Author

# Kumbakonam R. Rajagopal

Other affiliations: Kent State University, University of Wisconsin-Madison, University of Minnesota ...read more

Bio: Kumbakonam R. Rajagopal is an academic researcher from Texas A&M University. The author has contributed to research in topic(s): Constitutive equation & Viscoelasticity. The author has an hindex of 77, co-authored 659 publication(s) receiving 23443 citation(s). Previous affiliations of Kumbakonam R. Rajagopal include Kent State University & University of Wisconsin-Madison.

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01 Oct 1995

TL;DR: In this article, a discussion of a mixture of immiscible fluids is given, and the status of Darcy's law within the context of mixture theory is discussed. And the entropy inequality constitutive theory steady state problems diffusing singular surface wave propagation in solids infused with fluids are discussed.

Abstract: Kinematics partial stress and total stress balance laws and the entropy inequality constitutive theory steady state problems diffusing singular surface wave propagation in solids infused with fluids epilogue some results from differential geometry status of Darcy's law within the context of mixture theory a brief discussion of a mixture of immiscible fluids.

592 citations

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TL;DR: In this paper, the authors provide an extended analysis of the genesis and development of fluids of differential type, and show that certain ideas of flow retardation and model approximation have been consistently misinterpreted.

Abstract: Thermodynamics, in the form of a dissipation inequality and the commonly accepted idea that the stored energy should have an extremum in equilibrium, is used to find restrictions on the response functions for the stress and the stored energy in incompressible fluids of differential type. For those special fluids of differential type known as grade n fluids these thermodynamic restrictions have been a source of some controversy and much confusion not withstanding the fact that they are in complete harmony with results achieved by either linear or nonlinear stability analysis. In order to clarify the issues that seem to underlie this controversy, we provide an extended analysis of the genesis and development of fluids of differential type. As part of our analysis, we will show that certain ideas of flow retardation and model approximation have been consistently misinterpreted. Additionally, we establish several new results concerning the thermodynamics of these materials. A special application of our results reveals that work of Joseph [1, 2] and Renardy [3] on the instability of the rest state for certain, very special grade n fluids is in fact inapplicable to all those grade n fluids that are consistent with thermodynamics.

589 citations

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TL;DR: It was thought that the most important characteristics of soft tissues were their complex mechanical properties: they often exhibit nonlinear, anisotropic, nearly incoherent, and often incoherent properties as discussed by the authors.

Abstract: Not long ago it was thought that the most important characteristics of the mechanics of soft tissues were their complex mechanical properties: they often exhibit nonlinear, anisotropic, nearly inco...

587 citations

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TL;DR: In this paper, the flow of an incompressible second-order fluid past a stretching sheet is studied, and the authors present a study of the flow in the presence of a stretch sheet.

Abstract: This paper presents a study of the flow of an incompressible second-order fluid past a stretching sheet. The problem has a bearing on some polymer processing application such as the continuous extrusion of a polymer sheet from a die.

452 citations

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TL;DR: In this paper, the authors develop a thermodynamic approach for modeling a class of viscoelastic fluids based on the notion of an evolving natural configuration, where the material has a family of elastic responses governed by a stored energy function that is parametrized by the ''natural configurations''. Changes in the current natural configuration result in dissipative behavior that is determined by a rate of dissipation function.

Abstract: In this paper, we develop a thermodynamic approach for modeling a class of viscoelastic fluids based on the notion of an `evolving natural configuration'. The material has a family of elastic (or non-dissipative) responses governed by a stored energy function that is parametrized by the `natural configurations'. Changes in the current natural configuration result in dissipative behavior that is determined by a rate of dissipation function. Specifically, we assume that the material possesses an infinity of possible natural (or stress-free) configurations. The way in which the current natural configuration changes is determined by a `maximum rate of dissipation' criterion subject to the constraint that the difference between the stress power and the rate of change of the stored energy is equal to the rate of dissipation. By choosing different forms for the stored energy function ψ and the rate of dissipation function ξ, a whole plethora of energetically consistent rate type models can be developed. We show that the choice of a neo-Hookean type stored energy function and a rate of dissipation function that is quadratic, leads to a Maxwell-like fluid response. By using this procedure with a different choice for the rate of dissipation, we also derive a model that is similar to the Oldroyd-B model. We also discuss several limiting cases, including the limit of small elastic strains, but arbitrarily large total strains, which leads to the classical upper convected Maxwell model as well as the Oldroyd-B model.

407 citations

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TL;DR: In this paper, the authors developed a constitutive law for the description of the (passive) mechanical response of arterial tissue, where the artery is modeled as a thick-walled nonlinearly elastic circular cylindrical tube consisting of two layers corresponding to the media and adventitia.

Abstract: 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.

2,617 citations

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TL;DR: In this paper, 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.

Abstract: 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.

2,137 citations

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TL;DR: The variable-order Adams method (SIVA/DIVA) package as discussed by the authors is a collection of subroutines for solution of non-stiff ODEs.

Abstract: 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.

1,785 citations

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TL;DR: A structural continuum framework that is able to represent the dispersion of the collagen fibre orientation is developed and 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.

Abstract: 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.

1,703 citations