-section between the middle and the end section of each tube. The curves depart at the bifurcation point, which is near the corresponding maximum normalized pressure. . . . . . . . . . . . . . . .. 4.6. 30. Average stresses σθθ and σzz of cross-sections along their position in the axial direction. The values are extracted at their corresponding bending bifurcation points for models with L/R = 30 and β = 2.0. . . . . . . . . .. 4.7. Normalized pressure values at bifurcation points obtained via models with β = 2.0, L/R = 30 and for axial pre-stretches 1.0 ≤ λz ≤ 2.0. . . . .. 4.8. 33. Normalized pressure values at bifurcation points obtained with models of L/R = 30 and for axial pre-stretches 1.0 ≤ λz ≤ 2.0. . . . . . . . . . . .. 4.9. 32. 34. Values of normalized pressure for the onset of bifurcation for β = 2.0 and 1.0 ≤ λz ≤ 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35. 4.10 Values of normalized pressure at their bifurcation points relative to the values of transition λz . Each value is obtained in terms of a certain β value indicated near its position in the generated curves. The curves are obtained for two distinct ratii L/R. . . . . . . . . . . . . . . . . . . . . . .. 35. 4.11 Variation of β and the values of transition λz for two models. . . . . . . .. 36. 4.12 Analysis sequence of a cylinder tube with a bulging mode of bifurcation with λz = 1.40, L/R = 30 and β = 2.0. . . . . . . . . . . . . . . . . . . . .. 37. 4.13 Contour of von Mises stresses at the bifurcation point for parameter values λz = 1.40, L/R = 30 and β = 2.0. . . . . . . . . . . . . . . . . . . . . .. 38.

(18) 4.14 Contour of von Mises stresses at a post-bifurcation stage for an analysis with parameter values λz = 1.40, L/R = 30 and β = 2.0. . . . . . . . . . .. 38. 4.15 Analysis sequence of a cylinder tube with a bending mode of bifurcation with λz = 1.08, L/R = 30 and β = 2.0. . . . . . . . . . . . . . . . . . . . .. 39. 4.16 Contour of von Mises stresses at the bifurcation point for parameter values λz = 1.08, L/R = 30 and β = 2.0. . . . . . . . . . . . . . . . . . . . . .. 40. 4.17 Contour of von Mises stresses at a post-bifurcation stage for an analysis with parameter values λz = 1.08, L/R = 30 and β = 2.0. . . . . . . . . . .. 40. 4.18 The figures 4.18a and 4.18b are from a bifurcation point and a postbifurcation stage in the BC01 condition with restricted radial displacements in the step 3. The figures 4.18c and 4.18d are from a similar analysis under a BC02 condition with free displacements in the radial direction. The models feature the parameters L/R = 30 and β = 2.0. . . . . .. xiv. 41.

(19) Chapter 1 Introduction In the field of biomechanics it is possible to analyze the mechanical behavior of a blood vessel as a boundary value problem based on the finite element method. This approach allows the inclusion of actions, specific geometry and boundary restrictions that emulate the real characteristics in which arteries function within the human body. It is necessary to use a material model which is an invariant based formulation based on continuum mechanics theory; a constitutive equation of an hyper-elastic material emulates the response of soft tissue under mechanical stresses. These models are focused on the behavior of arteries during instabilities or mechanical bifurcations; where with specific analyses the formation of aneurysms can be simulated. With bases in other experimental studies, it constitutes itself in a tool for research in the medical field. For this project, the material model incorporates residual stresses, based on a particular constitutive equation defined from a strain-energy equation. The effect of residual stress needs to be accounted as it has been demonstrated both their presence, in a loadfree artery ring, and their influence in the mechanical response. The performed analysis applied on the material model consists of a three step mechanical loading. Their sequence goes in the following order, first the application of residual stress, then a predetermined axial stretch and finally an uniform internal pressure load. This outline is appropriate as is similar to the conditioning of an average artery.. 1.

(20) 2. 1.1. Chapter 1. Introduction. Motivation. The mechanical behavior of arteries described with thick-walled cylinder models is a topic of study in the field of biomechanics. Most notably, this includes the formation and propagation of aneurysms and the influence of the related parameters (material, anisotropy, residual stresses, geometry, etc.). The purpose of this project to reproduce this phenomena via a finite element model while taking into account the influence of residual stress in its formulation. The fidelity of the obtained results from the finite element model depends on the invariant based formulation applied, which is similar to some previous successful research on the subject. The effects of the introduced residual stress has an influence in the stress and strain distribution in the model and requires an evaluation in terms of the numerical results. The generated model itself needs to show a certain stability during the study; as an loading-unloading problem the evaluation will be done with the arc length method, it is important to obtain reasonable responses for different instabilities.. 1.2. Objectives. The main objective of this project is the realization of a bifurcation analysis, that contemplates both bulging and bending type of instabilities, for a generated finite element model of a cylinder tube that replicates the mechanical behavior of arterial tissue with residual stresses in the initial configuration. Each type of bifurcation requires an understanding of the necessary conditions to obtain their propagation. For this purpose is important to describe the parameters that take place in the definition of the model and its properties. The project will also see to carry a sensitivity analysis to evaluate the stability in the behavior of the cylinder with the variation of the magnitude of the residual stresses applied in the material model. Different options will be explored in terms of the geometrical dimensions, boundary conditions and restrictions applied in the analyses. The evolution of stresses and other relevant data of the analysis in certain cross sections will sustain the corresponding observations..

(21) 1.3. Methodology. 1.3. 3. Methodology. The finite element model for the cylinder tube will be generated in the commercial software Abaqus/Standard. The material model is implemented as a subroutine (UMAT) in which the constitutive equation is an invariant based formulation for an incompressible material that includes the effect of residual stresses. The numerical method for solving the problem is the modified Risk method, which is appropriate given the non-linear nature of the problem. The formulation is in concordance to the cylindrical coordinates of the cylinder tube model as the residual stresses are applied in radial and azimuthal dimensions. A sensitivity analysis for the residual stresses will be conducted for a couple of reasons. First, to evaluate the effect in the resulting stresses and equilibrium of the model in the initial configuration; and also to find an stable configuration for the following bifurcation analysis. For the aforementioned analysis, the model is to be subjected to axial stretching and internal constant pressure. The bifurcation analysis contemplates two different instabilities which are the bulging and bending types. In order to capture the bulging bifurcation the cross sections along the axial direction of the cylinder need to maintain their circular shape. The movement of nodes in the hoop direction, in both ends of the cylinder, are restricted through the analysis. For the case of bending, it is necessary to introduce a small imperfection in the geometry of the problem to onset the bifurcation. An evaluation of the existing conditions at the bifurcation points in terms of axial stretch is the starting point to understand the influence of the parameters that define this analysis..

(22) Chapter 2 Theoretical Framework 2.1. Arterial histology. Elastic arteries are those who receive blood directly from the heart (aorta). These arterial walls contain large amounts of elastine, an elastic protein which confers them a necessary elastic behavior; the walls also contain a fibrous protein named collagen and their molecules contribute to tensile strength. The combination of these fibers limit possible overstretch of the artery [22]. The media or middle layer of the artery, as shown in 2.1, is composed of muscle cells and the previously mentioned fibers arranged in an almost circumferential position. This layout conferes high strength and resistance to artery sections, necessary for the mechanical solicitations which are subjected to [13]. The other layers are the intima, located in an interior position, and the adventitia which is located in an external position 2.1. Unless affected with certain pathologies (atherosclerosis) or other factors, the intima is comparably thin in a healthy condition. The adventitia also contains fibers and serves as reinforcement of the wall that gains stiffness when the collagen is straightened at high values of pressure. The mechanical behavior of an elastic artery can be observed in its corresponding stress-strain curve, represented in figure 2.2. Studies show that the behavior of the material subjected to loading-unloading cycles is elastic after the effects of stress softening passed in the initial cycles which is noticeable in the curve that goes up to point I. Inelastic deformations appear for solicitations above the limit of point I and the curve ends up in point II; for additional cycles the same stress softening effect appears again and the curve reaches point III. Finally, the behavior is characterized by the black line that passes 4.

(23) 2.1. Arterial histology. 5. Figure 2.1: Structural model of the composing layers for an artery in healthy conditions [13].. through the origin, point I, point III and then returns to a null stress value. A certain magnitude of inelastic deformation is a result of the dissipation of energy, observed in the generated area under the curve, in a loading cycle that over-stretches the material. The non-linear behavior encountered can be attributed to the response mechanism of the fibers. The elastine fibers are the ones activated for low stretching at low pressures. In physiological levels of pressure, more collagen fibers participate in the response to stretching; while at higher values of pressure, collagen has the most contribution [23]. This causes the observed exponential shape of the curve. For physiological solicitations the arterial tissue can be considered as an incompressible material. This assumption has been proved in a past research [2] on aortic segments subjected to internal pressure and longitudinal stresses; for practical purposes the volumetric strain is negligible. It has been demonstrated that the inner walls (intima and media regions) of blood vessels are subjected to compressive residual stresses in the azimuthal direction while stresses of tensile nature act in the exterior (adventitia region) [9]. An initial stress state has an influence in the mechanical response of arterial tissue when subjected to axial stretches and internal pressuring from blood flow..

(24) 6. Chapter 2. Theoretical Framework. Figure 2.2: Stress-strain curve for loading-unloading cycles of an elastic artery. Ultimately a non-linear elastic behavior is observed after the initial cycles. [13].. 2.2. Geometry of the problem. The initial reference configuration, as defined in continuum mechanics, geometry of the problem is of a cylinder tube; defined with a circular cross section of diameter D, length L and thickness with a dimension H equal to the difference between the internal and external radii A and B. A ≤ R ≤ B, 0 ≤ Θ ≤ 2π, −. L L ≤Z≤ 2 2. (2.1). Appropriately, cylindrical coordinates are used through the entire project. A position vector X in the initial configuration is defined through the unit vectors ER , EΘ and EZ in their corresponding directions. X = RER (Θ) + ZEZ. (2.2). As the cylinder dimensions change during mechanical solicitations, the position vector x is defined according in terms of the new geometrical dimensions. l l a ≤ r ≤ b, 0 ≤ Θ ≤ 2π, − ≤ z ≤ 2 2. (2.3). x = ree (θ) + zez. (2.4). The displacements u between each configuration are defined as u = x − X..

(25) 2.3. Continuum mechanics. 2.3. 7. Continuum mechanics. In order to characterize the elastic properties of the material to be modeled, concepts and equations on the subject of continuum mechanics and elasticity are mentioned in this chapter. An important amount of related literature has been published which serves as a foundation for research in the field of biomechanics. Studies on the subjects of modeling of soft tissue bodies were developed in the latest years [20] and similar formulations will serve as a main frame for the models to be developed. It is appropriate to express some of the equations referred to cylindrical coordinates. As a way for introduce the effects of residual stress into the constitutive equations, a series of papers analyzing the effects have been published by [17] and [13]. The invariant formulation can incorporate the effects of residual stress into the formulation based on these studies.. 2.3.1. Kinematics and strain. A continuous body B is a set of elements which occupies an open subset or region of a three-dimensional Euclidean point space. These elements are called particles and a certain subset B is called a configuration, that changes as the body moves during time [8]. An arbitrary fixed configuration called Br is defined as the reference configuration, in which each particle is referenced by its corresponding position vector X relative to an origin O. A given configuration Bt, at a certain time t, has each of its particles defined by a position vector x relative to an origin o; then it is referred as the current or deformed configuration. The configuration Bt is a result of the deformation of the body, the deformation is defined as the mapping χ : Br → Bt. As each posterior configuration is dependable on time t, the expression 2.5 is obtained [19]. x = χ(X, t). (2.5). The deformation gradient tensor F is defined as the gradient operator for a current configuration with respect to the reference configuration. This second-order tensor describes the deformation of small line elements dX positioned in X into dx at their correspond-.

(26) 8. Chapter 2. Theoretical Framework. Figure 2.3: Mapping effect. Finite lines in the reference and current configurations [19].. ing position x(Lagrangian representation). F(X, t) = Grad x ≡ Grad χ(X, t). (2.6). dx = FdX. (2.7). The deformation is locally invertible which requires F to be non-singular. The polar decomposition allows the representation of F in symmetric and positive definite tensors U (right stretch tensor) and V (left stretch tensor) with the use of R, an orthogonal tensor. J ≡ detF > 0. (2.8). F = RU = VR. (2.9). This allows the identification of principal stretches λi in their corresponding principal axes u (Lagrangian representation) in U, it also applies similarly for v (Eulerian representation) in V. 3 X U= λi u(i) × u(i) (2.10) i=1. Then F is expressed, as a diagonal matrix cylindrical coordinates.  λr  F=0 0. in terms of the principal stretches 2.10, in  0 0  λθ 0  0 λz. The principal stretches can be expressed in terms of the dimensions of the cylinder in their respective configurations as seen in 2.1 and 2.3. λθ = r/R, λθ = l/L. (2.11). As the volume of material remains constant due to its incompressible nature, it is possible to express the stretch in the radial direction in terms of the other two. −1 λr = λ−1 θ λr. (2.12).

(27) 2.3. Continuum mechanics. 9. Figure 2.4: Stress t and normal unit direction n in surface S [19]. The tensors C and B are the right and left Cauchy-Green deformation tensors based on F, with the use of the polar decomposition. Then the measure of strain is obtained via the Green strain tensor E where I is the identity tensor.. 2.3.2. C = FT F = U2 , B = FFT = V2. (2.13). 1 1 E = [FT F − I] = [C − I] 2 2. (2.14). Motion equations and balance laws. The mass density of the material that composes B defined as ρ is considered a differentiable, continuous scalar field. The mass of a defined region remains constant in time. The mechanical behavior of a certain part of a configuration, enclosed inside a surface S, is defined on the basis of Cauchy’s stress principle. The stresses occurring are dependable on the normal unit direction n of S, and are defined as a vector field t(n) of stresses.. In continuum mechanics, contact and body forces are considered and these take part in the law of balance of linear momentum, defined in Euler’s first axiom for any given configuration. Z Z Z d ρ × vdv = ρ × bdv + t(n) da (2.15) dt Rt Rt ∂Rt The stress vectors denoted by t are dependable of the Cauchy stress tensor σ, a secondorder tensor which is independent of n and proven symmetrical. Using the divergence operator, referred to time t, the expression 2.15 can be transformed into the equilibrium equation which is satisfied by σ. σ = σT (2.16).

(28) 10. Chapter 2. Theoretical Framework div σ + ρb = ρa. (2.17). The state where no body forces are applied leads to the equilibrium equation, based on the expression 2.17, to be satisfied by the cylinder to be developed. Similarly S satisfies the equation when the Div operator is applied. div σ = 0. (2.18). It is possible to define the nominal stress tensor S, relative to the reference configuration, in function of σ with Nanson’s formula [19]. S = JF-1 σ. 2.3.3. (2.19). Constitutive equation. The constitutive behavior of an elastic material is characterized according to its strainenergy function W which is dependent on the deformation gradients F. In the case of incompressible materials S can be expressed in terms of W by taking into account the multiplier q denoted as the arbitrary incompressible pressure. Finally, it is possible to define σ based on equation 2.19. W = W(F). (2.20). σ=F. ∂W − qI ∂F. (2.21). The constitutive model considers that the material properties are independent of the reference frame (rigid-body rotations or translations) under the principle of material objectivity [11]. For an isotropic material, it is possible to express W as a function of U; thus being dependable on the principal stretches λi [18]. It is appropriate to point out that in an undeformed reference configuration W takes a null value. W(I) = 0. (2.22). A the objectivity principle applies, W is defined in terms of C through F as seen in 2.21. Therefore is also possible to express the effects of the residual stresses via the tensor τ . W = W(C,τ ). (2.23). The principal invariants of C are respectively I 1 , I 2 and I 3 . 1 I 1 = tr(C), I 2 = [I 1 2 − tr(C2 )], I 3 = det(C) ≡ det(F)2 2. (2.24).

(29) 2.3. Continuum mechanics. 11. The third invariant I 3 remains constant in the case of materials of incompressibility property such as soft tissue [18]. J ≡ detF = λ1 λ2 λ3 = 1. 2.3.4. (2.25). Residual stress in the reference configuration. The modeling of materials for soft tissue analysis requires the inclusion of the effects of residual stress to accomplish an approximation of its mechanical behavior. According to previous experimentation, the aorta while in a state free of external forces and maintaining a circumferential configuration is subjected to residual strain and stress [27]. The residual stress appears in the reference configuration while no body forces nor surface tension are applied; then, in an initial state of the model considering 2.23, the stress is given by τ . Also, both the equilibrium equation 2.18 and the boundary condition are satisfied in Br. Div τ = 0. (2.26). τ n = 0 on ∂Br. (2.27). It is possible to satisfy 2.27 when not considering any residual shear components. Considering τ ZZ equal to 0 is also compatible with a necessary boundary condition, as t is null at the ends of the cylinder (in the axial direction n = 0,0,1).   τRR 0 0   τ =  0 τΘΘ 0 0 0 0. (2.28). The equilibrium equation 2.26 in cylindrical coordinates is reduced to its radial component and it is expressed in terms of τ RR and τ ΘΘ [15]. The boundary surface of the cylinder has to comply the expression 2.27. This transformation (further explained in [19] ) is possible since the considered material is incompressible. 1 dτ RR + (τ RR − τ ΘΘ ) dR R. (2.29). τ RR = 0 on R = A, B. (2.30). The strain-energy equation formulation incorporates the residual stresses τ condition. It is applied into its reference configuration as a second order tensor, with similar properties as C, where the material maintains its objectivity principle. Finally, it is then possible to express W based on C, via its invariants, and also τ [17]..

(30) 12. Chapter 2. Theoretical Framework. For instance, in a case of a situation where F = I the expression 2.21 still applies. In this case the pressure qr is relative to the reference configuration. τ =. ∂W − qr I ∂τ. (2.31). The invariants related to τ are similarly defined as I 4 ; this invariant encompasses I 41 , I 42 and I 43 1 I 41 = tr(τ ), I 42 = [I 41 2 − tr(τ 2 )], I 43 = det(τ ) (2.32) 2 Finally, in order to incorporate the effects of residual stresses it is necessary to define the invariant I 5 , as it depends on both C and τ . I 5 = tr(τ C). 2.4. (2.33). Finite element analysis. Studies on membrane tubes models subjected to axial stretch, torsion and internal pressure were done with a theoretical framework in the past. These analyses often require the use of complex mathematical systems and are limited to simple configurations and some idealizations. The use of numerical methods to analyze material models gives the possibility of extend the range of factors that come to affect arteries, such as initial stresses, finite and varying layers in the thickness. A description of the fundamentals based on the process explained in [1] take part in this section. In order to simulate the behavior of an artery, material models will be generated and analyzed with the use of the commercial software Abaqus/Standard. This section discusses general properties such as geometry, type of elements and boundary conditions. It is important to distinguish the effects that each parameter, besides the material itself, plays into the nodes displacements during the different steps of the analysis. The modified Riks method or arc length method is the one to be implemented for the analysis; this iterative method is provided by the software and it is recommended given the hyper-elastic nature of the subject. One of the main objectives for these type of analysis is to capture a bifurcation during the mechanical actions. It is necessary to establish a correlation between the resulting load proportionality factor plot and the criteria of capturing a bifurcation point, indicator of an instability condition..

(31) 2.4. Finite element analysis. 13. Figure 2.5: Outline of a boundary value problem. In a incompressible material V 0 = V [28].. 2.4.1. The finite element method. The purpose of the finite element method, oriented to solid mechanics, is to find an approximate solution in a boundary value problem. A continuum body is subjected to a combination of external body forces b (per unit mass). The boundary conditions are the stress forces t, acting in a defined surface S t , and the restricted or known displacements of another surface S u . In large strain theory where each configuration is time dependent, the equilibrium equations are expressed in terms of the Cauchy stress tensor σ and the motion of particles is given by a deformation rate tensor. The deformable material has a constitutive law related to the objective stress rate of the stress tensor. The continuum body is defined as a closed domain Ω with a defined boundary Γ; where Γ = Γu ∪ Γu while ∅ = Γu ∩ Γt . The equilibrium of forces and momentum of its particles is denoted as the strong formulation. The external forces b : Ω, the known displacements ū : Γu and stresses t̄ : Γt complete the definition of the strong formulation, or equilibrium equation. div σ + b = 0 on Ω. σn = t̄ on Γt. (2.35). (2.34). u = ū on Γu. (2.36). It is possible to adapt the differential equilibrium equations 2.34 into an equivalent condition based on the virtual work principle denoted as the weak formulation of the problem. A solution comes from a scalar function where a virtual, arbitrary, velocity field δv is.

(32) 14. Chapter 2. Theoretical Framework. multiplied to the equilibrium condition over the volume and then integrated. The velocity field is continuous and compatible with the kinematics defined for the material where the virtual velocity gradient is δL = ∂δv . ∂x Z. ∂δv − b · δv)dΩ = (σ : ∂x Ω. Z t · δvdΓ. (2.37). Γ. The finite element method theory and formulation is covered in [12]. It allows for the discretization of the solid body into finite elements defined by nodes and governed by a constitutive law which satisfy a system of compatible equations.. 2.4.2. The arc length method. As for the purposes of this work, it is relevant to describe the concepts of the arc length method, used as a solving method. The program Abaqus/Standard incorporates this method [12] in order to find a converging solution in a non-linear system of equations. The external loads Fext can be related to proportionality factor λ. This numerical method allows a simultaneous variation of both displacements and external forces related to the quantities ∆u and ∆λ in an loading-unloading problem. It is the appropriate method for a problem with high non-linearity, in which the solution is the equilibrium between the response of a material model of hyper-elastic nature and the external loads, these solutions form the equilibrium path. Fint (u0 + ∆u) − (λ0 + ∆λ)q = 0. (2.38). The method incorporates a complex method to find the correct increments δu and δλ needed to reach the converging solution that satisfies 2.38 which is described in [28]. Abaqus/Standard is able to implement a modified Riks method with an analysis that involves a finite number of steps in order to find the equilibrium path. In each iteration the current load depends on previous ones; those not redefined in the current iteration are described as dead loads P0 , and those redefined in the step are reference loads P0 . Ptotal = P0 + λ(Pref − P0 ). (2.39). While the solving method is still the Newton one (linear increments), the increments of the arc length values (quantity related to the nodal variables of the model such as displacements and rotations) are automatically done in limited amounts by the software (algorithm for static case in Abaqus/Standard [12])..

(33) 2.4. Finite element analysis. 15. Figure 2.6: Graphical outline of the arc length method [12] The method is based on the assumption that the loads are proportional to a certain parameter (for this project there are uniform axial stretch and load pressuring in different steps) and that the response presents a smooth behavior. The equilibrium path is a generated curve in each step, it can be expressed in terms of the load parameter and arc length. It is then possible to identify bifurcation points during the analysis by an evaluation of the generated curves that illustrate the mechanical response..

(34) Chapter 3 Methodology 3.1. Material description. 3.1.1. Energy-strain equation. The material model has two important characteristics of an elastic artery that needs to be addressed; these are the nature of an elastic incompressible material and also the influence of residual stresses. The formulation of W is based in a simple neo-Hookean isotropic energy function with an additional term that introduces τ [16]. The first term depends of the, defined positive, shear modulus µ. 1 1 W = µ(I 1 − 3) + (I 5 − trτ ) 2 2. (3.1). In the second term of 3.1 is possible to see that the effects of τ depends on the single parameter β.. 3.1.2. Residual stress. The formulation for the residual stresses is expressed in terms of τ RR and τ ΘΘ as these define the tensor τ as seen in 2.28. The magnitude of the residual stress β̄, dependable of β, is related to the material constant µ. β̄ =. µβ 2BT. 16. (3.2).

(35) 3.2. Finite element analysis. 17. The equations compatible with the boundary conditions are as seen in [16].. 3.1.3. τ RR = β̄(R − A)(R − B). (3.3). τ ΘΘ = β̄[3R2 − 2(A + B)R + AB]. (3.4). Invariant formulation. The tensor τ can be represented in current configurations in the form of Σ. Σ = Fτ FT. (3.5). As stated in 2.23, W can be expressed as a combination of both tensors through its invariants, the dependency is written as W = W(I1 ,I5 ). The partial derivatives of the energy-strain function uses the notation Wi = ∂W/∂I i in order to define σ as seen in 2.21. The left Cauchy-Green deformation tensor B was previously seen in 2.13. σ = 2W1 B + 2W1 Σ − qI. 3.2 3.2.1. (3.6). Finite element analysis Geometry and elements. The dimensions of the cylinder tube in the reference configuration are a length L of 150 mm, an outer diameter of 10 mm and a thickness of 0.5 mm. The discretization leaves the model, according to the cylindrical dimensions, with 100 elements in the longitudinal and 7 elements in the thickness; as for the angular dimension, 32 elements are generated when bulging instability is of interest in the analysis and 64 up to 80 for bending. The mesh is composed of three dimensional 8-node elements, noted as C3D8RH in the Abaqus library; these elements are hybrid with constant pressure. The hybrid formulation is necessary since it would not be possible to compute the pressure stress when the element maintains a constant volume, these type of elements include an additional degree of freedom or this matter. The elements also present reduced integration, necessary in order to prevent locking as the material is incompressible..

(36) 18. 3.2.2. Chapter 3. Methodology. Material model. The mechanical constitutive behavior is represented with a formulation code, it is introduced with an Abaqus user subroutine (UMAT) as it can be used with incompressible elastic materials [3]. The material parameters are introduced in the definition of the Cauchy stress tensor σ in 3.6. For the hybrid element used, the pressure stress calculated by the software is derived from the Lagrange multiplier, modifying the Jacobian. Then, the material model depends on the ratio between µ and β, magnitudes of the neo-Hookean material and residual stresses, which can be manipulated for each analysis.. 3.2.3. Analysis description. The nonlinear analysis is a sequence conformed by three distinct steps applied in increments. In an initial step 1, the residual stresses of the material model defined in 3.6 is gradually applied in the initial reference configuration. The increments in this step stops once a stable compatibility is achieved in the deformed cylinder tube. Once step 1 is completed, the cylinder tube ends with initial stresses. For the most part, the introduced magnitudes will be positive which corresponds to tensional stresses in the outer walls. The distribution of radial and azimuthal stresses in the cross-section are similar when compared to the corresponding theoretical formulation in 3.4 and 3.4 as seen in figure 3.1. The cylinder tube will begin with the step 2 of the analysis in the same conditions as it finished in the previous one, this means that the nodes have initial displacements and corresponding stresses. Physiologically, there is a certain degree of pre-stretching in most arteries which is described in terms of the axial stretch ratio λz . The ratio compares the in situ length with an excised one, which ends up being relatively shorter. The in situ conditions are further described in [29]; for reference, the value of this ratio along the length of the aorta has a value of 1.1 < λz < 1.6. In this analysis, the following step 2 sees the introduction of the effects of λz as seen in 2.10. The axial stretch is also applied incrementally starting from the configuration of the previous step 1 and it stops once the final length is reached. Finally, the uniform internal pressure is gradually applied in step 3 until convergence is.

(37) 3.3. Bifurcation analysis. 19. Figure 3.1: Residual stresses τθθ and τrr in the hoop and radial directions along the thickness in the middle cross-section for a value of β = 2.0. The analytical values of these stresses are stated in equations 3.3 and 3.4 and similar results are obtained when compared with the numerical ones. achieved. The mechanical response is represented in terms of the load proportionality factor plot against arc length. For the intended types of instabilities, the boundary conditions at both ends of the model are restricted for hoop displacements in order to establish a necessary symmetry. In the case of arteries, it is appropriate to assume that the axial stretch is fixed [7], thus the boundary conditions in this direction are also restricted. In this project and for step 3, the radial displacement is also considered restricted on both ends of the cylinder tubes, this is independent of the intended type of bifurcation mode in the analyses.. 3.3. Bifurcation analysis. During the analysis, the cylinder tube withstands the combination of a gradual axial stretch up to a given λz in a fist stage and then an internal uniform pressure p in a second one. The structural response is represented in the load proportionality factor (LPF) plot against the arc length. The analysis reaches a bifurcation point when a small increment of the loading actions returns a relative large deformation. Some theoretical formulation, based on equilibrium equations for infinitesimal portions of a membrane subjected to uniform pressure, was developed in [24]. While the.

(38) 20. Chapter 3. Methodology. formulation is based on the membrane approximation, it serves as a way to illustrate the instability conditions. A brief description of the analyzed post-bifurcation modes are part of this section.. 3.3.1. Bulging mode. Regarding the arterial behavior, this type of instability is usually related with the initiation of aneurysms. This instability has certain geometrical properties, as the cross sections of the model do not depart from the initial circular shape, so the it remains a cylinder. The occurring displacements are independent of the angular coordinate θ and these are given as an displacement field. δu = δur (z)er + δuz (z)ez (3.7) Since the formulation considers an incompressible solid, it is possible to implement the notation Ŵ for the energy-strain equation. Ŵ(λz , λθ ) = W(λθ -1 lλz -1 , λθ , λz ). (3.8). According to the membrane approximation, it is possible to simplify the radial stress σrr = 0. Following 2.21 it is possible to write the remaining stress values in terms of the principal stretches (cylindrical coordinates) and the corresponding partial derivative of W. ∂ Ŵ ∂ Ŵ , σzz = λz (3.9) σθθ = λθ ∂λθ ∂λz Then, the compatible uniform pressure p is defined. p=. H Ŵλθ Rλθ λz. (3.10). In an infinitesimal section 3.2 of membrane the equations of equilibrium are established in the axial and radial directions, the system of equations and results are further explained in [16]. The instabilities are a result of complex solutions of the differential equations; the bulging type is given considering a finite length for a first mode with no incremental displacement in the axial direction at either ends of the cylinder. λ2z Ŵλz λz (λ2θ Ŵλθ λθ − λθ Ŵλθ ) − (λθ λz Ŵλθ λz − λθ Ŵλθ )2 + (. 2πR 2 2 ) λθ λz Ŵλz Ŵλz λz = 0 (3.11) L.

(39) 3.3. Bifurcation analysis. 21. Figure 3.2: Acting forces in an infinitesimal section of a membrane. Basis for equilibrium equations in the radial and axial directions [16].. 3.3.2. Bending mode. An instability criteria for the bending motion is also developed with the membrane approximation. For a condition where flexural stiffness is neglected, a section of a cylinder tube with a relative virtual displacement δu0r between the cross sections of both ends is studied. The cylinder and acting forces in the cut sections are represented in figure 3.3. There is an axial load N = 2πhrσzz that generates a moment N δu0r ; there is also the moment of the pressure of the fluid −pπr2 in the cut section. The resultant moment is M . M = N δu0r − pπr2 δu0r (3.12) A stable equilibrium condition exists when M = 0 and a different result implies an instability. The membrane approximation lets us express the force and pressure in terms of the axial and radial components. N = 2πrσzz. (3.13). p = σθθ. h r. (3.14). A bifurcation criteria based on 3.12 is defined for the bending motion [16]. σθθ − 2σzz > 0. 3.3.3. (3.15). Bifurcation in the arc length method. The theoretical formulation 3.11 evidently involves complex mathematical operations in order to solve the problem. In this proposed numerical methodology, a maximum value of the equilibrium curve in the load proportionality factor (LPF) plot against.

(40) 22. Chapter 3. Methodology. Figure 3.3: Interpretation of forces acting in a infinite length membrane for the bending mode [16]. the arc length represents the bifurcation point for a posterior bulging motion. The following plots are already a part of the numerical results obtained but it is important to include a description of the reasoning behind their interpretation. A bulging motion can be seen in 3.4 in which a decrease of the equilibrium path curve is expected after the bifurcation point is reached. The LPF value at the bifurcation point is close to the maximum value of the normalized pressure p that the cylinder tube can withstand before a collapse, in this case is an idealization of an aneurysm rupturing. The obtained curve should be smooth in order to properly capture the conditions of the instability. p = 10 ∗. µ∗H D. (3.16). In order to identify the bifurcation point, a comparison between the hoop displacements of two different points along the length of the cylinder, one situated before and another in the generated bulge needs to be made. These measurements are included in the numerical results chapter. The LPF vs arc length plot also indicates the bifurcation point for a bending mode. With the numerical method, a bending motion is obtained due to an introduced geometrical imperfection consisting of minimal nodal displacements in the middle section of the tube. The triggered instability is noticeable in the corresponding load proportionality factor plot, as seen in figure 3.5, where the bifurcation point does not correspond with a maximum value of the curve. A sudden loss of stiffness is noticeable as the equilibrium path curve deviates from an initial, more regular behavior, that stage corresponds to the instability point where the bifurcation occurs..

(41) 3.3. Bifurcation analysis. Figure 3.4: Load proportionality factor plot for a bulging motion with λz = 1.0.. Figure 3.5: Load proportionality factor plot for a bending motion with λz = 1.0.. 23.

(42) Chapter 4 Numerical Results 4.1. Parameters and conditions. The project aims to illustrate the effects of the principal parameters that define the cylinder tube models and also of the conditions in which their analyses are carried. These parameters are the magnitude β chosen for the material model, the λz stretch to be applied in step 2, an introduced geometrical imperfection in order to capture bending bifurcation and also the discretization for each cylindrical dimension. While the cylinder tube geometry was previously defined with an initial length of 150 mm, alternative models will be generated varying this dimension in order to see how L/R affects the mechanical response. As a result of the proposed formulation of the material model in 3.2, 3.3 and 3.4 the values of stresses obtained will be normalized in terms of µ/2.. 4.1.1. Sensitivity analysis. The analyses for this section were solely focused on the application of step 1, where the magnitude of β is the subject of a sensitivity analysis for cylinder tubes of L = 150 mm; the internal and external radii A and B are maintained constant throughout the project. The discretization of the mesh is such that the thickness is composed of 11 elements. In this analysis, it is possible for Abaqus/Standard to complete the step until a value of β = 5.0, usually in 6 increments. This maximum value could be slightly larger by increasing the number of iterations in each step at the expense of a larger computational cost. 24.

(43) 4.1. Parameters and conditions. 25. Figure 4.1: Values of τθθ along the normalized thickness in the middle cross-section for variable magnitudes of residual stress β. The sensitivity analysis serves to evaluate the limits in which the finite element model can complete the analyses without major inconveniences. It is reasonable to work with lower magnitudes than the mentioned limit considering the remaining steps 2 and 3, therefore a range of 0.0 < β < 3.0 shall be considered in this project. A comparative development of residual stresses along the thickness can be noticed in figure 4.1, the linear formulation is maintained and some negative (compressive) values are also taken into account. Positive values of β implies initial tensile stress values on the exterior walls.. 4.1.2. Effects of boundary conditions. As established in the methodology chapter, the displacements are restricted on both ends of the cylinder tube in the radial direction during the application of internal pressure (step 3); however, the expected physiological behavior should be between a null and a total limitation of displacements. A couple of comparative models are performed for different ratios of L/R in order to observe the variation of stresses considering an alternative boundary condition. The curves included in figure 4.2 the development of stresses in the angular and axial directions for models of two different ratios L/R and in terms of different λz plotted along their length. The average values and distribution are quite similar. The values near the ends of the cylinder are different as an effect of the type of boundary condition applied in the radial direction; however the effect is very localized regarding the general results..

(44) 26. Chapter 4. Numerical Results. (a) L/R = 30 and λz = 1.00.. (b) L/R = 30 and λz = 1.08.. (c) L/R = 30 and λz = 1.12..

(45) 4.1. Parameters and conditions. 27. (d) L/R = 20 and λz = 1.00.. (e) L/R = 20 and λz = 1.08.. Figure 4.2: Average stresses σθθ and σzz of cross-sections along their position in the axial direction. In continuous lines the analyses are performed with restricted displacements (BC 01) in the radial direction, while in dashed lines the displacements are permitted (BC 02). The values are extracted at the corresponding bending bifurcation point for a model with β = 2.0 .. The mechanical response of the models are similar enough for sufficiently long cylinders as the purpose of the project is focused on bifurcation capturing..

(46) 28. 4.2 4.2.1. Chapter 4. Numerical Results. Bifurcation criteria Membrane approximation criteria. A bifurcation criteria was referenced for the membrane approximation of a cylinder tube in 3.15. While some assumptions such as the finite length and thickness of the tube differ in the model, the value of M can still be obtained for a certain analysis in terms of average stresses σθθ and σzz in the middle cross-section. Two analyses are. Figure 4.3: Values of the criteria M = σθθ − 2 × σzz , as seen in equation 3.15, for an average stresses in the middle cross-section. The analysis considers β = 2.0, for a first model λz = 1.0 and a second one λz = 1.40. Positive values σθθ − 2 × σzz > 0 indicate an instability of bending motion while negative ones where obtained for the bulging one.. performed with different values of λz in step 2, in order to obtain the values of M . The results can be seen in figure 4.3 where the development of this criteria through the analysis in terms of the arc-length is plotted. The criteria takes positive values for λz = 1.00, suggesting that the bending mode of bifurcation is expected; the obtained bifurcation is indeed of the bending mode. Then, this numerical result is an indicator of the mechanical behavior but not a defining criteria. The second model at λz = 1.40 gives negative results of the criteria in which the bulging mode of bifurcation was obtained..

(47) 4.2. Bifurcation criteria. 4.2.2. 29. Onset of bifurcation. A qualitative description of each bifurcation mode was given in the previous chapter in terms of the load proportionality factor (LPF) vs arc length plots in figures 3.4 and 3.5 as an expression of the mechanical response during the step 3 of the analyses. The. Figure 4.4: Load proportionality factor against arc length plot for a cylinder tube model at β = 2.0 and λz = 1.0. The first equilibrium path showcases an instability of bending motion, that separates itself from the second one, which is the bulging one in its bifurcation point. The noticed bifurcation point for bulging modes are close to the maximum values of LPF. equilibrium paths included in figure 4.4 illustrate the differences between both bifurcation modes. The first equilibrium path is related to a model restricted to capture bulging that reaches a maximum value of LPF near the bifurcation point. However, when the analysis is restricted to capture the bending mode its equilibrium path deviates from the first one at its bifurcation point, which occurs at a lower LPF value. Under these conditions, bending is the onset of bifurcation for this model. The hoop strain, defined as λθ = r/R when considering the average nodal displacements in the thickness, is an appropriate measurement of the bulging phenomena as described in the methodology chapter. Figure 4.5 describes how this strain varies during a bulging mode analysis in two different sections of the cylinder tube; one of them is located in the middle section and the other one is between the end of the tube and the bulge itself. The strains are similar along the tubes indicating uniform radial displacements, until the normalized pressure values peak near the bifurcation points before the propagation of the bulge. The models included showcase an association between the instability.

(48) 30. Chapter 4. Numerical Results. Figure 4.5: Normalized pressure development against hoop strain λθ for bulging mode analyses at λz = 1.30. The hoop strains considered are in two distinct points along the axial direction, one inside the bulge (in) located in the middle cross-section and another one outside of the bulge (out) in an intermediate cross-section between the middle and the end section of each tube. The curves depart at the bifurcation point, which is near the corresponding maximum normalized pressure. point and a maximum normalized pressure (therefore LPF value). After the bifurcation point, λθ keeps increasing for the bulge following the instability but decreases in the section outside of it that tends to return to values similar to pre-bifurcation results.. 4.3. Distribution of azimuthal and axial stresses. The stress distribution along the length at the bifurcation point is another interest in this project. Given the geometric properties of the models, the radial stresses are considerably smaller and not included in this study. In figure 4.6 a series of plots showcasing different stress distributions in terms of the pre-stretch ratio λz illustrate some differences between the bifurcation modes. The effects that the selected boundary conditions (restricted radial displacements at both ends of the cylinder tube) has on the stress distribution near the end are clearly localized as seen in the previous section of this chapter, specially for the azimuthal stresses which are associated to radial displacements. The stress distributions for models in which their onset of bifurcation is the bending mode, as seen in figures 4.6a, 4.6b, 4.6c and 4.6d tend to be monotonic along the cylinder tube length. The average stresses tend to increase with higher values of λz as is.

(49) 4.3. Distribution of azimuthal and axial stresses. (a) λz = 1.00.. (b) λz = 1.04.. (c) λz = 1.08.. 31.

(50) 32. Chapter 4. Numerical Results. (d) λz = 1.12.. (e) λz = 1.40.. (f) λz = 1.84.. Figure 4.6: Average stresses σθθ and σzz of cross-sections along their position in the axial direction. The values are extracted at their corresponding bending bifurcation points for models with L/R = 30 and β = 2.0..

(51) 4.4. Bifurcation in terms of the axial stretch. 33. related to higher normalized pressure values at the corresponding bifurcation point. Following the same model, the analyses with a bulging mode of bifurcation are pictured in figures 4.6e and 4.6f. A change in the general distribution of stresses is noticeable as peak values for both directions are encountered in the middle cross-section, this is an effect of the radial displacements associated to the propagation of the bulge. The values of σzz tend to increase, relative to σθθ as higher values of λz are used in the analyses. It can be noticed how the peak values switch from stresses in the azimuthal directions to the axial ones between λz = 1.40 and λz = 1.84. This effect is also present for models where bending bifurcation is obtained, but less notorious for the parameter values used in these analyses, particularly the values of λz used as a comparison.. 4.4. Bifurcation in terms of the axial stretch. The existing pre-stretch ratio is the next parameter to be addressed from the problem. The step 2 of the analyses are defined by the chosen λz values. The figure 4.7 plots. Figure 4.7: Normalized pressure values at bifurcation points obtained via models with β = 2.0, L/R = 30 and for axial pre-stretches 1.0 ≤ λz ≤ 2.0. the normalized pressure at each bifurcation point, including models restricted to both bifurcation modes, in terms of λz . The obtained bending bifurcation curve exists in a region defined by 1.0 ≤ λz ≤ 1.12, has an increasing tendency and is the onset of bifurcation. The obtained bulging mode curve has a decreasing tendency. When the bending bifur-.

(52) 34. Chapter 4. Numerical Results. cation curve becomes higher than the bulging one in λz > 1.12, it is the bulging mode that becomes the onset of bifurcation. The pre-stretch value that divides the two bifurcation regions is referred as the transition λz in this project. This numerical value depends on the model parameters and it can be used to describe the general behavior shown in 4.7.. Figure 4.8: Normalized pressure values at bifurcation points obtained with models of L/R = 30 and for axial pre-stretches 1.0 ≤ λz ≤ 2.0.. Figure 4.8 illustrates the onset of bifurcations in terms of the axial stretch for a couple of models at different values of β, the corresponding pressure values have increased in both bifurcation regions for the added curve β = 0.5. The transition value λz is also relatively higher.. 4.4.1. Transition axial stretch. Figure 4.9 includes additional result curves for models of different initial lengths with similar results when bulging is the onset of bifurcation, a constant β used for all the curves. The numerical results are different in the bending mode domain as the transition λz tends to decrease for lower ratios L/R. A decrease in the critical pressure values obtained for increasing values of β can be also noticed. The critical pressure values are similar between analyses that share the values of L/R..

(53) 4.4. Bifurcation in terms of the axial stretch. 35. Figure 4.9: Values of normalized pressure for the onset of bifurcation for β = 2.0 and 1.0 ≤ λz ≤ 2.0.. 4.4.2. Effects of residual stresses. In order to describe the general influence of residual stresses, the following section studies the effects on the analyses in terms of the transition λz values. This approach is chosen as the behavior for the cylinder tube models, of sufficient length, follows a similar pattern as seen in 4.9.. Figure 4.10: Values of normalized pressure at their bifurcation points relative to the values of transition λz . Each value is obtained in terms of a certain β value indicated near its position in the generated curves. The curves are obtained for two distinct ratii L/R.. Figure 4.10 showcases how the magnitude of residual stresses β affect the normal-.

(54) 36. Chapter 4. Numerical Results. ized pressure values at their bifurcation points. The curves are plotted in terms of the transition λz in order to relate this with the previously discussed mechanical response pattern of the models. The curves illustrate that the problem reaches instabilities at lower values of normalized pressure for increasing values of β. It is also noticeable that the pressure values at that bifurcation point decreases in the model with higher L/R ratio. The critical pressure values in terms of the axial stretch are inversely proportional to β, as they decrease for higher values of applied residual stresses as seen in [21].. Figure 4.11: Variation of β and the values of transition λz for two models.. Based on results from the same analyses used for 4.10, a visual explanation of the existing relationship between β and the transition λz is seen in figure 4.11. The curves demonstrate how lower values of transition λz are to be expected as positive residual stresses become higher in magnitude.. 4.5. History of configurations. The numerical results included in the project have been mostly focused on the bifurcation points of the analysis. Additionally, is appropriate to understand the mechanical response of the models on the later stages, therefore a descriptive history of configurations is developed for both bifurcation modes by using a couple of representative analyses. The history of configurations for a model with bulging mode as its onset of bifurcation.

(55) 4.5. History of configurations. 37. (a). (b). (c). (d). (e). Figure 4.12: Analysis sequence of a cylinder tube with a bulging mode of bifurcation with λz = 1.40, L/R = 30 and β = 2.0.. is presented in figure4.12. The increasing radial displacements in the middle of the length translate to higher values of von Mises stresses in the model presented in figure 4.13 which is representative for similarly conditioned analyses. The bulge is enlarged after the bifurcation point is reached as the radial displacements become higher near the middle section. Figure 4.14 showcases the relative size of it as well as the contour of von Mises stresses which in general are lower after the maximum pressure was already sustained in the previous stage shown in 4.13. The critical geometrical state of the cylinder tube indicates an aneurysm rupturing. The models where the onset of bifurcation is bending presents lateral displacements, from the start of the analyses. The peak values appear in the middle cross-section as bending is associated to mechanical buckling for the mode m=1 as seen in 4.15..

(56) 38. Chapter 4. Numerical Results. Figure 4.13: Contour of von Mises stresses at the bifurcation point for parameter values λz = 1.40, L/R = 30 and β = 2.0.. Figure 4.14: Contour of von Mises stresses at a post-bifurcation stage for an analysis with parameter values λz = 1.40, L/R = 30 and β = 2.0. An asymmetric distribution of stresses already exists in the corresponding cross-section before the instability at the bifurcation point. This effect is pictured in the analysis shown in figure 4.15c, with a stress distribution pictured in 4.6c, and contour of von Mises stresses in figure . It also serves as a representation for analyses under similar conditions. Following the lateral displacements, a secondary lateral bulge appears in that direction, this is noticeable in figure 4.15. The bending sequence eventually switches to a more centered bulge for higher values of λz as it becomes nearer to the transition one; in that condition the instabilities are presented as symmetrical bulges in the bulging bifurcation mode domain. The figure 4.17 showcases that the maximum values of von Mises stresses are also present in the generated bulge. A qualitatively observation is that the corresponding LPF values at this stage are higher than the bifurcation values, but do not reach the ones obtained in similar models restricted to bulging mode. Finally, two sequences are included in figure 4.18 that illustrates the similar behavior.

(57) 4.5. History of configurations. (a). (b). 39. (c). (d). (e). Figure 4.15: Analysis sequence of a cylinder tube with a bending mode of bifurcation with λz = 1.08, L/R = 30 and β = 2.0. of models with different radial restrictions, described as BC01 and BC02 from a previous section. Differences can only be noticed at both ends of the cylinder tube as the radial stresses tend to decrease. However, these are not null as radial displacements are permitted during the first two steps of the analysis..

(58) 40. Chapter 4. Numerical Results. Figure 4.16: Contour of von Mises stresses at the bifurcation point for parameter values λz = 1.08, L/R = 30 and β = 2.0.. Figure 4.17: Contour of von Mises stresses at a post-bifurcation stage for an analysis with parameter values λz = 1.08, L/R = 30 and β = 2.0..

(59) 4.5. History of configurations. (a). (b). 41. (c). (d). Figure 4.18: The figures 4.18a and 4.18b are from a bifurcation point and a postbifurcation stage in the BC01 condition with restricted radial displacements in the step 3. The figures 4.18c and 4.18d are from a similar analysis under a BC02 condition with free displacements in the radial direction. The models feature the parameters L/R = 30 and β = 2.0..

(60) Chapter 5 Conclusions The project manages to describe the conditions in which a presented finite element model reaches an instability point during a mechanical analysis, in order to emulate the behavior of arterial tissue in a critical state caused by physiological activity. The material model addresses the dependence of certain parameters for muscle cells that compose arterial walls. Biomechanical properties from soft tissue such as arteries and blood vessel can be reproduced in a computational model as seen in [15], which allows us to identify the effects in their interaction trough a proposed mechanical analysis. This analysis is solved as an idealized boundary value problem, managed with the finite element method by using the software Abaqus/Standard that has the capacity to implement the idealized material model. This facilitates an approach to the problem under different conditions without the necessity of complex mathematical procedures. The initial steps from the analysis establish the required conditions to emulate the in vivo conditions in which arteries work in order to study the mechanical response to the physiological solicitation which is the blood flow. The stability of the model is analyzed with the arc-length method, as the equilibrium path illustrates the mechanical response in terms of loads and displacements. The methodology used in the project relies on the interpretation of the equilibrium curve this powerful criteria allows us to identify the bifurcation points (instabilities) that can be captured with this model. Bulging mode is an instability associated with the propagation of a large bulge in the middle section of a cylinder tube and is related to aneurysm rupturing. After this instability occurs, larger values of applied pressure cannot be sustained. The numerical results explain the conditions in which this mode becomes the onset of bifurcation. The corresponding history of configurations describes a sequence of the radial propagation of the bulge. The maximum normalized pressure values are associated to the. 42.