$$ \newcommand{\partd}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\partdd}[2]{\frac{\partial^{2} #1}{\partial {#2}^{2}}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\Int}{\int\limits} \newcommand{\D}{\displaystyle} \newcommand{\ie}{\textit{i.e. }} \newcommand{\dA}{\; \mbox{dA}} \newcommand{\dz}{\; \mbox{dz}} \newcommand{\tr}{\mathrm{tr}} \renewcommand{\eqref}[1]{Eq.~(\ref{#1})} \newcommand{\reqs}[2]{\req{#1} and \reqand{#2}} \newcommand{\rthreeeqs}[3]{Eqs.~(\ref{#1}), (\ref{#2}), and (\ref{#3})} \newcommand{\norm}[1]{\left\lvert#1\right\rvert} $$

 

 

 

Cardiovascular biomechanics

Leif Rune Hellevik


Nov 17, 2015


Table of contents

            Nomenclature
Dynamics
      Kinematics
            Extensive and intensive properties
            Notation and conventions
            Reynolds transport theorem of a moving control volume
            The material derivative of an extensive property
      Conservation of mass
      Equations of motion
            Coordinate stresses
            Example 1: Uniaxial state of stress
            Example 2: Pure shear stress state
            Cauchy's stress theorem and the Cauchy stress tensor
            Example 3: Example: Fluid at rest: Isotropic state of stress
      Cauchy's equations of motion
            Derivation of Cauchy's equations of motion
            Example 4: The hydrostatic pressure distribution
            Example 5: Cauchy equations in cylindrical coordinates
      Stress analysis
            Principal stresses
            Maximum shear stress
            Planar stress
            Example 6: Biaxial state of stress for thin-walled structures
Deformation
      Measures of strain
      The Green strain tensor
      Small strains and small deformations
            Small strains
            Small deformations
            Principal strains
            Small strains in a surface
            Example 7: Strain rosettes
      Strain rates and rates of rotation
            Example 8: Simple shear flow. Rectilinear rotational flow
Elasticity
            Fundamental properties of elastic materials
      Isotropic and linearly elastic materials
            The Hookean solid
            Navier equations
            2D theory of elasticity
            Example 9: Spherical shell of steel
            Plane displacement
            Example 10: Plane displacements for a thick walled cylinder
      Mechanical energy balance
      Hyperelastic materials and strain energy
            Hyperelasticity for large derformations
            Stress tensors for large deformations
            Isotropic hyperelastic materials
            Example 11: Thin sheet of incompressible material
            Example 12: The Mooney-Rivlin material
      Anisotropic Materials
      Waves in elastic materials
            Plane elastic waves
            Exercise 1: The generalized Hooke's law
            Exercise 2: Invariants
            Exercise 3: Shear modulus
Fluid mechanics
      Introduction
            Fundamental concepts in fluid mechanics
      Conservation of mass
      Inviscid fluids
            Example 13: Sound waves
      Linear viscous fluids
            Simple shear flow
            The Navier-Stokes equations
            About the NS equations
            Example 14: Flow between parallel planes
            Example 15: Laminar pipe flow
      Generalized Newtonian model
            Example 16: Stationary pipeflow for GNF
            Example 17: Power law for steady pipeflow
      Pulsatile flow in straight tubes
            Wall shear stress for pulsatile flow in straight tubes
            Longitudinal impedance for pulsatile flow in straight tubes
The cardiovascular system
      Pressure and flow in the cardiovascular system
            Arterial anatomy
            Compliance and distensibility
            Mathematical representation of periodic pressure and flow
            Vascular impedance
      Lumped models
            The Windkessel model
            The three-element Windkessel model
            Methods for estimation of total arterial compliance
            Example 18: Estimation of total arterial compliance with the TDM method
            Exercise 4: Windkessel model
Blood flow in compliant vessels
      Poiseuille flow in a compliant vessel
      Infinitesimal derivation of the 1D governing equations for a compliant vessel
            Conservation of mass
            The momentum equation
      Integral derivation of the 1D governing equations for a compliant vessel
            1D transport equation
            Mass conservation
            Momentum equation
            Example 19: Momentum equations for invicid flow
            Example 20: Momentum equations for polynomial velocity profiles
            Linearized and inviscid wave equations
            Characteristic impedance
            Progressive waves superimposed on steady flow
      Input impedance
      Wave reflections
      General equations with reflection and friction
      Wave separation
      Wave travel and reflection
      Networks 1D compliant vessels
            Lumped heart model: varying elastance model
            Nonlinear wave separation
      Fluid structure interaction for small deformations in Hookean vessel
            The governing equations for the Hookean vessel
            The governing equations for the fluid
            Coupling of structure and fluid
Appendix
      Trigonometric relations
      Vectors
      Orthogonal Coordinates
            Gradient, divergence and rotation in general orthogonal coordinates
      Integral Theorems
      Properties of Bessel functions
      Bibliography

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