In this paper we develop a new constitutive law for the description of the (passive) mechanical response of arterial tissue. The artery is modeled as a thick-walled nonlinearly elastic circular cylindrical tube consisting of two layers corresponding to the media and adventitia (the solid mechanically relevant layers in healthy tissue). Each layer is treated as a fiber-reinforced material with the fibers corresponding to the collagenous component of the material and symmetrically disposed with respect to the cylinder axis. The resulting constitutive law is orthotropic in each layer. Fiber orientations obtained from a statistical analysis of histological sections from each arterial layer are used. A specific form of the law, which requires only three material parameters for each layer, is used to study the response of an artery under combined axial extension, inflation and torsion. The characteristic and very important residual stress in an artery in vitro is accounted for by assuming that the natural (unstressed and unstrained) configuration of the material corresponds to an open sector of a tube, which is then closed by an initial bending to form a load-free, but stressed, circular cylindrical configuration prior to application of the extension, inflation and torsion. The effect of residual stress on the stress distribution through the deformed arterial wall in the physiological state is examined. The model is fitted to available data on arteries and its predictions are assessed for the considered combined loadings. It is explained how the new model is designed to avoid certain mechanical, mathematical and computational deficiencies evident in currently available phenomenological models. A critical review of these models is provided by way of background to the development of the new model.

https://hal.archives-ouvertes.fr/hal-01297725/document

A new constitutive framework for arterial wall mechanics and a comparative study of material models

Recent advances in theory and experimen- tation motivate a thorough reassessment of the physics of debris flows. Analyses of flows of dry, granular solids and solid-fluid mixtures provide a foundation for a com- prehensive debris flow theory, and experiments provide data that reveal the strengths and limitations of theoret- ical models. Both debris flow materials and dry granular materials can sustain shear stresses while remaining stat- ic; both can deform in a slow, tranquil mode character- ized by enduring, frictional grain contacts; and both can flow in a more rapid, agitated mode characterized by brief, inelastic grain collisions. In debris flows, however, pore fluid that is highly viscous and nearly incompress- ible, composed of water with suspended silt and clay, can strongly mediate intergranular friction and collisions. Grain friction, grain collisions, and viscous fluid flow may transfer significant momentum simultaneously. Both the vibrational kinetic energy of solid grains (mea- sured by a quantity termed the granular temperature) and the pressure of the intervening pore fluid facilitate motion of grains past one another, thereby enhancing debris flow mobility. Granular temperature arises from conversion of flow translational energy to grain vibra- tional energy, a process that depends on shear rates, grain properties, boundary conditions, and the ambient fluid viscosity and pressure. Pore fluid pressures that exceed static equilibrium pressures result from local or global debris contraction. Like larger, natural debris flows, experimental debris flows of ;10 m 3 of poorly sorted, water-saturated sediment invariably move as an unsteady surge or series of surges. Measurements at the base of experimental flows show that coarse-grained surge fronts have little or no pore fluid pressure. In contrast, finer-grained, thoroughly saturated debris be- hind surge fronts is nearly liquefied by high pore pres- sure, which persists owing to the great compressibility and moderate permeability of the debris. Realistic mod- els of debris flows therefore require equations that sim- ulate inertial motion of surges in which high-resistance fronts dominated by solid forces impede the motion of low-resistance tails more strongly influenced by fluid forces. Furthermore, because debris flows characteristi- cally originate as nearly rigid sediment masses, trans- form at least partly to liquefied flows, and then trans- form again to nearly rigid deposits, acceptable models must simulate an evolution of material behavior without invoking preternatural changes in material properties. A simple model that satisfies most of these criteria uses depth-averaged equations of motion patterned after those of the Savage-Hutter theory for gravity-driven flow of dry granular masses but generalized to include the effects of viscous pore fluid with varying pressure. These equations can describe a spectrum of debris flow behav- iors intermediate between those of wet rock avalanches and sediment-laden water floods. With appropriate pore pressure distributions the equations yield numerical so- lutions that successfully predict unsteady, nonuniform motion of experimental debris flows.

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The physics of debris flows

羟氨苄青霉素引起大疱性类天疱疮样疹

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

Constitutive relations are fundamental to the solution of problems in continuum mechanics, and are required in the study of, for example, mechanically dominated clinical interventions involving soft biological tissues. Structural continuum constitutive models of arterial layers integrate information about the tissue morphology and therefore allow investigation of the interrelation between structure and function in response to mechanical loading. Collagen fibres are key ingredients in the structure of arteries. In the media (the middle layer of the artery wall) they are arranged in two helically distributed families with a small pitch and very little dispersion in their orientation (i.e. they are aligned quite close to the circumferential direction). By contrast, in the adventitial and intimal layers, the orientation of the collagen fibres is dispersed, as shown by polarized light microscopy of stained arterial tissue. As a result, continuum models that do not account for the dispersion are not able to capture accurately the stress–strain response of these layers. The purpose of this paper, therefore, is to develop a structural continuum framework that is able to represent the dispersion of the collagen fibre orientation. This then allows the development of a new hyperelastic free-energy function that is particularly suited for representing the anisotropic elastic properties of adventitial and intimal layers of arterial walls, and is a generalization of the fibre-reinforced structural model introduced by Holzapfel & Gasser (Holzapfel & Gasser 2001 Comput. Meth. Appl. Mech. Eng. 190, 4379–4403) and Holzapfel et al. (Holzapfel et al. 2000 J. Elast. 61, 1–48). The model incorporates an additional scalar structure parameter that characterizes the dispersed collagen orientation. An efficient finite element implementation of the model is then presented and numerical examples show that the dispersion of the orientation of collagen fibres in the adventitia of human iliac arteries has a significant effect on their mechanical response.

https://royalsocietypublishing.org/doi/pdf/10.1098/rsif.2005.0073

Hyperelastic modelling of arterial layers with distributed collagen fibre orientations

Preface- General References- Bodies, Configurations, and Motions- Kinematics and Basic Laws- Constitutive Equations, Reduced Constitutive Equations, and Internal Constraints- Simple Fluids- Flows of Incompressible Fluids in General- Some Flows of Particular Nonlinear Fluids- Some Flows of Fluids of Grade 2- Navier-Stokes Fluids- Incompressible Euler Fluids- Compressible Euler Fluids- Singular Surfaces and Waves- Some Elementary Results from Real Analysis- Index

An Introduction to the Mechanics of Fluids

When a sheet of polymer is extruded continuously from a die, it entrains the ambient fluid and a boundary layer develops. Such a boundary layer is markedly different from that in the Blasius flow past a flat plate in that the boundary layer grows in the direction of the motion of the sheet, starting at the die. Sakiadis [1]–[3] was the first to study such a boundary layer flow due to a rigid flat continuous surface moving in its own plane. Later, Erickson, Fan & Fox [4] studied the boundary layer due to the motion of a porous flat plate when the transverse velocity at the surface is non-zero.

On the Uniqueness of Flow of a Navier-Stokes Fluid Due to a Stretching Boundary

Both biochemical and mechanical factors have to be taken into account if a meaningful model for the formation, growth, and lysis of clots in flowing blood is to be developed. Most models that are curr

/pdf/a-model-for-the-formation-and-lysis-of-blood-clots-npvf1a68jz.pdf

A Model for the Formation and Lysis of Blood Clots

This chapter primarily deals with internal, isothermal, unsteady flows of a class of incompressible fluids with both constant and shear or pressure dependent viscosity that includes the Navier–Stokes fluid as a special subclass. We begin with a description of fluids within the framework of a continuum. We then discuss various ways in which the response of a fluid can depart from that of a Navier–Stokes fluid. Next, we introduce a general thermodynamic framework that has been successful in describing the disparate response of continua that includes those of inelasticity, solid-to-solid transformation, viscoelasticity, granular materials, blood and asphalt rheology, etc. Here, it leads to a novel derivation of the constitutive equation for the Cauchy stress for fluids with constant, or shear and/or pressure, or density dependent viscosity within a full thermomechanical setting. One advantage of this approach consists in a transparent treatment of the constraint of incompressibility. We then concentrate on the mathematical analysis of three-dimensional unsteady flows of fluids with shear dependent viscosity that includes the Navier–Stokes model and Ladyzhenskaya's model as special cases. We are interested in the issues connected with mathematical self-consistency of the models, i.e., we are interested in knowing whether (1) flows exist for reasonable, but arbitrary initial data and all instants of time, (2) flows are uniquely determined, (3) the velocity is bounded and (4) the long-time behavior of all possible flows can be captured by a finite-dimensional, small (compact) set attracting all flow trajectories exponentially. For simplicity, we eliminate the choice of boundary conditions and their influence on the flows by assuming that all functions are spatially periodic with zero mean value over a periodic cell. All these results can however be extended to internal flows wherein the tangential component of the velocity satisfies Navier's slip at the boundary. Most of the results also hold for the no-slip boundary condition. While the mathematical consistency understood in the above sense for the Navier–Stokes model in three dimensions has not been established as yet, we will show that Ladyzhenskaya's model and some of its generalization enjoy all above characteristics for a certain range of parameters. We also discuss briefly further results related to generalizations of the Navier–Stokes equations.

Mathematical Issues Concerning the Navier–Stokes Equations and Some of Its Generalizations

The purpose of this brief note is to develop fully Eulerian, implicit constitutive equations for the mechanical response of a class of materials that do not dissipate mechanical work in any process. We show that such materials can be modelled by obtaining a form for the Helmholtz potential as a function of the current mass density, the Cauchy stress and certain other parameters that capture anisotropic response. The resulting constitutive equations are of the form ![Graphic][1] , where ![Graphic][2] and ![Graphic][3] are functions of the state variables of the system. The class of materials that can be obtained from such a constitutive relation is considerably more general than conventional Green-elastic hyperelastic materials. Such response functions may be suitable for the modelling of biological tissue where, due to the constant remodelling that takes place, there may be no physical meaning to a ‘reference configuration’.

 [1]: /embed/inline-graphic-1.gif
 [2]: /embed/inline-graphic-2.gif
 [3]: /embed/inline-graphic-3.gif

Kumbakonam R. Rajagopal

Papers

An Introduction to the Mechanics of Fluids

On the Uniqueness of Flow of a Navier-Stokes Fluid Due to a Stretching Boundary

A Model for the Formation and Lysis of Blood Clots

Mathematical Issues Concerning the Navier–Stokes Equations and Some of Its Generalizations

On a class of non-dissipative materials that are not hyperelastic