Showing papers by "Kumbakonam R. Rajagopal published in 2013"

Anti-plane stress state of a plate with a V-notch for a new class of elastic solids

Abstract: The main purpose of this study is to investigate the efficacy and usefulness of a class of recently proposed models that could be reasonable candidates for describing the response of brittle elastic materials. The class of models that are considered allows for a non-linear relationship between the linearized elastic strain and the Cauchy stress, and this allows one to describe situations wherein the stress increases while the strain yet remains small. Thus one would be in a position to model the response of brittle elastic bodies in the neighborhood of the tips of cracks and notches. In this paper we study the behavior of such models in a plate with a V-notch subject to a state of anti-plane stress. This geometrical simplification enables us to characterize the governing equation for the problem by means of the Airy stress function, though the constitutive relation is a non-linear relation between the linearized strain and the stress. We study the problem numerically by appealing to the finite element method. We find that the numerical solutions are stable. We are able to provide some information regarding the nature of the solution near the tip of the V-notch. In particular we find stress concentration in the vicinity of the singularity.

A new development and interpretation of the Navier–Stokes fluid which reveals why the “Stokes assumption” is inapt

Abstract: In this short paper I present an alternate way of describing fluids in general, and the Navier–Stokes fluid in particular, from a phenomenological point of view, that shows clearly that the putative assumption conjectured by Stokes is not a reasonable assumption. I also show that the procedure presented here is more suited for incorporating constraints such as that of incompressibility, as well as having other advantages. The approach also helps to pinpoint several serious errors in the justifications that are provided in classical texts for the development of the Navier–Stokes model. Finally, from the point of view of the role that causality plays in Newtonian mechanics, the approach suggested is the preferable approach.

On a new class of electroelastic bodies. I

Abstract: Implicit constitutive relations are proposed for large deformations of electroelastic bodies, and approximations to these are developed within the context of small displacement gradients. The resul...

A new class of quasi-linear models for describing the nonlinear viscoelastic response of materials

Abstract: In this short note, we develop a new class of “quasi-linear” viscoelastic models wherein the linearized strain is expressed in terms of a nonlinear measure of the stress. The class of models that is developed could be regarded as counterpart to the class of models referred to popularly as “quasi-linear” models, proposed by Fung to describe the response of viscoelastic bodies; however, now the strain is expressed as an integral of a nonlinear measure of the stress. The class of models that are developed can describe response that cannot be described by the class of models proposed by Fung, and moreover, these models are more reasonable from the point of view of causality.

Shear flows of a new class of power-law fluids

Abstract: We consider the flow of a class of incompressible fluids which are constitutively defined by the symmetric part of the velocity gradient being a function, which can be non-monotone, of the deviator of the stress tensor. These models are generalizations of the stress power-law models introduced and studied by J. Malek, V. Průsa, K.R. Rajagopal: Generalizations of the Navier-Stokes fluid from a new perspective. Int. J. Eng. Sci. 48 (2010), 1907–1924. We discuss a potential application of the new models and then consider some simple boundary-value problems, namely steady planar Couette and Poiseuille flows with no-slip and slip boundary conditions. We show that these problems can have more than one solution and that the multiplicity of the solutions depends on the values of the model parameters as well as the choice of boundary conditions.

Flow of a fluid through a porous solid due to high pressure gradients

Abstract: It is well known that the viscosity of fluids could vary by several orders of magnitude with pressure. This fact is not usually taken into account in many important applications involving the flow of fluids through a porous media, like the problems of enhanced oil recovery or carbon dioxide sequestration where very high pressure differentials are involved. Another important technical problem where such high pressure differentials are involved is that of extracting unconventional oil deposits such as shale which is becoming ever so important now. In this study, we show that the traditional approach that ignores the variation of the viscosity and drag with the pressure greatly over-predicts the mass flux taking place through the porous structure. While taking the pressure dependence of viscosity and drag leads to a ceiling flux, the traditional approaches lead to a continued increase in the flux with the pressure difference. In this study, we consider a generalization of the classical Brinkman equation that takes the dependence of the viscosity and the Drag coefficient on pressure. To our knowledge, this is the first study to carry out such an analysis.

On a new class of electro-elastic bodies.II. Boundary value problems

Abstract: In part I of this two-part paper, a new theoretical framework was presented to describe the response of electro-elastic bodies. The constitutive theory that was developed consists of two implicit c...

A thermodynamic framework to model thixotropic materials

Abstract: Thixotropic materials are widely used in a variety of industrial applications. The constitutive relations to describe these materials are based on one-dimensional experiments in which the material is subjected to a shear motion and there is no unique methodology to obtain proper three-dimensional models. The path towards generalization to a three-dimensional framework is invariably carried out in a ad hoc manner. Here we propose a three-dimensional model that stems from a general thermodynamic framework that has proved to be quite robust in the development of constitutive relations, namely the application of the second law of thermodynamics together with the maximization of the entropy production. This leads to a constitutive equation that has the same form of a generalized Upper Convected Maxwell equation, if we require that changes of microstructure due to the deformation of each Maxwell element that comprises the model are reversible. Changes in microstructure are governed by a potential that is a measure of the difference between the current structure and the equilibrium structure associated with it. The equilibrium structure associated with the current structure is determined by the current value of stress, considered the main break up agent. We assume that the state of equilibrium would be achieved in a Motion With Constant Stress History, starting from the current stress state, until a steady state where the kinematics is not changing.

On models for viscoelastic materials that are mechanically incompressible and thermally compressible or expansible and their oberbeck–boussinesq type approximations

Abstract: Viscoelastic fluid like materials that are mechanically incompressible but are compressible or expansible with respect to thermal stimuli are of interest in various applications ranging from geophysics and polymer processing to glass manufacturing. Here we develop a thermodynamical framework for the modeling of such materials. First we illustrate the basic ideas in the simpler case of a viscous fluid, and after that we use the notion of natural configuration and the concept of the maximization of the entropy production, and we develop a model for a Maxwell type viscoelastic fluid that is mechanically incompressible and thermally expansible or compressible. An important approximation in fluid mechanics that is frequently used in modeling buoyancy driven flows is the Oberbeck–Boussinesq approximation. Originally, the approximation was used for studying the flows of viscous fluids in thin layers subject to a small temperature gradient. However, the approximation has been used almost without any justification...

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] .

A model for the flow of rock glaciers

Abstract: A constitutive model is developed for describing the response of rock glaciers that takes into account the effect of the shear rate, pressure and the volume fraction of the rocks and sand grains trapped within the interstices of the rock glacier on the viscosity of the rock glacier. Using the model, we describe the motion of the Murtel-Corvatsch rock glacier by corroborating the predictions of the model with the experimental data available for the displacement of a vertical borehole, by appealing to a simple semi-inverse solution for the velocity, pressure and volume fraction fields. The predictions of the model agree quite well with the experimental results.

Fidelity of the Estimation of the Deformation Gradient From Data Deduced From the Motion of Markers Placed on a Body That is Subject to an Inhomogeneous Deformation Field

Abstract: Practically all experimental measurements related to the response of nonlinear bodies that are made within a purely mechanical context are concerned with inhomogeneous deformations, though, in many experiments, much effort is taken to engender homogeneous deformation fields. However, in experiments that are carried out in vivo, one cannot control the nature of the deformation. The quantity of interest is the deformation gradient and/or its invariants. The deformation gradient is estimated by tracking positions of a finite number of markers placed in the body. Any experimental data-reduction procedure based on tracking a finite number of markers will, for a general inhomogeneous deformation, introduce an error in the determination of the deformation gradient, even in the idealized case, when the positions of the markers are measured with no error. In our study, we are interested in a quantitative description of the difference between the true gradient and its estimate obtained by tracking the markers, that is, in the quantitative description of the induced error due to the data reduction. We derive a rigorous upper bound on the error, and we discuss what factors influence the error bound and the actual error itself. Finally, we illustrate the results by studying a practically interesting model problem. We show that different choices of the tracked markers can lead to substantially different estimates of the deformation gradient and its invariants. It is alarming that even qualitative features of the material under consideration, such as the incompressibility of the body, can be evaluated differently with different choices of the tracked markers. We also demonstrate that the derived error estimate can be used as a tool for choosing the appropriate marker set that leads to the deformation gradient estimate with the least guaranteed error.

Modeling bodies that can only undergo isochoric motions subject to mechanical stimuli but are compressible or expansible with respect to thermal stimuli

Abstract: In this short note, we discuss a new constitutive approach to describing the response of bodies, both solid and fluid, that can only undergo isochoric motions in isothermal processes but which can undergo non-isochoric motions in arbitrary processes. Within this new framework, one finds that conditions that were perceived as constraints on the response of the body now arise naturally within the frame work of the constitutive definition of these bodies. For instance, a central approximation in fluid mechanics that is of great utility in the analysis of fluid flow problems in geophysics and astrophysics is that due to Oberbeck (Ann Phys Chem 1:271–292, 1879; Uber die bewengungsercheinungen der Atmosphare, Sitz Ber K Preuss Akad Miss, pp 383–395, 1129–1138, 1888) and Boussinesq (Theorie Analytique de la Chaleur. Gauthier-Villas, Paris, 1903) for describing the flow of fluids, which can undergo only isochoric motions in isothermal processes but which are otherwise capable of non-isochoric motions. A similar demand can be made concerning the response of solid bodies wherein one could carry out an approximation similar to that of the Oberbeck–Boussinesq equations, and such an approach might be of great value in the study of technologically relevant problems.

A note on the modeling of incompressible fluids with material moduli dependent on the mean normal stress

Abstract: In incompressible materials, both fluids and solids, a part of the stress is not prescribed by constitutive specification, that is, the part of the stress is not determined in terms of kinematical quantities, temperature, et cetera. This “indeterminate” part of the stress is variously referred to as the “constraint stress”, the “reaction stress” or the “Lagrange multiplier” enforcing the constraint. In the case of an incompressible Navier–Stokes fluid, the part of the stress, that is a consequence of the constraint, also happens to coincide with the mean value of the stress which is referred to as the “mechanical pressure”. However, in general non-Newtonian fluids this is not the case, and, unfortunately, in view of the widespread use of the Navier–Stokes equation, the terminology “pressure” is used interchangeably for both the part of the stress that is not constitutively specified and the mean value of the stress, leading to considerable confusion with regard to important issues concerning non-Newtonian fluids. Recognizing the distinction between the mean value of the stress and the part of the stress that is not constitutively specified becomes critical in materials whose moduli depend on the mean value of the stress. An example of the same concerns the viscosity, which depending on whether it is a function of the indeterminate part of the stress or the mean value of the stress could lead to different flow characteristics. In this short note we discuss an error that is a consequence of not recognizing the distinction between these different quantities but misidentifying them as being the same, the mechanical “pressure”.

Restrictions placed on constitutive relations by angular momentum balance and Galilean invariance

Abstract: In this note, we will show that for describing the response of a wide class of bodies, it is sufficient to invoke only the balance of angular momentum to obtain the restrictions on the constitutive functions that one obtains by appealing to frame indifference. While this result is known for hyperelastic materials (although it is not found in any standard text on the subject), we extend this result to classes of elasto-plastic and viscoelastic materials as well as for a class of implicit constitutive equations for viscous fluids. In particular, we show that for a class of bodies capable of instantaneous elastic response that is dictated by a stored energy function, the symmetry of the Cauchy stress alone is enough to obtain all the necessary restrictions. The result is related to Noether’s theorem; if we know that there is a conserved quantity (i.e., angular momentum), we can then show that the energy function must be invariant under a group of transformations. For a class of generalized Newtonian fluids (including the Navier Stokes fluid and the Bingham fluid), the symmetry of the stress and Galilean invariance of the response functions are all that are required to obtain restrictions that are usually obtained by enforcing frame indifference.

A Pressure Based Solver for an Incompressible Laminar Newtonian Fluids

Abstract: A pressure based solver is developed for an incompressible laminar Newtonian fluid using finite volume method. In the present work the coupling between density and pressure is removed, as well as the coupling between the energy equations for compressible flows. And also the effect of increased mesh resolution and mesh grading towards the walls is investigated. The solver is validated with the existing solver in the literature. The results are analyzed for standard test case driven cavity flow for different mesh sizes.

A methodology to assess the safety of automatically controlled vehicles

Abstract: Automated driving is getting a step closer to reality. However, systematic methods for analyzing the safety benefits of deploying automated vehicles on roads are lacking. In this paper, we provide an outline of a methodology that may be adopted to assess the safety benefits of automatic vehicle following in an emergency braking scenario. Automated vehicles travel in a single straight lane and the maximum deceleration of every vehicle in the lane is a random variable with a known probability distribution. The lead vehicle in the lane brakes at its maximum value of deceleration. The deceleration of each following vehicle is limited by its maximum deceleration and is specified by a vehicle following law. If the vehicle following law commands a deceleration greater than or equal to the maximum deceleration, the controlled vehicle can only brake at its maximum deceleration. Clearly, the deceleration of every controlled vehicle in this case is also a random variable and we provide a methodology to compute its probability distribution. The importance of the probability distribution of deceleration of a vehicle can be readily seen from its variance; for a given initial following distance, if the initial relative velocity is small, variance in their deceleration is small and the mean is the same, then the probability of a collision is correspondingly small. Since the deceleration of every vehicle is a random variable, the following distance and relative velocity of a following vehicle with respect to its immediate predecessor in the lane are also random variables. We say that a violation occurs if the following distance of a vehicle is zero and the corresponding severity is the value of its relative velocity with respect to its predecessor. The notion of violation is a surrogate for the notion of a collision and does not depend on any model of collision or on detailed vehicle dynamical models. We then provide a methodology to compute metrics of safety in terms of the probability of a violation, expected number of violations and severity of a violation.

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

Abstract: 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.

Unsteady Three-Dimensional Mhd Flow Due to Impulsive Motion with Heat and Mass Transfer Past a Stretching Sheet in a Saturated Porous Medium

Abstract: The development of velocity, temperature and concentration fields of an incompressible viscous electrically conducting fluid, caused by impulsive stretching of the surface in two lateral directions and by suddenly increasing the surface temperature from that of the surrounding fluid in a saturated porous medium is studied. The partial differential equations governing the unsteady laminar boundary layer flow are solved analytically. For some particular cases, closed form solutions are obtained, and for large values of the independent variable asymptotic solutions are found. The surface shear stress in x and y directions and the surface heat transfer and surface mass transfer increase with the magnetic parameter and with permeability parameter and the stretching ratio, and there is a smooth transition from the short-time solution to the long-time solution.

Combinatorial Motion Planning Algorithms for a Heterogeneous Collection of Unmanned Vehicles

Abstract: : The project dealt with two classes of core decision-making algorithms related to operator-UV collaboration; the first class involves the routing of UVs through the set of targets nominated by the operator and the second class of problems involves decision-making algorithms for UVs to accommodate uncertainty. We have developed approximation, lower bounding and exact algorithms to address the two classes of problems. We have also implemented these algorithms in simulations to corroborate the performance of these algorithms. In the ensuing discussion, we will summarize our work for the project, and our main results.

Solution Approach for Coupled Diffusion-Reaction-Deformation Problems in Anisotropic Materials

Abstract: A capability to model oxidizing carbon-fiber polyimide matrix composites has evolved over the past number of years at Air Force Research Laboratory [1]. Quoting [1] regarding a unidirectional non-woven fibrous layer, without cracks, “The [finite element model] requires mesh sizes in the 1-μm scale and time increments in 1-s steps. A 200-h oxidation simulation with 100-μm oxidation zone size typically requires problem sizes in the order of 100,000 degrees of freedom (DOF) and 720,000 time steps.” Because of interest in a number of related problem classes including structural component scales, desire to incorporate process restrictions offered by thermodynamics, and the possible involvement of finite deformations, a mixture theory approach was developed by Hall and Rajagopal [2]. The theory is based on two constituents, an anisotropic viscous fluid and an anisotropic hyperelastic solid, which react with each other. The model considers the comparatively simple cases where conversions of species, including the associated masses, linear and angular momenta, energies and entropies, are limited to interchanges between the original fluid and solid.