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

A basic premise in the classical theory of elasticity (see Truesdell and Noll [1]) is that the body has a unique “natural configuration” modulo rigid motion, the “natural configuration” being usually understood as the stress free state. Such a premise is not generally true for most if not all materials, there being numerous other “configurations” in which they can exist naturally. Not all such states may be accessed by a body in a specific process, but that is not to say that such natural configurations do not exist. For instance, a virgin specimen of metal (if such a specimen is possible for after all such a specimen is produced via a complicated manufacturing process), that is in a stress free state, could be subject to sufficiently small deformations by the application of sufficiently small loads, which on the removal of the loads could return to its original stress free state (or “natural configuration”). However, the same body when subjected to a sufficiently large homogenous deformation would not return to its original stress free configuration on the removal of the load, as it might have undergone “plastic” deformation. A piece of steel, for instance, can undergo classical slip, or twin, or undergo solid to solid phase change on the application of loads, or it could melt due to an increase in temperature. At the end of any one of these above mentioned processes, the piece of steel would be associated with a different natural configuration from its initial natural configuration. In order to model the response of materials undergoing such processes, it is imperative to explicitly take cognizance of the fact that different natural configurations are accessed during such processes.

On the Modelling of Dissipative Processes

Electronic version of an article published as "IMA Journal of applied mathematics", vol. 79, no 5, 2014, p. 778-789. DOI: 10.1093/imamat/hxt050

/pdf/analysis-of-the-equations-governing-the-motion-of-a-o8l7c167nx.pdf

Analysis of the equations governing the motion of a degrading elastic solid due to diffusion of a fluid

A thermodynamic framework for additive manufacturing of crystallizing polymers, Part II: Simulation of the printing of a stent

In this paper, we study the stress concentration around a circular rigid inclusion in an incompressible anisotropic homogeneous nonlinearly elastic thin sheet subjected to in-plane finite deformation uniaxial extension. Two classes of material symmetry are considered: transverse isotropy and orthotropy. For the case of transverse isotropy, a three-parameter constitutive relation that under infinitesimal deformations reduces appropriately to the classical incompressible linearized elastic transversely isotropic constitutive relation is studied. Similarly, a six-parameter constitutive relation is considered for the case of orthotropic material symmetry. These constitutive relations may be suitable to describe the response of transversely isotropic and orthotropic bodies subjected to moderate deformations.

Stress concentration around a circular rigid inclusion in incompressible nonlinearly elastic transversely isotropic and orthotropic thin sheets under uniaxial extension

Classical hydrodynamics and aerodynamics concern fluids having linear viscosity or none at all. We defined these fluids in Chapter 4; in Chapters 5 and 7 we referred to them many times as special instances. Thus, in one sense, we have studied them already, but now we consider some of their properties that seem, as yet at least, peculiar to them, not to be understood easily as being merely the most special examples of more general ideas and flows.

Kumbakonam R. Rajagopal

Papers

On the Modelling of Dissipative Processes

Analysis of the equations governing the motion of a degrading elastic solid due to diffusion of a fluid

A thermodynamic framework for additive manufacturing of crystallizing polymers, Part II: Simulation of the printing of a stent

Stress concentration around a circular rigid inclusion in incompressible nonlinearly elastic transversely isotropic and orthotropic thin sheets under uniaxial extension

Navier-Stokes Fluids