Showing papers by "Kumbakonam R. Rajagopal published in 2003"

On Implicit Constitutive Theories

Abstract: In classical constitutive models such as the Navier-Stokes fluid model, and the Hookean or neo-Hookean solid models, the stress is given explicitly in terms of kinematical quantities. Models for viscoelastic and inelastic responses on the other hand are usually implicit relationships between the stress and the kinematical quantities. Another class of problems wherein it would be natural to develop implicit constitutive theories, though seldom resorted to, are models for bodies that are constrained. In general, for such materials the material moduli that characterize the extra stress could depend on the constraint reaction. (E.g., in an incompressible fluid, the viscosity could depend on the constraint reaction associated with the constraint of incompressibility. In the linear case, this would be the pressure.) Here we discuss such implicit constitutive theories. We also discuss a class of bodies described by an implicit constitutive relation for the specific Helmholtz potential that depends on both the stress and strain, and which does not dissipate in any admissible process. The stress in such a material is not derivable from a potential, i.e., the body is not hyperelastic (Green elastic).

A constrained mixture model for arterial adaptations to a sustained step change in blood flow

Abstract: A sustained change in blood flow results in an arterial adaptation that can be thought to consist of two general steps: an immediate vasoactive response that seeks to return the wall shear stress to its homeostatic value, and a long-term growth and remodeling process that seeks to restore the intramural stresses and, if needed, the wall shear stress toward their homeostatic values. Few papers present mathematical models of arterial growth and remodeling in general, and fewer yet address flow-induced changes. Of these, most prior models build upon the concept of “kinematic growth” proposed by Skalak in the early 1980s (Skalak R (1981) In: Proceedings of the IUTAM Symposium on finite elasticity. Martinus Nijhoff, The Hague, pp 347–355). Such approaches address important consequences of growth and remodeling, but not the fundamental means by which such changes occur. In this paper, therefore, we present a new approach for mathematically modeling arterial adaptations and, in particular, flow-induced alterations. The model is motivated by observations reported in the literature and is based on a locally homogenized, constrained mixture theory. Specifically, we develop a 3-D constitutive relation for stress in terms of the responses of the three primary load-bearing constituents and their time-varying mass fractions, with the latter accounting for the kinetics of the turnover of cells and extracellular matrix in changing, stressed configurations. Of particular importance is the concept that the natural configurations of the individual constituents can evolve separately and that this leads to changes in the overall material properties and empirically inferred residual stress field of the vessel. Potential applications are discussed, but there is a pressing need for new, theoretically motivated data to allow the prescription of specific functional forms of the requisite constitutive relations and the values of the associated material parameters.

A Model Incorporating Some of the Mechanical and Biochemical Factors Underlying Clot Formation and Dissolution in Flowing Blood

Abstract: Multiple interacting mechanisms control the formation and dissolution of clots to maintain blood in a state of delicate balance. In addition to a myriad of biochemical reactions, rheological factors also play a crucial role in modulating the response of blood to external stimuli. To date, a comprehensive model for clot formation and dissolution, that takes into account the biochemical, medical and rheological factors, has not been put into place, the existing models emphasizing either one or the other of the factors. In this paper, after discussing the various biochemical, physiologic and rheological factors at some length, we develop a model for clot formation and dissolution that incorporates many of the relevant crucial factors that have a bearing on the problem. The model, though just a first step towards understanding a complex phenomenon, goes further than previous models in integrating the biochemical, physiologic and rheological factors that come into play.

On an inconsistency in the derivation of the equations of elastohydrodynamic lubrication

Abstract: Reynolds's lubrication approximation, one of the cornerstones of fluid mechanics, is constructed on the assumption that the viscosity is independent of the pressure. This assumption is reasonable at low pressures and is appropriate for a large class of applications. However, in an important instance that appeals to the approximation (elastohydrodynamic lubrication (EHL)), the liquid lubricant is subjected to extremely high pressures and the assumption that the viscosity is independent of the pressure no longer holds. On the contrary, pressure dependence of viscosity is severe and the viscosity can increase by several orders of magnitude due to pressure increase. Nevertheless, in the current literature the pressure dependence of viscosity in the derivation of the governing equations for EHL is only recognized a posteriori , that is, after the Reynolds equation has been stated under the assumption of constant viscosity. A consistent derivation of the equations of EHL that takes into account the pressure dependence of viscosity right from the outset leads to additional and hitherto neglected terms in the governing equations. Consequently, construction of a single pressure equation, analogous to the Reynolds equation, is no longer possible without additional, drastic, assumptions. In this study,we provide a consistent derivation of the equations of motion for EHL and, with additional, simplifying assumptions, derive a modified Reynolds equation. We then provide a comparison between the solutions to the classical equation of Reynolds's and our modified equation. The modified equation results in slightly higher pressures, but at significantly higher viscosities, than the classical Reynolds equation.

Diffusion through polymeric solids undergoing large deformations

Abstract: The intention of this overview is to review the literature on the diffusion of liquids through polymers that are undergoing large deformations, and to discuss the difficulties associated with modelling such a process. The inadequacy of simple equations such as Fick's equation or Darcy's equation to describe the problem with regard to characterising the stresses and strains in the polymer through which the liquid is diffusing, and the change in its material properties during swelling and the need for an alternate approach are discussed. Finally, the problem of swelling is studied within the context of the theory of mixtures and it is shown that such an approach is very well suited for the analysis of the problem.

Biological Growth and Remodeling: A Uniaxial Example with Possible Application to Tendons and Ligaments

Abstract: Recent discoveries in molecular and cell biology reveal that many cell types sense and respond (via altered gene expression) to changes in their mechanical environment. Such mechanotransduction mechanisms are responsible for many changes in structure and function, including the growth and remodeling process. To understand better, and ultimately to use (e.g., in tissue engineering), biological growth and remodeling, there is a need for mathematical models that have predictive and not just descriptive capability. In contrast to prior models based on reaction-diffusion equations or the concept of volumetric growth, we examine here a newly proposed constrained mixture model for growth and remodeling. Specifically, we use this new model to present illustrative computations in a representative, transversely-isotropic soft tissue subjected to homogeneous deformations under uniaxial loading. Consequences of various assumptions for the kinetics of mass production and removal are discussed, as are open problems in this important area of biomechanics.

Modeling the pneumatic subsystem of a s-cam air brake system

Abstract: The air brake system is one of the critical components in ensuring the safe operation of any commercial vehicle. This work is directed towards the development of a fault-free model of the pneumatic subsystem of the air brake system. This model can be used in brake control and diagnostic applications. Current enforcement inspections are done manually and hence are time consuming and subjective. The long-term objective is to develop a model-based, performance-based diagnostic system that will automate enforcement inspections and help in monitoring the condition of the air brake system. Such a diagnostic system can update the driver on the performance of the brake system during travel and with recent advancements in communication technology, this information can be remotely transferred to the brake inspection teams. The model of the pneumatic subsystem correlates the pressure transients in the brake chamber with the brake pedal actuation force and the brake valve plunger displacement. An experimental test bench was set up at Texas A&M University and the experimental data is used to corroborate the results obtained from the model.

On the effect of dissipation in shape-memory alloys

Abstract: After reviewing the phenomena of martensitic phase transformation in shape-memory alloys, the conventional models that take into consideration viscosity-like and capillarity-like response are investigated for vanishing "dissipative effects". It is shown that they do approach the fully conservative case. Experimental evidence indicates that the response is rate independent and thus a model incorporating a phenomenological rate-independent plasticity-type dissipation related with an activated phase-transformation process is investigated.

Shear Waves in a Class of Nonlinear Viscoelastic Solids

Abstract: We investigate the propagation of shear waves in a special class of nonlinear viscoelastic solids of differential type. We determine several useful exact solutions for such models for initial data with compact and non-compact support. Moreover, we find some explicit examples of blow-up for boundary-value problems with smooth initial data.

On some unresolved issues in non-linear fluid dynamics

Abstract: This paper is a discussion of some open problems concerning generalizations of Navier-Stokes fluids and other non-linear fluids, which are in need of thorough investigation. The first topic is an interesting generalization of the Navier-Stokes model which is appropriate, in particular, in the description of fluids like water subjected to processes in which the normal stresses are very large. Discussed next is the modelling of non-Newton fluids at a distance from the boundary of the volume occupied by the fluid. Attention is then given to the necessity of a careful revision of conditions obtained at the boundary of the fluid volume, and finally, the modelling of "turbulent" flows is discussed.

On the Role of Inhomogeneities in the Deformation of Elastic Bodies

Abstract: It is quite common to approximate “mildly” inhomogeneous bodies as homogeneous bodies belonging to a certain constitutive class in view of the simplification that such an approximation accords. In this study, we investigate the consequences of such an assumption and we show that it is clearly inappropriate for many classes of inhomogeneous bodies. We choose specific boundary value problems to illustrate the fact that we could be grossly in error, both qualitatively and quantitatively, with regard to local measures such as stresses and strains. In the examples considered, we find that, for global quantities such as applied forces and moments, the error could be significant. Not only could the material parameters found from, say, an extension test and torsion test, which neglect the inhomogeneity of the body, be quite different from one that incorporates the inhomogeneity, but also the values for the material parameter in the homogenized approximation gleaned from these different experiments could be differ...

A Comparison of the Response of Isotropic Inhomogeneous Elastic Cylindrical and Spherical Shells and Their Homogenized Counterparts

Abstract: All real bodies are inhomogeneous, though in many such bodies the inhomogeneity is “mild” in that the response of the bodies can be “approximated” well by the response of a homogeneous approximation. In this study we explore the status of such approximations when one is concerned with bodies whose response is nonlinear. We find that significant departures in response can occur between that of a “mildly” inhomogeneous body and its homogeneous approximation (if the approximate model is restricted to a certain class), both quantitatively and qualitatively. We illustrate this fact within the context of a specific boundary value problem, the inflation of an inhomogeneous spherical shell. We also discuss the inappropriateness of homogenization procedures that lead to a homogenized stored energy for the body when in fact what is required is a homogenized model that predicts the appropriate stresses as they invariably determine the failure or integrity of the body.

The natural viscosity of turbulence

Abstract: In this work, we extend the notion of natural viscosity introduced by Truesdell in 1964 for the simple fluid in a viscometric flow to the study of homogeneous turbulent shear flow of a Navier–Stokes fluid. Within the framework of Navier–Stokes equations, first, we prove the existence of the natural viscosity of turbulence of a Navier–Stokes fluid in the limit of zero shear rate in homogeneous turbulent shear flow and derive its exact expression. Secondly, we show that, in contrast with the simple fluid in which case its natural viscosity is a material constant, the natural viscosity of turbulence is a function of time and always non-negative, different from the eddy viscosity introduced by Boussinesq in 1877, which could take on negative values as shown in experiments and differs from one model to another in turbulence modelling. In addition, we analyse Prandtl's mixing length model and a few linear and non-linear K–e models related to the concept of eddy viscosity and compare their performance in approxi...

Gas mixing for achieving suitable conditions for single point aerosol sampling in a straight tube: experimental and numerical results.

Abstract: Experimental measurements of velocity and tracer gas concentration are taken in a straight tube to evaluate the effectiveness of mixing in achieving conditions as required by ANSI N13.1-1999 for single point extractive sampling from stacks and ducts of nuclear facilities. Mixing is evaluated for inlet turbulent intensities of 1.5%, 10%, and 20%, achieved by introducing various bi-plane grids, and for conditions generated by a commercial static gas mixer. The data obtained (at Reynolds number = 15,000) highlight the importance of inlet turbulence intensity in the process of turbulent dispersion of a dilute gas. The gas mixer does not introduce significant pressure losses and unlike bi-plane grids, the turbulence downstream of the mixer is not homogenous. A judicious choice of the release location that uses the large scale eddies and inhomogeneity of the turbulence ensures that the specified ANSI N13.1-1999 criteria are attained within 7 diameters downstream of the duct inlet. This is significantly more effective than a bi-plane grid where even with 20% inlet intensity the criteria are met only at 21 diameters downstream. The predictions of a proposed semi-empirical correlation match favorably with data. For example, at 18 diameters downstream with inlet intensities of 1.5% and 10%, the predicted coefficients of variation (COVs) of 150% and 65% are close to the actual values of 154% and 50%; where the COV of a set of measurements is the ratio of the standard deviation of the set to its mean value. The corresponding results obtained using commercially available software are 141% and 12%. Results from a particle-tracking model show good qualitative trends, but they should not be used to determine compliance with the requirements of the ANSI standard.

Inhomogeneous deformations of a cone of a neo-Hookean solid

Abstract: We study the inhomogeneous deformations of a cone due to squeezing or expanding the cone. Reminiscent of solutions to Jefferey–Hamel flows between intersecting planes and the deformation of elastic wedges, we find that depending on the cone angle we can have solutions that correspond to radial filaments being purely stretched or contracted, or solutions in which the displacement does not have the same sense, certain subparts being stretched, while others are contracted.