Kumbakonam R. Rajagopal

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

Further remarks on simple flows of fluids with pressure-dependent viscosities

Abstract: Recently Suslov and Tran (2008) [1] claimed to have found an error in one of the solutions given in the paper by Hron (2001) et al. [2] concerning the flows of fluids with pressure-dependent viscosities. We show that their arguments are related to the question of the continuity of the pressure, and we show that the original solution, although it is not a classical one, can be interpreted as a solution in a generalized sense. Mathematical and physical implications of such generalizations are briefly discussed. The discussion in the paper highlights the importance of recognizing what is meant by a “solution” to a partial differential equation, whether by a solution we mean a classical solution, a weak solution, or a solution in some other sense.

On the Deformation of the Circumflex Coronary Artery During Inflation Tests at Constant Length

Abstract: Here we investigate whether the deformation observed in an experiment in which the porcine circumflex coronary artery is subjected to inflation at constant length included in the class, $$r$$ $$=$$ $$r(R)$$ , $$\theta$$ $$=$$ $$\Theta$$ , $$z$$ $$=$$ $$\Lambda Z$$ . We find that this is not the case and discuss its implications in the study of the mechanics of this artery. Moreover, we identify and quantify the uncertainty in the value of the invariants of the left Cauchy–Green tensor inferred from the 2D motion of markers affixed to the surface of the test specimen, and suggest that 3D tracking of markers is needed due to inherent bending and twisting induced by pressurization in vitro.

On the modeling of the non-linear response of soft elastic bodies

Abstract: In this short note we articulate the need for a new approach to develop constitutive models for the non-linear response of materials wherein one is interested in describing the Cauchy–Green stretch as a non-linear function of the Cauchy stress, with the relationship not in general being invertible. Such a material is neither Cauchy nor Green elastic. The new class of materials has several advantages over classical elastic bodies. When linearized under the assumption that the displacement gradient be small, the classical theory leads unerringly to the classical linearized model for elastic response, while the current theory would allow for the possibility that the linearized strain be a non-linear function of the stress. Such bodies also exhibit a very desirable property when viewed within the context of constraints. One does not need to introduce a Lagrange multiplier as is usually done in the classical approach to incompressibility and the models are also more suitable when considering nearly incompressible materials. The class of materials considered in this paper belongs to a new class of implicit elastic bodies introduced by Rajagopal [19] , [20] . We show how such a model can be used to interpret the data for an experiment on rubber by Penn [18] .

Spherical inflation of a class of compressible elastic bodies

Abstract: Using a special model that belongs to a new class of elastic bodies wherein the Cauchy–Green stretch is given in terms of the Cauchy stress and its invariants, within the context of the spherical inflation of a spherical annulus, we show that interesting phenomena like the development of “stress boundary layers” manifest themselves. We consider two cases of boundary value problems, one in which there is a cavity in a sphere and the other in which there is a rigid spherical inclusion in a sphere. We show that in the case of a rigid inclusion, it is possible for a pronounced “stress boundary” layer to develop, in that the values of the stresses within this boundary layer that is adjacent to a spherical inclusion are much larger than external to it. We also show that in the case of both the cavity and a rigid inclusion, the stress concentration is an order of magnitude higher than the increase in the deformation gradient, that is, the stress and the stretch do not scale in a similar manner. While the stress adjacent to a rigid inclusion can be 2500 times the applied radial stress, the maximum stretch, which occurs at the rigid inclusion is about 10. While the variation in the stresses are linear in thin walled annular regions, we find that in thick walled annular regions, the variation of the stresses is non-linear.

On the anti-plane state of stress near pointed or sharply radiused notches in strain limiting elastic materials: closed form solution and implications for fracture assessements

Abstract: In this paper a comprehensive investigation is carried out with regard to the state of the stress and strain in the neighbourhood of notches in bodies subjected to an anti-plane state of shear stress, within the context of a strain limiting theory of elasticity. Taking advantage of a unified analytical framework, the strain-limiting theory of elasticity is used to determine the full stress and strain field close to a pointed or radiused notch with any notch opening angle. An extensive discussion is provided that highlights the main features of stress and strain distributions, and the implications of the new theory for fracture assessments. In particular, it is proved that the obtained stress and strain solution predicts finite strains at the notch tip and allows the intensity of the stress field to be written as a function of the elastic Notch Stress Intensity Factor $$K_{3}$$ , as in the case of conventional linearized elasticity theory. This makes the strain limiting elasticity an excellent vehicle for justifying theoretically a K based-approach to the fracture of brittle elastic solids, within the context of a self consistent theory, unlike the classical linearized theory that predicts singularities for the strain at crack tips.

A new constitutive framework for arterial wall mechanics and a comparative study of material models

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.

The physics of debris flows

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.

Solving Ordinary Differential Equations

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.

Hyperelastic modelling of arterial layers with distributed collagen fibre orientations

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.