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

In this paper we study the state of stress and strain in infinite elastic slabs of nonlinear viscoelastic solids containing elliptic holes subject to an uni-axial as well as a bi-axial state of stress. The geometry affords one to get some inkling concerning the states of stress and strain in bodies containing a crack by obtaining the limit of the solutions as the aspect ratio (in this case the ratio of the minor axis to the major axis) of the ellipse tends to zero. We consider two classes of non-linear viscoelastic bodies, the classical incompressible Kelvin–Voigt solid (Thomson in R Soc Lond 14:289–297, 1865; Voigt in Ann Phys 283(12):671–693, 1892) and a generalization of a compressible model due to Gent (Rubber Chem Technol 69(1):59–61, 1996). We also study for the sake of comparison the case of a nonlinear neo-Hookean elastic solid with an elliptic hole.

Deformations of infinite slabs of non-linear viscoelastic solids containing an elliptic hole

This manuscript presents a systematic investigation of the response of a Burgers’ viscoelastic fluid model with stress-dependent material parameters. Such a model has been used extensively in geomechanics as well as to describe the response of materials like asphalt. The stress, strain, and time relation of Burgers’ fluid model is expressed with second order differential operators applied to the stress and strain. The nonlinearity is due to the stress dependence of the material parameters, i.e., the fluid viscosity and the parameter related to the characteristic time. We impose discontinuity conditions, whose necessity was not recognized until the recent work of Prusa and Rajagopal (2011), for the stress and strain and also for the stress- and strain rates such that we satisfy the following assumptions: if there is a jump discontinuity in strain there should be a jump discontinuity in the corresponding stress, and if there is a small change in strain there ought to be a small change in the corresponding stress. These assumptions are also applied when a stress history is considered as input. We present constraints on the stress-dependent material functions in order to obtain a physically meaningful solution that describes the viscoelastic response of materials. We also allow different responses for tension and compression and perform parametric studies geared towards obtaining an understanding of the effect of nonlinear stress-dependent functions on the stress-relaxation and creep deformation under various loading histories. It is important to recognize that methods such as time–temperature superposition or the use of Laplace transforms that are useful in the case of the classical linear viscoelastic material will not work in the case of the non-linear model considered in this paper.

On the response of Burgers’ fluid and its generalizations with pressure dependent moduli

Studying the mechanics of aortic tissue is a crucial component in understanding its behavior under healthy as well as diseased conditions. Wall shear stress and circumferential stress have been largely accepted as significant factors in arterial growth and remodeling as a response to changes in flow and pressure. But, experimental studies on aortic tissues have largely focused on uniaxial and biaxial tests which are more suited for investigating circumferential stress. This may be explained by the inherent convenience of gripping tissue and then applying deformations in the uniaxial and biaxial modes. As such, the behavior of aortic tissue under shear has been left relatively unstudied. We propose to study the response of porcine aortic wall tissue under cyclic constant torsional shear strain rate for high amplitude (50%) and at different shear strain rates (4%/s and 40%/s). Three to four 12.5 mm diameter samples are obtained from the descending porcine aorta. Initial results clearly indicate a non linear response for the moment as a function of the angle of twist while many popular models predict a linear response for the arterial wall even under large shear strain.

High Amplitude Torsional Shear of Porcine Thoracic Aorta

We formulate and consider the problem of an inextensible, unshearable, viscoelastic rod, with evolving natural configuration, moving on a plane. We prove that the dynamic equations describing quasistatic motion of an Eulerian strut, an infinite dimensional dynamical system, are globally well-posed. For every value of the terminal thrust, these equations contain a smooth embedded curve of static solutions (equilibrium points). We characterize the spectrum of the linearized equations about an arbitrary equilibrium point, and using this information and a convergence result for dynamical systems due to Brunovský and Polácik, we prove that every solution to the quasistatic equations of motion converges to an equilibrium point as time goes to infinity. 

/pdf/on-evolving-natural-curvature-for-an-inextensible-crbsu5o8.pdf

On Evolving Natural Curvature for an Inextensible, Unshearable, Viscoelastic Rod

The research described in this dissertation deals with an open problem concerning the synthesis of controllers of fixed order and structure. This problem is encountered in a variety of applications. Simply put, the problem may be put as the determination of the set, S of controller parameter vectors, K = (k 1,k2,...,kl), that render Hurwitz a family (indexed by F ) of complex polynomials of the form {P0( s,α) + i=1
l
Pi(s,α) ki, α ∈ F }, where the polynomials Pj(s,α), j = 0,...,l are given data. They are specified by the plant to be controlled, the structure of the controller desired and the performance that the controllers are expected to achieve. Simple examples indicate that the set S can be non-convex and even be disconnected. 
While the determination of the non-emptiness of S is decidable and amenable to methods such as the quantifier elimination scheme, such methods have not been computationally tractable and more importantly, do not provide a reasonable approximation for the set of controllers. Practical applications require the construction of a set of controllers that will enable a control engineer to check the satisfaction of performance criteria that may not be mathematically well characterized. The transient performance criteria often fall into this category. 
From the practical viewpoint of the construction of approximations for S , this dissertation is different from earlier work in the literature on this problem. A novel feature of the proposed algorithm is the exploitation of the interlacing property of Hurwitz polynomials to provide arbitrarily tight outer and inner approximation to S . The approximation is given in terms of the union of polyhedral sets which are constructed systematically using the Hermite-Biehler theorem and the generalizations of the Descartes' rule of signs.

Kumbakonam R. Rajagopal

Papers

Deformations of infinite slabs of non-linear viscoelastic solids containing an elliptic hole

On the response of Burgers’ fluid and its generalizations with pressure dependent moduli

High Amplitude Torsional Shear of Porcine Thoracic Aorta

On Evolving Natural Curvature for an Inextensible, Unshearable, Viscoelastic Rod

A new computational approach to the synthesis of fixed order controllers