Showing papers by "Kumbakonam R. Rajagopal published in 2014"

On the nonlinear elastic response of bodies in the small strain range

Abstract: The classical linearized approximation to describe the elastic response of solids is the most widely used model in solid mechanics. This approximate model is arrived at by assuming that the norm of the displacement gradient is sufficiently small so that one can neglect the square of the norm in terms of the norm. Recent experimental results on Titanium and Gum metal alloys, among other alloys, indicate with unmistakable clarity a nonlinear relationship between the strain and the stress in the range of strain wherein one would have to use the classical linearized theory of elasticity, namely wherein the square of the norm of the strain can be ignored with regard to the value of the strain, leading to a dilemma concerning the modeling of the response, as the classical nonlinear Cauchy elastic model would collapse to the linearized elastic model in this range. A novel and important generalization of the theory of elastic materials has been suggested by Rajagopal in Appl Math 48: 279–319, 2003 and Zeit Angew Math Phys 58: 309–317, 2007 that allows for an approximation wherein the linearized strain can be a nonlinear function of the stress. In this paper, we show how this new theory can be used to describe the new experiments on Titanium and Gum metal alloys and also clarify several issues concerning the domain of application of the classical linearized theory.

On elastic solids with limiting small strain: modelling and analysis

Abstract: In order to understand nonlinear responses of materials to external stimuli of different sort, be they of mechanical, thermal, electrical, magnetic, or of optical nature, it is useful to have at one’s disposal a broad spectrum of models that have the capacity to describe in mathematical terms a wide range of material behavior. It is advantageous if such a framework stems from a simple and elegant general idea. Implicit constitutive theory of materials provides such a framework: while being built upon simple ideas, it is able to capture experimental observations with the minimum number of physical quantities involved. It also provides theoretical justification in the full three-dimensional setting for various models that were previously proposed in an ad hoc manner. From the perspective of the theory of nonlinear partial differential equations, implicit constitutive theory leads to new classes of challenging mathematical problems. This study focuses on implicit constitutive models for elastic solids in general, and on its subclass consisting of elastic solids with limiting small strain. After introducing the basic concepts of implicit constitutive theory, we provide an overview of results concerning modeling within the framework of continuum mechanics. We then concentrate on the mathematical analysis of relevant boundary-value problems associated with models with limiting small strain, and we present the first analytical result concerning the existence of weak solutions in general three-dimensional domains.

A thermodynamic basis for the derivation of the Darcy, Forchheimer and Brinkman models for flows through porous media and their generalizations

Abstract: In this study we use a general thermodynamic framework which appeals to the criterion of maximal rate of entropy production to obtain popular models due to Darcy and Brinkman and their generalizations, to describe flow of fluids through porous solids. Such a thermodynamic approach has been used with great success to describe various classes of material response and here we demonstrate its use within the context of mixture theory to obtain the classical models for the flow of fluids through porous media and more general models which are all consistent with the second law of thermodynamics.

Circularly polarized wave propagation in a class of bodies defined by a new class of implicit constitutive relations

Abstract: In this paper, we show that circularly polarized transverse stress waves, standing shear stress waves, and oscillatory shear stress waves can propagate in a new class of viscoelastic solid bodies which are a subclass of bodies described by implicit constitutive theories. The class of models that is being considered includes as sub-classes, the classical Kelvin–Voigt model, the new models introduced by Rajagopal wherein the Cauchy–Green tensor is a non-linear function of the stress, and the Navier–Stokes fluid model. The solutions established in this paper are generalizations of solutions that have been established within the context of nonlinear elasticity by Carroll, and Destrade and Saccomandi, to the new class of elastic and viscoelastic bodies that are being considered.

Flow of a Burgers fluid due to time varying loads on deforming boundaries

Abstract: In this paper we study three boundary-initial value problems within the context of four rate type viscoelastic constitutive models, the Maxwell model, the Oldroyd-B model, Burgers model and the generalized Burgers model. We consider challenging problems wherein the boundary is deforming with time. The flows lead to a complex system of partial differential equations that require the development of a robust numerical procedure based on the arbitrary Lagrangian–Eulerian method.

On the natural structure of thermodynamic potentials and fluxes in the theory of chemically non-reacting binary mixtures

Abstract: A theory describing the behavior of chemically non-reacting binary mixtures can be based on a detailed formulation of the governing equations for the individual components of the mixture or on treating the mixture as a single homogenized continuous medium. We argue that if we accept that both approaches can be used to describe the behavior of the given mixture, then the requirement on the equivalence of these approaches places restrictions on the possible structure of the internal energy, entropy, Helmholtz potential, and also of the diffusive, energy, and entropy fluxes. (The equivalence of the approaches is understood in the sense that the quantities used in one approach can be interpreted in terms of the quantities used in the other approach and vice versa. Further, both approaches must lead to the same predictions concerning the evolution of the physical system under consideration). In the case of a general chemically non-reacting binary mixture of components at the same temperature, we show that these restrictions can indeed be obtained by purely algebraic manipulations. An important outcome of this analysis is, for example, a general form of the evolution equation for the diffusive flux. The restrictions can be further exploited in the specification of thermodynamically consistent constitutive relations for quantities such as the interaction (drag) force or the Cauchy stress tensor. As an example of the application of the current framework, we derive, among others, a generalization of Fick’s law and we recover several non-trivial results obtained by other techniques. The qualitative features of the derived generalization of Fick’s law are demonstrated by a numerical experiment.

Universal relations for a new class of elastic bodies

Abstract: Universal relations hold in all members that belong to a certain class of bodies, and they are therefore useful in designing experiments in which all members belonging to the particular class of bodies can be tested. It has been shown recently that the class of elastic bodies is much larger than the classical Cauchy elastic bodies. It has also been shown that such elastic bodies have firm thermodynamic underpinnings. In this short paper, we discuss universal relations that hold for a large sub-class of bodies which belong to this new class of elastic bodies. To be more precise, we consider a class of compressible isotropic elastic solids that are a sub-class of the new class of elastic bodies. We show that practically all the universal solutions which are possible in classical Cauchy elastic bodies are also possible within the context of the sub-class of elastic bodies that we consider.

Thermo-inelastic Response of Polymeric Solids

Abstract: : We the study the impact response of a large layered viscoelastic plate by subjecting it to a variety of inputs at one end of plate while the other end is fixed. The nonlinear response of the material is modeled using an implicit two network theory. A finite volume technique is used to solve the momentum equation together with a one-step explicit scheme for the time evaluation and in order to capture the wave propagation phenomena. Polymethylmethacrylate (PMMA) and Polycarbonate (PC) are used in this study. Results indicate that the stress on the wall was lowest when using a trilayer (PMMA/PC/PMMA). Such problems have relevance to several applications including blast protection and wave attenuation. The parameters characterizing the model were obtained from the creep compliance data found in the literature. The reflected and the transmitted wave characteristics for the layered materials were studied using the model. The study can be viewed as a significant step in the determination the shock absorbing characteristics of composite layers made of polymeric materials and the model and the method used in this study serve as a tool for selecting better impact resistant composite bodies.

Analysis of the equations governing the motion of a degrading elastic solid due to diffusion of a fluid

Abstract: Electronic version of an article published as "IMA Journal of applied mathematics", vol. 79, no 5, 2014, p. 778-789. DOI: 10.1093/imamat/hxt050