# K. R. Rajagopal's research while affiliated with Texas A&M University and other places

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## Publications (534)

The objective of this study is to understand the formation of vortices and other flow characteristics associated with the three-dimensional motions of an incompressible Navier–Stokes fluid in tubes containing a sinusoidal extension. The study has some bearing on two conjectures that da Vinci made concerning the flow of blood through the aortic root...

In Part I, an implicit constitutive relation was proposed to describe stress softening of solids. In this second part, we use the constitutive relation developed earlier to study different boundary value problems, namely the homogeneous compression of a cylinder without radial stresses, the homogeneous compression of a cylinder with radial stress (...

An implicit constitutive relation is proposed to describe stress softening exhibited by solids, using as a basis the second law of thermodynamics, and appropriate choices for the specific Helmholtz potential and the rate of entropy production function. The implicit constitutive relation that is developed is a generalization of the earlier one-dimen...

Despite the tremendous impact that a good constitutive relation for the response of arterial tissues can have with regard to advances in cardiovascular science and medicine, and notwithstanding the intense effort to put a felicitous constitutive relation into place, no reliable constitutive relation is available in the literature. In this review ar...

Motivated by recent strain-limiting models for solids and biological fibers, we introduce the first intrinsic set of nonlinear constitutive relations, between the geometrically exact strains and the components of the contact force and contact couple, describing a uniform, hyperelastic, strain-limiting special Cosserat rod. After discussing some att...

In this paper we study the deformation of a body with a notch subject to an anti-plane state of stress within the context of a new class of elastic models. These models stem as approximations of constitutive response functions for an elastic body that is defined within the context of an implicit constitutive relation between the stress and the defo...

In this article, we consider the motion planning of a rigid object on the unit sphere with a unit speed. The motion of the object is constrained by the maximum absolute value, $U_{max}$ of geodesic curvature of its path; this constrains the object to change the heading at the fastest rate only when traveling on a tight smaller circular arc of radiu...

Phase transition in continua is, by its very definition, a dynamic process. Micro-structural changes take place, entropy is produced, and the body's natural configuration (the configuration the body would take when all external stimuli are removed) evolves in time. Motivated by these considerations, we formulate and consider the problem of an inext...

In this paper, a model for the phase change process due to irradiation with an Ultraviolet (UV) light in a mold filled with a viscoelastic fluid in roll-to-roll nanoimprinting lithography is developed. Employing a thermomechanical approach, constitutive equations for the phase change of a mixture of viscoelastic fluid and solid constituents of the...

In this paper we study the demolding process of a viscoelastic solid feature at nano scale after being imprinted by a roll-to-roll Nanoimprinting Lithography (R2R NIL) process. We assume that the feature has undergone some previous processes from which a state of initial internal stresses and strains are present and influence the behavior of the de...

In this short note we study finite amplitude stability of the rest state of a class of fluids which form a natural complement to generalized Newtonian fluids in that the symmetric part of the velocity gradient is a function of the stress. We find that the rest state of the fluid is asymptotically stable to finite amplitude disturbances in that the...

In this paper, we study the deformation and the stress concentration factor due to a small circular hole in a thin nonlinearly elastic large sheet reinforced by two families of fibers under a state of plane stress undergoing large deformations. We restrict our attention to the class of materials wherein the fibers are mechanically equivalent, rende...

In this short note, we develop a constitutive relation that is linear in both the Cauchy stress and the linearized strain, by linearizing implicit constitutive relations between the stress and the deformation gradient that have been put into place to describe the response of elastic bodies (Rajagopal, KR. On implicit constitutive theories. Applicat...

In this paper, we use an implicit generalization of the celebrated Kelvin–Voigt solid constitutive equation to study the problem of circumferential shearing of a cylindrical annulus of viscoelastic material. This generalization of the Kelvin–Voigt model allows one to take into consideration the possibility of the body undergoing shear thinning or s...

In this paper a model for the phase change process due to irradiation with an Ultraviolet (UV) light in a mold filled with a viscoelastic fluid in roll-to-roll nanoimprinting lithography is developed. Employing a thermomechanical approach, constitutive equations for the phase change of a mixture of viscoelastic fluid and solid constituents of the p...

In this note we consider 1D problems within the context of a new class of elastic bodies. Under suitable conditions on the constitutive equations we prove instability and nonexistence of solutions similar to those in place for the linearized theory. The last section is devoted to describing the spatial behavior of the solutions.

Most solid bodies are porous and in such bodies we expect the material properties to vary with porosity and hence with the density. Such bodies whose properties depend on the density cannot be described by the classical linearized elastic constitutive relation when they are undergoing small deformations, as the material moduli cannot depend on dens...

In this note, we study the response of a viscoelastic body whose stress relaxation modulus and creep compliance depend on the density of the body in such a manner that the stress and strain appear linearly in the constitutive equation. Such models would be useful to study the response of porous viscoelastic bodies undergoing small deformations, as...

An implicit constitutive relation is proposed to study transversely isotropic bodies. The relation is obtained assuming the existence of a Gibbs potential that depends on the second Piola–Kirchhoff stress tensor, from which the Green Saint-Venant strain tensor is obtained as the derivative with respect to the stress. The responses of unconstrained...

Viscoelastic fluids are non-Newtonian fluids that exhibit both “viscous” and “elastic” characteristics in virtue of the mechanisms used to store energy and produce entropy. Usually, the energy storage properties of such fluids are modeled using the same concepts as in the classical theory of nonlinear solids. Recently, new models for elastic solids...

Viscoelastic fluids are non-Newtonian fluids that exhibit both "viscous" and "elastic" characteristics in virtue of mechanisms to store energy and produce entropy. Usually the energy storage properties of such fluids are modelled using the same concepts as in the classical theory of nonlinear solids. Recently new models for elastic solids have been...

Wet granular materials develop both shear stresses and normal stresses during simple shear. Over a certain range of shear rate levels, both the shear stress and the normal stress that develop are linearly proportional to the shear rate. In addition, such materials exhibit permanent increase in volume during simple shear. In this article, a model is...

A fluid occupying a mechanically isolated vessel with walls kept at spatially non-uniform temperature is in the long run expected to reach the spatially inhomogeneous steady state. Irrespective of the initial conditions the velocity field is expected to vanish, and the temperature field is expected to be fully determined by the steady heat equation...

We consider flows of an incompressible Navier-Stokes fluid in a tubular domain with Navier’s slip boundary condition imposed on the impermeable wall. We focus on several implementational issues associated with this type of boundary conditions within the framework of the standard Taylor-Hood mixed finite element method and present the computational...

A thermodynamic framework has been developed for a class of amorphous polymers used in fused deposition modeling (FDM), in order to predict the residual stresses and the accompanying distortion of the geometry of the printed part (warping). When a polymeric melt is cooled, the inhomogeneous distribution of temperature causes spatially varying volum...

We present a mathematical model to study the dissipation of energy in a composite helmet that is subjected to blast loading. In addition to assuming a constitutive relation for the material of the helmet, we also assume models for the skull, and brain, all of them being treated as homogeneous isotropic bodies. A rate type model with non-linear diss...

In this short paper, I provide a brief discussion of the rationale and need for implicit constitutive relations for describing the response of many real materials. Classical theories to describe the response of materials such as the Cauchy theory of elasticity and the Navier–Stokes equations turn the demands of causality on its head and provide an...

Damage in concrete has been modelled using various approaches such as fracture mechanics, continuum damage mechanics and failure envelope theories. This study proposes a new approach to model the initiation of damage in concrete that addresses some limitations associated with the existing approaches. The proposed approach defines damage in terms of...

In Part I, a density driven damage mechanics (D3-M) approach and its application to model mechanical damage in concrete are presented. In this study, chemical and chemo-mechanical damage in concrete were modelled using the D3-M approach. It is proposed that reductions in local density in certain regions, created when concrete is subjected to chemic...

A constitutive relation was developed in Part I for describing the response of a class of visco-elastic bodies, wherein the left Cauchy Green tensor, the symmetric part of the velocity gradient, and the Cauchy stress tensor are related through an implicit constitutive relation. Here, we study a boundary value problem within the context of the model...

We derive the constitutive equations for a viscoelastic fluid that is undergoing a continuous mold filling process at the nanoscale within the context of a thermomechanical framework. The governing equations are obtained by substituting the constitutive equations into the appropriate balance laws. From the general characteristics inherent to the ro...

A constitutive relation is proposed for viscoelastic bodies that is a generalization of the classic Kelvin-Voigt model, wherein the left Cauchy Green tensor, the symmetric part of the velocity gradient, and the Cauchy stress tensor are implicitly related. The model developed includes several models that are being used in the literature to describe...

Aortic dissection occurs predominantly in the thoracic aorta and the mechanisms for the initiation and propagation of the tear in aortic dissection are not well understood. We study the tearing characteristics of the porcine thoracic aorta using a peeling test and we estimate the peeling energy per unit area in the ascending and the descending segm...

We derive a representation formula for a class of solids described by implicit constitutive relations between the Cauchy stress tensor and the Hencky strain tensor. Using a thermodynamic framework, we show that the Hencky strain tensor can be obtained as the derivative of the specific Gibbs free energy with respect to a stress tensor related to the...

Light-activated shape memory polymers (LASMPs) are a new generation of active materials that undergo phase transformations upon light irradiation at different wavelength and intensities. This type of material shows much promise in its use in adaptive and shape reconfiguration structures for applications in biomedical and aerospace engineering. LASM...

We consider the deformation of a nonlinearly elastic wedge in the case of a stored energy function that is given by a power law. Numerical solutions that are not unidirectional, in that the displacement is towards the apex in certain regions of the wedge and away from the apex in other regions, are obtained. Boundary conditions describing the exten...

In this paper we study the motion of a finite composite cylindrical annulus made of generalized neo-Hookean solids that is subject to periodic shear loading on the inner boundary. Such a problem has relevance to several problems of technological significance, for example blood vessels can be idealized as finite anisotropic composite cylinders. Here...

The kinematics of an inelastic solid established in terms of Laplace stretch
and its rate are decomposed into one stretch that describes an elastic response,
another stretch that describes an inelastic response, and their rates.
These kinematics are a direct consequence of Laplace stretch belonging to
the group of all real, 3x3, upper-triangular ma...

In this paper we study the deformation of a body with a notch subject to an anti-plane state of stress within the context of a new class of elastic models. These models stem as approximations of constitutive response functions for an elastic body that is defined within the context of an implicit constitutive relation between the stress and the defo...

In this paper we study the motion of a finite composite cylindrical annulus made of generalized neo-Hookean solids that is subject to periodic shear loading on the inner boundary. Such a problem has relevance to several problems of technological significance, for example blood vessels can be idealized as finite anisotropic composite cylinders. Here...

In the first part of the paper we provide a new classification of incompressible fluids characterized by a continuous monotone relation between the velocity gradient and the Cauchy stress. The considered class includes Euler fluids, Navier-Stokes fluids, classical power-law fluids as well as stress power-law fluids, and their various generalization...

We consider the deformation of a nonlinearly elastic wedge in the case of a stored energy function that is given by a power law. This model, introduced by Knowles, reduces to the classical neo-Hookean model when the power-law exponent is unity, and for certain values of the exponent the equations of anti-plane strain lose ellipticity allowing one t...

The working temperature of a bituminous pavement can typically range from 75 C to C. Bitumen shows a wide spectrum of mechanical behaviour in this temperature range and these include those of a viscoelastic fluid, a viscoelastic solid and an elastic solid. Due to the amorphous nature of the material, the transitions between such mechanical response...

The last sentence of [1] on page 1612, namely: “In fact no Helmholtz potential for an isotropic material can reduce this 1D equation of Fung.”

In their study “Reversal of flow of non-Newtonian fluid in an expanding channel” (Harley et al., 2018), the authors assume a similarity transformation to study the problem of the flow between two intersecting planes. After deriving their equations, the authors purport to obtain a numerical solution to the problem. This solution is incorrect as a si...

In this paper we study the response of bodies that are residually stressed within the context of a new class of constitutive relations, wherein the strains are assumed to be functions of the stresses. Such bodies are said to have residual stresses if there are stresses within the bodies even though the bodies are unstrained in the configuration of...

We extend the methodology introduced for the initiation of damage within the context of a class of elastic solids to a class of viscoelastic solids (Alagappan et al. 2016 Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 472, 20160231. (doi:10.1098/rspa.2016.0231)). In a departure from studies on damage that consider the body to be homogeneous, with ini...

We consider the flow of a rate-type fluid defined by an implicit constitutive equation in a channel with non flat walls. We also assume that the channel characteristic width is small in comparison to the channel length, so that the lubrication approximation can be applied. The model developed is mainly motivated by the evidence that many lubricants...

The flow of an incompressible power-law fluid through convergent–divergent channels is considered where the choice of the viscosity is such that the stress tensor is not degenerate in the sense that the zero shear rate viscosity is neither zero nor infinity for any finite value of the power-law exponent in contrast to the earlier study by Mansutti...

In this paper, a nonlinear constitutive relation is proposed to model the behaviour of sandstone. The model is based on a relatively new class of constitutive relations proposed recently in the literature, which cannot be classified as Cauchy or Green elastic bodies. A specific expression for the constitutive relation is proposed on the basis of so...

Structures made of viscoelastic materials generate sufficient heat during relatively long exposure to cyclic loading thereby perceptibly altering their body temperature. Temperature changes influence the rate of stress relaxation (or the rate of creep) in viscoelastic materials which in turn affects the deformations of the structures. This study is...

A methodology for obtaining implicit constitutive representations involving the Cauchy stress and the Hencky strain for isotropic materials undergoing a non-dissipative process is developed. Using this methodology, a general constitutive representation for a subclass of implicit models relating the Cauchy stress and the Hencky strain is obtained fo...

We discuss the development of a method for the determination of the material properties of rubber and rubberlike materials within the context of a novel constitutive framework that has been put into place recently. The new constitutive framework leads to fewer material moduli than the models that are currently in vogue. We corroborate the predictio...

Within the context of the non-linear theory of Cauchy elastic bodies (hence Green elastic bodies which are a sub-set of Cauchy elastic bodies wherein the stress is derivable from a potential), linearization with regard to the gradient of displacement, in the sense that the squares of the norms of the gradient of displacement can be neglected in com...

In this paper the state of stress and strain close to sharp cracks in bodies subjected to an anti-plane state of stress is studied within the context of a non-monotonic strain-stress relation within the context of a generalization of the Cauchy theory of elasticity, providing an exact analytical solution to the problem.
A discussion is provided to...

Recently, a thermodynamically consistent non-linear constitutive equation has been developed to describe the large deformation cyclic response of viscoelastic polyamides (see [17]). In this paper, two boundary value problems within the context of the above model, namely the stress relaxation of a right circular annular cylinder subject to twisting,...

This paper is Part 2 of a study of blood flow across cardiovascular stenoses. In Part 1, we developed a rigorous mathematical approach for deriving a pressure field from experimental data for a velocity field that can be obtained by direct measurement. In this Part, existing methods for quantifying stenoses, with specific reference to cardiac valve...

In this paper we generalize the recent implicit models that have been put into place to describe the elastic response of bodies when thermal effects come into play. The implicit constitutive relations for thermoelastic response presented here provide a very natural way to overcome a serious problem associated with the celebrated model due to Fourie...

This paper is Part 2 of a study of blood flow across cardiovascular stenoses. In Part 1, we developed a rigorous mathematical approach for deriving a pressure field from experimental data for a velocity field that can be obtained by direct measurement. In this Part, existing methods for quantifying stenoses, with specific reference to cardiac valve...

Fatigue and damage are the least understood phenomena in the mechanics of solids. Recently, Alagappan et al. (“On a possible methodology for identifying the initiation of damage of a class of polymeric materials”, Proc R Soc Lond A Math Phys Eng Sci 2016; 472(2192): 20160231) hypothesized a criterion for the initiation of damage for a certain class...

This paper is Part 2 of a study of blood flow across cardiovascular stenoses. In Part 1, we developed a rigorous mathematical approach for deriving a pressure field from experimental data for a velocity field that can be obtained by direct measurement. In this Part, existing methods for quantifying stenoses, with specific reference to cardiac valve...

In this article, we model the mechanical behavior of light-activated shape memory polymers with a view toward determining the effect of the viscoelasticity of the polymers with regard to their shape memory response. This study is a companion to our earlier investigation of both isotropic and anisotropic elastic light-activated shape memory polymers...

Many titanium alloys and even materials such as concrete exhibit a nonlinear relationship between strain and stress, when the strain is small enough that the square of the norm of the displacement gradient can be ignored in comparison to the norm of the displacement gradient. Such response cannot be described within the classical theory of Cauchy e...

In this paper, we extend the modeling effort of our earlier work concerning the development of an implicit elastic model for describing the response of biological fibers. We put into place a fractional-order viscoelastic (FOV) solid model that can quantify the properties of biological fibers comprised of collagen fibrils and elastic filaments. The...

In this paper we study the state of stress and strain in infinite elastic slabs of nonlinear viscoelastic solids containing elliptic holes subject to an uni-axial as well as a bi-axial state of stress. The geometry affords one to get some inkling concerning the states of stress and strain in bodies containing a crack by obtaining the limit of the s...

Many materials that have been developed recently such as titanium alloys and polymeric composites exhibit nonlinear elasticity in the “small” strain regime, and the linearized theory cannot be used to describe the response. Recently, Rajagopal (Appl Math 48(4):279–319, 2003) introduced a new implicit constitutive theory which can be used to develop...

The aim of this paper is to carry out a three dimensional finite deformation simulation of a layered polymeric structure subject to blast loading. Such a simulation may be of use to evaluate the relative capabilities of different layered composites at the design phase to identify suitable candidates for detailed testing. While this is a follow up t...

In this paper, we provide a possible methodology for identifying the initiation of damage in a class of polymeric solids. Unlike most approaches to damage that introduce a damage parameter, which might be a scalar, vector or tensor, that depends on the stress or strain (that requires knowledge of an appropriate reference configuration in which the...

In this short note we study special unsteady flows of a fluid whose viscosity depends on both the pressure and the shear rate. Here we consider an interesting dependence of the viscosity on the pressure and the shear rate; a power-law of the shear rate wherein the exponent depends on the pressure. The problem is important from the perspective of fl...

In this paper we carry out an experimental study and develop and corroborate a model for describing the response of viscoelastic solids undergoing microstructural changes. We adopt the theory of multiple natural configurations (multi network model) and a single integral constitutive representation to incorporate the effect of microstructural change...

There are many alloys used in orthopaedic applications that are nonlinear in the elastic regime even when the strains are ‘small’ (see Hao et al., 2005; Saito et al., 2003; Sakaguch et al., 2004). By using conventional theories of elasticity, either Cauchy or Green elasticity, it is impossible to systematically arrive at constitutive equations, whi...

The aim of this paper is to develop a new unified class of 3D nonlinear anisotropic finite deformation inelasticity model that (1) exhibits rate-independent or dependent hysteretic response (i.e., response wherein reversal of the external stimuli does not cause reversal of the path in state space) with or without yield surfaces. The hysteresis pers...

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 elast...

The final-to-initial stiffness ratio is very large (>100) for many biological fibers, and as such, these materials have been modeled as being strain limiting. We propose an unconventional structure for a stored energy function that leads to a constitutive relation capable of describing this observed strain-limiting behavior. The model can attain in...

In this short note, we study the stability of flows of a fluid through porous media that satisfies a generalization of Brinkman’s equation to include inertial effects. Such flows could have relevance to enhanced oil recovery and also to the flow of dense liquids through porous media. In any event, one cannot ignore the fact that flows through porou...

The consequences of the constraint of incompressibility is studied for a new class of constitutive relation for elastic bodies, for which the left Cauchy–Green tensor is a function of the Cauchy stress tensor. The requirement of incompressibility is imposed directly in the constitutive relation, and it is not necessary to assume a priori that the s...

This study aims to understand the effect of prestressing the inclusion on the overall mechanical performance of composites having nonlinear viscoelastic constituents. Particle reinforced composites are considered and each non-overlapping particle is assumed to be fully surrounded by polymeric matrix. We study different responses of polymeric matrix...

The non-invasive determination of the pressure (mean normal stress) in a flowing fluid has ramifications in a variety of important problems: the flow of blood in blood vessels, flows taking place in inaccessible locations in complex internal geometries that occur in mechanical systems, etc. In this paper we discuss a rigorous new mathematical proce...

The response of physical systems governed by linear ordinary differential equations to step input is traditionally investigated using the classical theory of distributions. The response of non-linear systems is however beyond the reach of the classical theory. The reason is that the simplest non-linear operation—multiplication—is not defined for di...