Showing papers by "Kumbakonam R. Rajagopal published in 2007"

On a hierarchy of approximate models for flows of incompressible fluids through porous solids

Abstract: The celebrated equations due to Fick and Darcy are approximations that can be obtained systematically on the basis of numerous assumptions within the context of mixture theory; the equations however not having been developed in such a manner by Fick or Darcy. Relaxing the assumptions made in deriving these equations via mixture theory selectively leads to a hierarchy of mathematical models and it can be shown that popular models due to Brinkman, Biot and many others can be obtained via various approximations. It is shown that a variety of other generalizations are possible in addition to those that are currently in favor, and these might be appropriate for describing interesting technological applications.

The elasticity of elasticity

Abstract: In this note we assert that the usual interpretation of what one means by “elasticity” is much too insular and illustrates our thesis by introducing implicit constitutive theories that can describe the non-dissipative response of solids. There is another important aspect to the introduction of such an implicit approach to the non-dissipative response of solids, the development of a hierarchy of approximations wherein, while the strains are infinitesimal the relationship between the stress and the linearized strain is non-linear. Such approximations would not be logically consistent within the context of explicit theories of Cauchy elasticity or Green elasticity that are currently popular.

On the response of non-dissipative solids

Abstract: In addition to being incapable of dissipation, in any process that it is subject to, there are other tacit requirements that a classical elastic body has to meet. The class of solids that are incapable of dissipation is far richer than the class of bodies that is usually understood as being elastic. We show that, unlike the case of a classical elastic body, the stress in non-dissipative bodies is not necessarily derivable from a stored energy that depends only on the deformation gradient.

Navier's slip and evolutionary Navier-Stokes like systems with pressure and shear-rate dependent viscosity

Abstract: There is compelling experimental evidence for the viscosity of a fluid to depend on the shear rate as well as the mean normal stress (pressure). Moreover, while the viscosity can vary by several orders of magnitude the density suffers very minor variation, when the range of the pressure is sufficiently large, thereby providing justification for considering the fluid as being incompressible while at the same time possessing a viscosity that is dependent on the pressure. In this article we investigate the mathematical properties of internal unsteady three-dimensional flows of such fluids subject to Navier's slip at the boundary. We establish the long-time existence of a weak solution for large data provided that the viscosity depends on the shear rate and the pressure in a suitably specified manner. This specific relationship however includes the classical Navier-Stokes fluids and power law fluids (with power law index r - 2, r ≤ 2) as special cases. Even for these special cases, the existence results that are being presented are new.

Towards an understanding of the mechanics underlying aortic dissection

Abstract: Acute aortic dissection and associated aortic catastrophes are among the most devastating forms of cardiovascular disease, with a remarkably high morbidity and mortality despite current medical and surgical treatment. The mechanics underlying aortic dissection are incompletely understood, and a further understanding of the relevant fluid and solid mechanics may yield not only a better appreciation of its pathogenesis, but also the development of improved diagnostic and therapeutic strategies. After illustrating some of the inadequacies with respect to the extant work on the mechanics of aortic dissection, we alternatively postulate that the clinical hemodynamic disturbances that render the aorta susceptible to the initiation of dissection are principally elevated maximum systolic and mean aortic blood pressure, whereas the hemodynamic disturbances that facilitate propagation of dissection are principally elevated pulse pressure and heart rate. Furthermore, abnormal aortic mechanical properties and/or geometry are requisite for dissection to occur. Specifically, we propose that the degree of anisotropy will directly influence the probability of future aortic dissection. Imaging of the aorta may provide information regarding aortic anisotropy and geometry, and in combination with a hemodynamic risk assessment, has the potential to be able to prospectively identify patients at high risk for future aortic dissection thereby facilitating prophylactic intervention. The aim of the paper is to identify the main mechanical issues that have a bearing on aortic dissection, and to suggest an appropriate mathematical model for describing the problem.

Modeling of Biological Materials

Abstract: Preface Rheology of Living Materials / R. Chotard-Ghodsnia and C. Verdier Biochemical and Biomechanical Aspects of Blood Flow / M. Thiriet Theoretical Modeling of Enlarging Intracranial Aneurysms / S. Baeck, K.R. Rajagopal, and J.D. Humphrey Theoretical Modeling of Cyclically Loaded, Biodegradeable Cylinders / J.S. Soares, J.E. Moore, Jr., and K.R. Rajagopal Regulation of Hemostatic System Function by Biochemical and Mechanical Factors / K. Rajagopal and J. Lawson Mechanical Properties of Human Mineralized Connective Tissues / R. De Santis, L. Ambrosio, F. Mollica, P. Netti, and L. Nicolais Mechanics in Tumor Growth / L. Graziano and L. Preziosi Inhomogeneities in Biological Membranes / R. Rosso and E.G. Virga

Chapter 7 - Mathematical Properties of the Solutions to the Equations Governing the Flow of Fluids with Pressure and Shear Rate Dependent Viscosities*

Abstract: A meaningful discussion of the mathematical properties of the equations governing the flow of fluids requires a proper understanding of what is meant by a fluid as well as a clear understanding of the nature of the specific fluid. It is impossible to provide a definition of fluid, without the definition's inadequacy being laid bare with an easy counterexample. Many of the definitions, including those in renowned dictionaries are circular; a fluid being defined as a material that flows and flow being defined as an innate property of the fluid. For the purposes of this chapter a body as a fluid is regarded, if in the time scale of observation of interest, it undergoes a flow that is discernible to the naked eye because of the application of a shear stress (that can be measured with the aid of reasonably unsophisticated instruments; i.e., the forces in question are robust, not mere picoNewtons). Based on how they are constituted, different fluids respond differently to the application of external stimuli. Many mathematical models have been developed to describe the diverse response exhibited by fluids, but Navier–Stokes fluid model enjoys a central place amongst them.

A note on the flow through porous solids at high pressures

Abstract: In this study we propose an equation for the flow through a porous solid that includes the equations due to Darcy and Brinkman as special sub-cases. The model is particularly well suited for describing the flow when the range of the pressure is sufficiently large, as it takes into account the possibility that the viscosity and the drag coefficient can be functions of the pressure. We can see a marked departure in the pressure field with respect to the classical Darcy solution with a narrow boundary layer within which the pressure suffers most of its drop.

Incompressible Rate Type Fluids with Pressure and Shear-Rate Dependent Material Moduli

Abstract: Rate type constitutive theories are developed for describing the response of inhomogeneous fluids whose material properties can depend upon the shear rate and the mean normal stress, within a general thermodynamic setting. The classical Navier–Stokes fluid and the power-law fluid are special subclasses of the rate type fluids that have been developed. The models that have been obtained are particularly useful in describing the behavior of biological and geological fluids, and food products in view of their inherent inhomogeneity.

The status of the K-BKZ model within the framework of materials with multiple natural configurations

Abstract: In this paper we show how the K-BKZ model fits in within a thermodynamic framework for describing the response of materials undergoing dissipative processes. The K-BKZ model is shown to arise naturally within this framework by choosing appropriate forms for the stored energy and the rate of dissipation for describing the material. It is also shown that by relaxing some of the assumptions that lead to a K-BKZ model, it is possible to derive a host of other fluid models that are generalizations of the K-BKZ model.

Theoretical Modeling of Cyclically Loaded, Biodegradable Cylinders

Abstract: The adaptation of fully biodegradable stents, thought to be the next revolution in minimally invasive cardiovascular interventions, is supported by recent findings in cardiovascular medicine concerning human coronaries and the likelihood of their deployment has been made possible by advances in polymer engineering. The main potential advantages of biodegradable polymeric stents are: (1) the stent can degrade and transfer the load to the healing artery wall which allows favorable remodeling, and (2) the size of the drug reservoir is dramatically increased. The in-stent restenotic response usually happens within the first six months, thus a fully biodegradable stent can fulfill the mission of restoring flow while mitigating the probability of long-term complications. However, it is a key concern that the stent not degrade away too soon, or develop structural instabilities due to faster degradation in key portions of the stent. We present here a preliminary model of the mechanics of a loaded, biodegradable cylindrical structure. The eventual goal of this research is to provide a means of predicting the structural stability of biodegradable stents

Modeling materials with a stretching threshold

Abstract: We examine the dynamics of materials characterized by the presence of a deformation threshold beyond which no deformation is possible. The class of bodies that we are interested in studying are described by an implicit constitutive relationship between the Cauchy stress and the deformation gradient. A specific one-dimensional dynamical problem is studied, showing that the mathematical model takes the form of a hyperbolic free boundary problem in which the free boundary conditions can be of two different types, selected according to whether the stress is continuous at the interface (separating the deformable region from the fully strained region), or whether it is discontinuous. Both situations have been analyzed. Sample numerical computations are carried out using data that are relevant to biological materials. A comparison with the problem obtained from a limiting procedure for a constitutive model with a piecewise linear elastic response is performed, showing a very interesting feature, namely that the limit does not lead to the solution of the model with a threshold. This is however not surprising as the latter exhibits dissipative behavior.

Regulation of Hemostatic System Function by Biochemical and Mechanical Factors

Abstract: The mammalian hemostatic system has evolved to accomplish the task of sealing defects in the cardiovascular system. Hemostasis occurs in and around a disruption in a vascular conduit through which blood normally flows, and is characterized by the localized formation of thrombus. Consequently, the process of hemostasis is influenced by: (1) the biochemical properties of the cellular and soluble components of the hemostatic system, counterregulatory networks, and the vascular conduit; (2) the local hemodynamic conditions, which regulate the influx and efflux of substrates, cofactors, and catalysts, and which also impose loads on the forming clot; and (3) the local mechanical properties of the vasculature. We review the components of the hemostatic and negative regulatory systems and their biochemical functions, and the roles that local hemodynamics play in the regulation of hemostasis

An Experimental Investigation into the Influence of Fillers on the Development of Normal Stresses and Stress Relaxation in Asphalt Mixtures Due to Torsion

Abstract: This study documents the measurement of normal stresses and stress relaxation in sand-asphalt mixtures fabricated with different fillers and asphalts during torsion. Hydrated lime and limestone fillers and asphalts graded as PG64-22 and AC-30 (from Sinclair (Wyoming), Crown (Nevada), and Crown (Canada)) are used in the fabrication of the sand-asphalt mixtures. The specimens are tested in a torsional rheometer. The experimental results clearly show that the normal stresses that are developed are quite significant even for specimens tested at very low rotational rates. Also, asphalts from different sources show differences in peak normal stresses and in their relaxation pattern. The measurement of significant normal stresses is a reflection of the nonlinear character of the material and warrants the development of nonlinear constitutive models for describing their behavior.

On Some Finite Deformations of Inhomogeneous Compressible Elastic Solids

Abstract: We study several simple boundary value problems for a special class of inhomogeneous, isotropie, compressible elastic solids: inflation of a spherical shell; bending, stretching and shearing of a rectangular block; and straightening, stretching and shearing of a sector of a hollow cylinder. The purpose of the study is twofold: to establish new exact solutions to boundary value problems concerning inhomogeneous, isotropie, compressible elastic solids that have technological relevance (to the deformation of layered composites); and to reaffirm a thesis concerning an inherent difficulty with regard to the homogenization techniques that are in vogue that appeal to the stored energy in the homogenized body being the same as that for the inhomogeneous body. In addition to establishing new exact solutions for a class of inhomogeneous elastic solids, the study reinforces the thesis of Saravanan and Rajagopal that great care ought to be exercised in approximating even a very mildly inhomogeneous body by an equivalent homogeneous body when large deformations are involved.

A new computational approach to the synthesis of fixed order controllers

Abstract: 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.

Constitutive Model of Biodegradable Non-Linear Polymeric Materials for Applications in the Biomedical Field

Abstract: Synthetic biodegradable polymers have seen a dramatic increase in their availability and utilization over the last few decades. The first reported biomedical application of biodegradable polymers was during the 70s in biodegradable sutures and to date, it remains as the most widespread usage of this family of materials. Biodegradable polymers have also been proven to be effective carriers in local drug delivery therapies and are widely used as a primary constituent of scaffolds in tissue engineering applications.© 2007 ASME

A model for the kinetics of thermal damage of biological membranes

Abstract: En The increased use of heating in many medical specialties is driven by the availability of new devices, not an understanding of the underlying physics. Whereas most prior studies have quantified material response characteristics before and after thermal damage, the goal of this work is to describe changing response characteristics during a particular class of thermal damage tests—biaxial isometric tests on collagenous biomembranes. Evolution equations are derived/postulated that provide a good fit to data but more generally provide direction for future work.

Theoretical Modeling of Enlarging Intracranial Aneurysms

Abstract: Rupture of intracranial aneurysms is the leading cause of spontaneous subarachnoid hemorrhage, which results in significant morbidity and mortality. The mechanisms by which intracranial aneurysms develop, enlarge, and rupture are unknown, and it remains difficult to collect the longitudinal patient-based information needed to improve our understanding. We suggest, therefore, that mathematical models hold considerable promise by allowing us to propose and test competing hypotheses on potential mechanisms of aneurysmal enlargement and to compare predicted outcomes with limited clinical information; in this way, we may begin to narrow the possible mechanisms and thereby focus experimental studies. Toward this end, we develop a constrained mixture model for evolving thin-walled, saccular, and fusiform aneurysms and illustrate its efficacy via computer simulations of lesions having idealized geometries. We also present a method to estimate linearized material properties over the cardiac cycle, which can be exploited when solving coupled fluid-solid interactions in a lesion

Dynamics of Materials with a Deformability Threshold

Abstract: Some biological tissues exhibit a sharp reduction of deformability beyond some stress threshold, below which they may be considered elastic. This behaviour can be seen as a limit situation in which the material becomes undeformable beyond some deformation threshold. As a model problem we have considered the motion of a layer of such a material in which one boundary is kept fixed while to the other a tangential stress beyond threshold is applied. The corresponding mathematical model is formulated as a hyperbolic free boundary problem in which at each time instant the interface is made of the points reaching the threshold stretching. [ DOI : 10.1685 / CSC06075] About DOI

Information Flow Requirements for the Stability of Motion of Vehicles in a Rigid Formation

Abstract: It is known in the literature on Automated Highway Systems that information flow can significantly affect the propagation of errors in spacing in a collection of vehicles. This chapter investigates this issue further for a homogeneous collection of vehicles, where in the motion of each vehicle is modeled as a point mass and is digitally controlled. The structure of the controller employed by the vehicles is as follows: \( U_i (z) = C(z)\sum olimits_{j \in S_i } {(X_i - X_j - \tfrac{{L_{ij} z}} {{z - 1}})} \), where U i(z) is the (z- transformation of) control action for the i th vehicle, X i is the position of the i th vehicle, L ij is the desired distance between the i th and the j th vehicles in the collection, C(z) is the discrete transfer function of the controller and S i is the set of vehicles that the i th vehicle can communicate with directly. This chapter further assumes that the information flow is undirected, i.e., i ∈ S j ⇔ j ∈ S i and the information flow graph is connected. We consider information flow in the collection, where each vehicle can communicate with a maximum of q(n) vehicles. We allow q(n) to vary with the size n of the collection. We first show that C(z) cannot have any zeroes at z = 1 to ensure that relative spacing is maintained in response to a reference vehicle making a maneuver where its velocity experiences a steady state offset. We then show that if the control transfer function C(z) has one or more poles located at z = 1, then the motion of the collection of vehicles will become unstable if the size of the collection is sufficiently large. These two results imply that C(1) ≠ 0 and C(1) must be well defined. We further show that if q(n)/n → 0 as n → ∞ then there is a low frequency sinusoidal disturbance of at most unit amplitude acting on each vehicle such that the maximum error in spacing response increase at least as \( \Omega \left( {\sqrt {\tfrac{{n^3 }} {{q^3 (n)}}} } \right) \). A consequence of the results presented in this chapter is that the maximum of the error in spacing and velocity of any vehicle can be made insensitive to the size of the collection only if there is at least one vehicle in the collection that communicates with at least Ω(n) other vehicles in the collection. We also show that there can be at most one vehicle that communicates with Ω(n) vehicles and that any other vehicle in the collection can only communicate with at most p vehicles, where p depends only on the chosen controller and the its sampling time.