Showing papers by "Kumbakonam R. Rajagopal published in 2015"
TL;DR: In this paper, the authors developed models within a thermodynamic standpoint that are very similar in form to the classical Maxwell and Oldroyd-B models but differ from them in one important aspect, the manner in which they unload instantaneously from the deformed configuration.
Abstract: In this paper we develop models within a thermodynamic standpoint that are very similar in form to the classical Maxwell and Oldroyd-B models but differ from them in one important aspect, the manner in which they unload instantaneously from the deformed configuration. As long as the response is not instantaneous, the models that are derived cannot be differentiated from the Maxwell and Oldroyd-B models, respectively. The models can be viewed within the context of materials whose natural configuration evolves, the evolution being determined by the maximization of the rate of entropy production of the material. However, the underpinnings to develop the model are quite different from an earlier development by Rajagopal and Srinivasa [8] in that while the total response of the viscoelastic fluid satisfies the constraint of an incompressible material, the energy storage mechanism associated with the elastic response is allowed to be that for a compressible elastic solid and the dissipative mechanism associated with the viscous response allowed to be that for a compressible fluid, the total deformation however being isochoric. The analysis calls for a careful evaluation of firmly held customs in viscoelasticity wherein it is assumed that it is possible to subject a material to a purely instantaneous elastic response without any dissipation whatsoever. Finally, while the model developed by Rajagopal and Srinivasa [8] arises from the linearization of the non-linear elastic response that they chose and leads to a model wherein the instantaneous elastic response is isochoric, here we develop the model within the context of a different non-linear elastic response that need not be linearized but the instantaneous elastic response not necessarily being isochoric.
52 citations
TL;DR: In this paper, the distinction between various significations of the word "pressure" and their implications with regard to response relations for bodies is discussed, and the distinction in the meanings of the above terms assumes paramount significance when discussing properties of materials, which could possibly depend on "pressure".
Abstract: In addition to understanding the various meanings attached to the word “pressure” one also has to comprehend the meanings of the phrases in which the term “pressure” appears. For instance one comes across the following combinations: “static-fluid pressure”, “thermodynamic pressure”, “mechanical pressure”, “contact pressure”, “stagnation pressure”, “vapor pressure”, “electro-osmotic pressure”, etc., One also often comes across the comment that “pressure is the Lagrange multiplier that enforces the constraint of incompressibility” and that “pressure is the mean normal stress”. In general the word “pressure” with different significations, is used with gay abandon without paying proper attention to its usage 1 . The distinction in the meanings of the above terms assumes paramount significance when discussing properties of materials, which could possibly depend on “pressure”. In this short note we discuss the distinction between various significations of the word “pressure”, and their implications with regard to response relations for bodies.
50 citations
TL;DR: In this article, the authors investigated the asymptotic behavior of the stress field at the tip of a straight plane-strain fracture and showed that the only cases satisfying the required boundary conditions correspond to bounded stresses.
50 citations
TL;DR: In this article, the authors established the existence of a weak solution to the antiplane stress problem on V-notch domains for a class of recently proposed new models that could describe elastic materials in which the stress can increase unboundedly while the strain yet remains small.
Abstract: The main purpose of this study is to establish the existence of a weak solution to the anti-plane stress problem on V-notch domains for a class of recently proposed new models that could describe elastic materials in which the stress can increase unboundedly while the strain yet remains small. We shall also investigate the qualitative properties of the solution that is established. Although the equations governing the deformation that are being considered share certain similarities with the minimal surface problem, the boundary conditions and the presence of an additional model parameter that appears in the equation and its specific range makes the problem, as well as the result, different from those associated with the minimal surface problem.
46 citations
TL;DR: In this paper, a model to describe the inelastic response of bodies that exhibit non-linear response even in the small strain regime was developed, exploiting the discontinuity of the functions that appear in the constitutive relations.
34 citations
TL;DR: In this article, the authors derive a novel and rigorous correction to the classical Reynolds lubrication approximation for fluids with viscosity depending upon the pressure, which leads to higher pressure and viscosities in the flow domain.
33 citations
TL;DR: This paper extends the development of implicit constitutive relations to describe the behaviour of elastic bodies that respond to magnetic stimuli and uses the linearized model to describe experimentally observed phenomena which the classical linearized magnetoelastic models are incapable of doing.
Abstract: Implicit constitutive relations that characterize the response of elastic bodies have greatly enhanced the arsenal available at the disposal of the analyst working in the field of elasticity. This class of models were recently extended to describe electroelastic bodies by the present authors. In this paper, we extend the development of implicit constitutive relations to describe the behaviour of elastic bodies that respond to magnetic stimuli. The models that are developed provide a rational way to describe phenomena that have hitherto not been adequately described by the classical models that are in place. After developing implicit constitutive relations for magnetoelastic bodies undergoing large deformations, we consider the linearization of the models within the context of small displacement gradients. We then use the linearized model to describe experimentally observed phenomena which the classical linearized magnetoelastic models are incapable of doing. We also solve several boundary value problems within the context of the models that are developed: extension and shear of a slab, and radial inflation and extension of a cylinder.
29 citations
TL;DR: In this article, a nonlinear constitutive model was proposed to characterize axial strain and circumferential strain in reinforced portland-cement concrete, where the linearized strain can be a non-linear function of the stress.
Abstract: Unreinforced portland-cement concrete exhibits a nonlinear relationship between applied stress and observed strain, even though the strains are at magnitudes that warrant the infinitesimal strain approximation (i.e., the norm of the displacement gradient is appropriately small). Previous efforts to model this nonlinear response of concrete express a dependence of stress on the deformation gradient (via the infinitesimal strain). However, models derived from the class of Cauchy elastic bodies do not allow a nonlinear relationship between the stress and linearized strain. Nonlinear constitutive relations that are implicit relations between the stress and a proper measure of strain, or nonlinear expressions of an appropriate measure of strain as a function of stress, lead to a logical linearization procedure wherein the linearized strain can be a nonlinear function of the stress. Using such a constitutive model, the authors accurately characterize both axial strain and circumferential strain in concr...
28 citations
TL;DR: In this paper, it was shown that when the stresses are small, the strains are also small which is in agreement with traditional elasticity, unlike the implications of Hooke's law.
Abstract: Motivated by the recent generalization of the class of elastic bodies by Rajagopal (Appl Math 48:279–319, 2003), there have been several recent studies that have been carried out within the context of this new class. Rajagopal and Srinivasa (Proc R Soc Ser A 463:357–367, 2007, Proc R Soc Ser A: Math Phys Eng Sci 465:493–500, 2009) provided a thermodynamic basis for such models and appealing to the idea that rate of entropy production ought to be maximized they developed nonlinear rate equations of the form $${\mathbf{A}\dot{\mathbf{T}}+\mathbf{B}\mathbf{D} = \mathbf{O}}$$
where T is the Cauchy stress and D is the stretching tensor as well as $${\mathbf{A}\dot{\mathbf{S}}+\mathbf{B}\dot{\mathbf{E}} = \mathbf{O}}$$
, where S is the Piola–Kirchhoff stress tensor and E is the Green–St. Venant strain tensor. We follow a similar procedure by utilizing the Gibb’s potential and the left stretch tensor V from the Polar Decomposition of the deformation gradient, and we show that when the displacement gradient is small one arrives at constitutive relations of the form $${\boldsymbol{\varepsilon} = \mathbf{f}(\mathbf{T})}$$
. This is, of course, in stark contrast to traditional elasticity wherein one obtains a single model, Hooke’s law, when the displacement gradient is small. By solving a classical boundary value problem, with a particular form for f(T), we show that when the stresses are small, the strains are also small which is in agreement with traditional elasticity. However, within the context of our model, when the stress blows up the strains remain small, unlike the implications of Hooke’s law. We use this model to study boundary value problems in annular domains to illustrate its efficacy.
28 citations
TL;DR: In this article, the authors consider a special class of elastic bodies that have a nonlinear relationship between the linearized strain and the stress and study boundary value problems, the first concerning the telescopic shearing and inflation of a tube and the second being the extension, inflation and circumferential shearing.
Abstract: There is considerable evidence that shows that for a large class of materials the relationship between the stress and the strain is nonlinear even in the range of strain that is considered small enough for the classical linearized theory of elasticity to be applicable (see Saito et al. in Science 300:464–467, 2003; Li et al. in Phys Rev Lett 98:105503, 2007; Talling et al. in Scr Mater 59:669–672, 2008; Withey et al. in Mater Sci Eng A 493:26–32, 2008; Zhang et al. in Scr Mater 60:733–736, 2009). A proper description of the experiments requires an alternative theory which when linearized would allow the possibility of such a nonlinear relationship between the stress and the strain. Recently, such a theory of elastic bodies has been put into place (see Rajagopal in Appl Math 48:279–319, 2003; Bustamante in Proc R Soc A 465:1377–1392, 2009; Rajagopal in Math Mech Solids 16:536–562, 2011). In this paper, we consider a special class of bodies that belong to the new generalization of response relations for elastic bodies that have a nonlinear relationship between the linearized strain and the stress. We use the special class of bodies that exhibit limited small strain to study two boundary value problems, the first concerning the telescopic shearing and inflation of a tube and the second being the extension, inflation and circumferential shearing of a tube. The results that we obtain for the models under consideration are markedly different from the predictions of the classical linearized elastic model with regard to the same boundary value problems.
24 citations
TL;DR: In this article, the authors developed a model from a thermodynamic standpoint that seems capable of describing the nonlinear response of asphalt binders and tested the efficacy of the model by comparing its predictions against two different sets of torsion experiments.
Abstract: In this paper, we develop a model from a thermodynamic standpoint that seems capable of describing the nonlinear response of asphalt binders. We test the efficacy of the model by comparing its predictions against two different sets of torsion experiments on asphalt binders. The first set of experiments that we use for corroborating the model was carried out by Narayan et al. [2012. Mechanics Research Communications, 43, 66–74] wherein for the first time it was found that the relaxation times associated with the torque and the normal forces, in a torsion experiment, are markedly different, and the second set of experiments that we use to corroborate the model documents the overshoot of torque in a torsion experiment [Krishnan and Narayan, 2007. Steady shear experiments on ashpalt. Chennai: IIT, Madras]. The model that is developed in this paper fits both sets of experiments well, and it seems to be a good candidate for describing the response of asphalt binders in general. As the deformation is nonlinear, ...
TL;DR: In this paper, the authors consider the definition of anisotropy with regard to the response of bodies described by implicit constitutive relations and show that implicit relations between the history of the stress, the density, and the deformation gradient can be defined as symmetry groups for simple materials.
TL;DR: In this paper, a nonlinear integral model for solid-like materials, which upon linearization reduces to a linear viscoelastic model, is defined by separating the normalized time function and nonlinear stress measure that describes the elastic response of the materials.
TL;DR: In this paper, a systematic and complete analysis of the overdetermined problem that one obtains while considering the balance equations of unconstrained isotropic nonlinear Cauchy elastic bodies undergoing anti-plane shear deformations is provided.
TL;DR: In this article, Rajagopal et al. defined implicit constitutive relations between the Piola stress and the Green-St. Venant strain in the context of non-dissipative solids.
Abstract: The class of elastic bodies, that is bodies incapable of dissipation in whatever motion that they undergo, has been significantly enlarged recently (see Rajagopal 2003, On implicit constitutive theories. Appl. Math., 48, 279–319; Rajagopal 2007, The elasticity of elasticity. Z. Angew. Math. Phys. 58, 309–317; Rajagopal, K. R. & Srinivasa, A. R. 2007, On the response of non-dissipative solids. Proc. R. Soc. Lond. A, 463, 357–367). The new classes include fully implicit constitutive relations for the stress and the deformation gradient, and the interesting sub-class wherein the Cauchy–Green tensor or the linearized strain tensor bears a non-linear relationship to the stress. While a fully thermodynamic treatment of such elastic bodies, when defined through implicit constitutive relations between the Piola stress and the Green–St. Venant strain, within a 3D framework has been carried out (see Rajagopal, K. R. & Srinivasa, A. R. 2007, On the response of non-dissipative solids, Proc. R. Soc. Lond. A, 463, 357–367), other possible implicit relationships between other stress and kinematic measures have not been investigated. This paper is devoted to the determination of the consequences of thermodynamics on the new class of elastic bodies, when they are expressed through implicit relations for different stress and stretch/strain measures.
TL;DR: The model developed in this paper is a generalization of the Oldroyd-B model and is used successfully to describe results of several experiments concerning the nonlinear response of asphalt binders, including nonlinear creep-recovery and stress-relaxation behavior, thinning behavior, and the appearance of normal forces perpendicular to the plane of shear in simple shear tests.
Abstract: Many researchers have asserted the importance of considering the nonlinearity of the mechanical behavior of asphalt binders for accurately estimating their performance under field conditions, and for comparing and ranking them accordingly. To do so, it is necessary to have a robust and reliable nonlinear viscoelastic theory and a model derived under its purview that can describe the mechanical response of asphalt binders reasonably well. The objective of this study is to develop such a model. A new Gibbs-potential-based thermodynamic framework is used for this purpose. The model developed in this paper is a generalization of the Oldroyd-B model. It is used successfully to describe results of several experiments from the literature concerning the nonlinear response of asphalt binders, including nonlinear creep-recovery and stress-relaxation behavior, thinning behavior, and the appearance of normal forces perpendicular to the plane of shear in simple shear tests.
TL;DR: In this paper, a generalization of the Oberbeck-Boussinesq approximation is proposed, where the volume change depends both on the temperature and on the pressure that the fluid is subject to.
Abstract: There has been considerable interest, ever since the development of the approximation by Oberbeck and Boussinesq concerning fluids that are mechanically incompressible but thermally compressible, in giving a rigorous justification for the same. For such fluids, it would be natural to assume that the determinant of the deformation gradient (which is a measure of the volume change of the body) depends on the temperature. However, such an assumption has the attendant drawbacks of the specific heat of the fluid at constant volume being zero and the speed of sound in the fluid being complex. In this paper, we consider a generalization of the Oberbeck–Boussinesq approximation, wherein the volume change depends both on the temperature and on the pressure that the fluid is subject to. We show that within the context of this generalization, the specific heat at constant volume can be defined meaningfully, and it is not zero.
TL;DR: In this paper, the authors studied the response of a spherical annular region of the same strain limiting elastic body considered in Part I, wherein the linearized strain is a nonlinear function of the stress.
Abstract: In Part II of the paper, we study the response of a spherical annular region of the same strain limiting elastic body considered in Part I, wherein the linearized strain is a nonlinear function of the stress. We study the response of the annular region due to a normal radial inflation, and as in Part I, we find the response to be strikingly different from that of the classical linearized elastic solid.
TL;DR: In this paper, an analysis of rutting prediction criteria was conducted in order to identify the virtues and shortcomings of each criterion, and each criterion represents the resistance offered by asphalt binders to permanent strain over a specific range of stresses.
Abstract: Several prediction criteria have been proposed to rank asphalt binders based on their rutting performance, and each one has been shown to be able to predict rutting behavior observed in either accelerated loading field tests or under highway traffic with some level of success. In this study, an analysis of these rutting prediction criteria was conducted in order to identify the virtues and shortcomings of each. Asphalt binders that were used in the accelerated loading facility (ALF) study conducted by the Federal Highway Administration (FHWA) were obtained and studied. The experimental data obtained on these binders were fit with a nonlinear viscoelastic model so that it can be used to predict their response under different loading conditions. By comparing predicted behavior based on the nonlinear viscoelastic model and the various rutting prediction criteria, the part of the complex nonlinear mechanical behavior of these binders that each criterion addresses was determined. Results show that each criterion represents the resistance offered by asphalt binders to permanent strain over a specific range of stresses. Results also show that a simple viscosity test might be sufficient to characterize the rutting performance of asphalt binders accurately.
TL;DR: In this article, a new exact solution for a variant of Stokes first problem for stress power-law fluids, when the exponent n=0 (Navier-Stokes fluid), is obtained.
Abstract: Stress power-law fluids are a special sub-class of fluids defined through implicit constitutive relations, wherein the symmetric part of the velocity gradient depends on a power-law of the stress (see Eq. (2.2) ), and were introduced recently to describe the non-Newtonian response of fluid bodies. Such fluids are counterparts to the classical power-law fluids wherein the stress is given in terms of a power-law for the symmetric part of the velocity gradient. Stress power-law fluids can describe phenomena that cannot be described by classical power-law fluids (see [1] ). In this paper, first a new exact solution for a variant of Stokes׳ first problem for stress power-law fluids, when the exponent n=0 (Navier–Stokes fluid), is obtained. Such an exact solution for the stress is in terms of a convolution integral, for which we establish bounds. We then compute the convolution integral using Gauss–Kronrod quadrature by ensuring that its value always lies within the bounds. Using the validated quadrature, we can accurately evaluate the exact solution and we the exact solution it to validate the numerical scheme employed in solving the governing equations for stress-power law fluids with arbitrary exponent n. Finally, for stress power-law fluids wherein the exponent n 0 (stress-thickening fluids), we obtain an approximate solution for the stress that agrees well with the numerical solution.
TL;DR: In this paper, the consequences of the constraint of inextensibility with regard to a class of implicit constitutive relations, where the strain is given as a function of the stress, are studied.
Abstract: In this paper, we study the consequences of the constraint of inextensibility with regard to a class of constitutive relations, where the strain is given as a function of the stress. Such constitutive equations belong to a wider class of implicit constitutive relations, which have been proposed recently in the literature.
TL;DR: A nonlinear crack problem subject to a non-penetration inequality is considered within the framework of the limiting small strain approach, which does not suffer from the inconsistency of infinite strain at the crack tip.
Abstract: A nonlinear crack problem subject to a non-penetration inequality is considered within the framework of the limiting small strain approach, which does not suffer from the inconsistency of infinite strain at the crack tip. Based on the concept of a generalized solution, sufficient conditions proving the well-posedness of the problem are established and analyzed.
TL;DR: In this article, the authors address a misconception that seems to be prevalent with regard to the use of compatibility conditions for the strains in both the linearized and non-linear theories of elasticity, when one uses a stress-based approach.
Abstract: This short note addresses a misconception that seems to be prevalent with regard to the use of compatibility conditions for the strains in both the linearized and non-linear theories of elasticity, when one uses a “stress-based” approach.
TL;DR: In this paper, the authors study the behavior of a particular lumped parameter system whose mechanical response is given by a non-invertible expression for the displacement in terms of the force, under harmonic external force.
Abstract: The standard setting concerning vibrations of lumped parameter systems is based on the assumption that the mechanical response of the elements of the system is given explicitly in terms of kinematical variables. In particular, the force in a spring element is assumed to be given as a function of the displacement from the equilibrium position. However, some simple mechanical systems such as linear springs with limited compressibility/extensibility do not fit into the standard setting. In this case the displacement must be written as a function of the force. In general, the mechanical response of such elements must be described by an implicit relation between the force and kinematical variables. We study the behaviour of a particular lumped parameter system whose mechanical response is given by a non-invertible expression for the displacement in terms of the force, under harmonic external force. We show that a solution to the original system wherein the displacement is given in terms of the force can be obtained as a limit of a sequence of approximate problems. The approximate problems are designed in such a way that they can be solved using standard numerical methods, and one can avoid using concepts such as set valued mappings. Moreover, we show that the “bounce back” behaviour of the system with linear spring with limited compressibility/extensibility is a direct consequence of the assumed constitutive relation. There is no need to a priori supply the rules for the bounce back (impact rules). Further, we show that the advocated approximation procedure is capable of describing the behaviour of the lumped parameter system even in the situations where the governing ordinary differential equation collapses to an algebraic equation. Representative results are demonstrated by a numerical experiment.
TL;DR: In this paper, a two network theory based on the maximization of entropy production is used to obtain the constitutive equations governing the response of a composite viscoelastic plate subjected to impact on one face of the plate.
TL;DR: In this article, the effect of stress relaxation in a component of the prestressed composite on the overall load-carrying capacity of the composite was examined. But the authors focused on the effect on the composites having linearized viscoelastic constituents that can exhibit fluidlike and solidlike behavior.
Abstract: Bodies are prestressed with the intention to enhance the load-carrying capacity of the body. The primary objective of this study is to understand the effect of prestressing the constituents in composite bodies on the overall mechanical performance of the composites. This paper considers composites having linearized viscoelastic constituents that can exhibit fluidlike and solidlike behavior. It also examines the effect of stress relaxation in a component of the prestressed composite on the overall load-carrying capacity of the composite. The properties of the composite, whether a brittle inclusion embedded in ductile matrix or a ductile inclusion in brittle matrix, are greatly influenced by the ratio of the induced prestress with respect to the external load and thereby influence the load-carrying capacity of the composite.
TL;DR: In this article, the authors considered fiber reinforced composite materials in which the fibers have more than just a stiffening function and developed general constitutive equations for the stress, polarization vector, current density vector and heat flux in terms of the deformation, electric field vector and temperature gradient.
Abstract: The present work considers fiber reinforced composite materials in which the fibers have more than just a stiffening function. The composite is assumed to be composed of a non-conducting matrix reinforced with electroelastic fibers that conduct both current and heat in addition to supporting an applied load. The material system is treated as equivalent homogenized material that is nonlinearly elastic and transversely isotropic with the fiber direction as the direction of transverse isotropy. General constitutive equations are developed for the stress, polarization vector, current density vector and heat flux in terms of the deformation, electric field vector and temperature gradient. From these the special constitutive equations are extracted for a non-conducting matrix with conducting reinforcing fibers.