arXiv subjects

K. R. Rajagopal

Publications and source records attributed to K. R. Rajagopal.

At least 19 recordsLinked to original sources

arXiv2025

Piecewise linear constitutive relations for stretch-limited elastic strings

This study proposes a simple and novel class of stretch-limiting constitutive relations for perfectly flexible elastic strings drawing from modern advances in constitutive theory for elastic bodies. We investigate strings governed by constitutive relations where stretch is a bounded, piecewise linear function of tension, extending beyond the traditional Cauchy elasticity framework. Our analysis includes explicit solutions for both catenaries and longitudinal, piecewise constant stretched motions.

math-ph↗

arXiv2024

A mathematical justification for nonlinear constitutive relations between stress and linearized strain

We present an asymptotic framework that rigorously generates nonlinear constitutive relations between stress and linearized strain for elastic bodies. Each of these relations arises as the leading order relationship satisfied by a one-parameter family of nonlinear constitutive relations between stress and nonlinear strain. The asymptotic parameter limits the overall range of strains that satisfy the corresponding constitutive relation in the one-parameter family while the stresses can remain large (relative to a fixed stress scale). This differs from classical linearized elasticity where a fixed constitutive relation is assumed, and the magnitude of the displacement gradient serves as the asymptotic parameter. Also unlike classical approaches, the constitutive relations in our framework are expressed as implicit relationships between stress and strain rather than requiring stress explicitly expressed as a function of strain, adding conceptual simplicity and versatility. We demonstrate that our framework rigorously justifies nonlinear constitutive relations between stress and linearized strain including those with density-dependent Young's moduli or derived from strain energies beyond quadratic forms.

math-ph↗

arXiv2024

The conclusion that metamaterials could have negative mass is a consequence of improper constitutive characterisation

The concept of "effective mass" is frequently used for the simplification of complex lumped parameter systems (discrete dynamical systems) as well as materials that have complicated microstructural features. From the perspective of wave propagation, it is claimed that for some bodies described as metamaterials, the corresponding "effective mass" can be frequency dependent, negative or it may not even be a scalar quantity. The procedure has even led some authors to suggest that Newton's second law needs to be modified within the context of classical continuum mechanics. Such absurd physical conclusions are a consequence of appealing to the notion of "effective mass" with a preconception for the constitutive structure of the metamaterial and using a correct mathematical procedure. We show that such unreasonable physical conclusions would not arise if we were to use the appropriate "effective constitutive relation" for the metamaterial, rather than use the concept of "effective mass" with an incorrect predetermined constitutive relation.

cond-mat.mtrl-sci↗

arXiv2023

Development of boundary layers in Euler fluids that on "activation'' respond like Navier-Stokes fluids

We consider the flow of a fluid whose response characteristics change due the value of the norm of the symmetric part of the velocity gradient, behaving as an Euler fluid below a critical value and as a Navier-Stokes fluid at and above the critical value, the norm being determined by the external stimuli. We show that such a fluid, while flowing past a bluff body, develops boundary layers which are practically identical to those that one encounters within the context of the classical boundary layer theory propounded by Prandtl. Unlike the classical boundary layer theory that arises as an approximation within the context of the Navier-Stokes theory, here the development of boundary layers is due to a change in the response characteristics of the constitutive relation. We study the flow of such a fluid past an airfoil and compare the same against the solution of the Navier-Stokes equations. We find that the results are in excellent agreement with regard to the velocity and vorticity fields for the two cases.

math.NA↗

arXiv2023

Special Cosserat rods with rate-dependent evolving natural configurations

We present a nonlinear, geometrically exact, and thermodynamically consistent framework for modeling special Cosserat rods with evolving natural configurations. In contrast to the common usage of the point-wise Clausius-Duhem inequality to embody the Second Law of Thermodynamics, we enforce the strictly weaker form that the rate of total entropy production is non-decreasing. The constitutive relations between the state variables and applied forces needed to close the governing field equations are derived via prescribing frame indifferent forms of the Helmholtz energy and the total dissipation rate and requiring that the state variables evolve in a way that maximizes the rate of total entropy production. Due to the flexibility afforded by enforcing a global form of the Second Law, there are two models obtained from this procedure: one satisfying the stronger form of the Clausius-Duhem inequality and one satisfying the weaker global form of the Clausius-Duhem inequality. Finally, we show that in contrast to other viscoelastic Cosserat rod models introduced in the past, certain quadratic strain energies in our model yield both solid-like stress relaxation and creep.

math-ph↗

arXiv2022

On evolving natural curvature for an inextensible, unshearable, viscoelastic rod

We formulate and consider the problem of an inextensible, unshearable, viscoelastic rod, with evolving natural configuration, moving on a plane. We prove that the dynamic equations describing quasistatic motion of an Eulerian strut, an infinite dimensional dynamical system, are globally well-posed. For every value of the terminal thrust, these equations contain a smooth embedded curve of static solutions (equilibrium points). We characterize the spectrum of the linearized equations about an arbitrary equilibrium point, and using this information and a convergence result for dynamical systems due to Brunovský and Polácik, we prove that every solution to the quasistatic equations of motion converges to an equilibrium point as time goes to infinity.

math-ph↗

arXiv2022

On an elastic strain-limiting special Cosserat rod model

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 attractive features of the constitutive relations (orientation preservation, transverse symmetry, and monotonicity), we exhibit several explicit equilibrium states under either an isolated end thrust or an isolated end couple. In particular, certain equilibrium states exhibit Poynting like effects, and we show that under mild assumptions on the material parameters, the model predicts an explicit tensile shearing bifurcation: a straight rod under a large enough tensile end thrust parallel to its center line can shear.

math-ph↗

arXiv2022

The state of stress and strain adjacent to notches in a new class of nonlinear elastic bodies

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 deformation gradient. Gum metal and many metallic alloys are described well by such constitutive relations. We consider the state of anti-plane stress of a body with a smoothened V-notch within the context of constitutive relations for the linearized strain in terms of a power-law for the stretch. The problem is solved numerically and the convergence and the stability of the solution is studied.

math.NA↗

arXiv2022

Optimal Geodesic Curvature Constrained Dubins' Paths on a Sphere

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 radius $r <1$, where $r$ depends on the bound, $U_{max}$. We show in this article that if $0 \frac{1}{2}$, while paths of the above type may cease to exist depending on the boundary conditions and the value of $r$, optimal paths may be concatenations of more than three circular arcs.

math.OC↗

arXiv2021

Implicit type constitutive relations for elastic solids and their use in the development of mathematical models for viscoelastic fluids

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 successfully developed by appealing to implicit constitutive relations, and these new models offer a different perspective to the old topic of elastic response of materials. In particular, a sub-class of implicit constitutive relations, namely relations wherein the left Cauchy-Green tensor is expressed as a function of stress is of interest. We show how to use this new perspective it the development of mathematical models for viscoelastic fluids, and we provide a discussion of the thermodynamic underpinnings of such models. We focus on the use of Gibbs free energy instead of the Helmholtz free energy, and using the standard Giesekus/Oldroyd-B models, we show how the alternative approach works in the case of well-known models. The proposed approach is straightforward to generalise to more complex setting wherein the classical approach might be impractical of even inapplicable.

physics.flu-dyn↗

arXiv2021

A Thermodynamic Framework for Additive Manufacturing, using Amorphous Polymers, Capable of Predicting Residual Stress, Warpage and Shrinkage

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 volumetric shrinkage resulting in the generation of residual stresses. Shrinkage is incorporated into the framework by introducing an isotropic volumetric expansion/contraction in the kinematics of the body. We show that the parameter for shrinkage also appears in the systematically derived rate-type constitutive relation for the stress. The solidification of the melt around the glass transition temperature is emulated by drastically increasing the viscosity of the melt. In order to illustrate the usefulness and efficacy of the derived constitutive relation, we consider four ribbons of polymeric melt stacked on each other such as those extruded using a flat nozzle: each layer laid instantaneously and allowed to cool for one second before another layer is laid on it. Each layer cools, shrinks and warps until a new layer is laid, at which time the heat from the newly laid layer flows and heats up the bottom layers. The residual stresses of the existing and newly laid layers readjust to satisfy equilibrium. Such mechanical and thermal interactions amongst layers result in a complex distribution of residual stresses. The plane strain approximation predicts nearly equibiaxial tensile stress conditions in the core region of the solidified part, implying that a pre-existing crack in that region is likely to propagate and cause failure of the part during service. The free-end of the interface between the first and the second layer is subjected to the largest magnitude of combined shear and tension in the plane with a propensity for delamination.

physics.class-ph↗

arXiv2021

An implicit constitutive relation in which the stress and the linearized strain appear linearly, for describing the small displacement gradient response of elastic solids

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 (see \cite{rajagopal2003implicit}), by assuming that the displacement gradient is small. These implicit equations include the classical linearized elastic constitutive approximation as well as constitutive relations that imply limiting strain, as special subclasses.

physics.class-ph↗

arXiv2020

Unconditional finite amplitude stability of a fluid in a mechanically isolated vessel with spatially non-uniform wall temperature

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. This simple observation is however difficult to prove using the corresponding governing equations. The main difficulties are the presence of the dissipative heating term in the evolution equation for temperature and the lack of control on the heat fluxes through the boundary. Using thermodynamically based arguments, it is shown that these difficulties in the proof can be overcome, and it is proved that the velocity and temperature perturbations to the steady state actually vanish as the time goes to infinity.

math.AP↗

arXiv2019

On the classification of incompressible fluids and a mathematical analysis of the equations that govern their motion

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 generalizations including the fluids that we refer to as activated fluids, namely fluids that behave as an Euler fluid prior activation and behave as a viscous fluid once activation takes place. We also present a classification concerning boundary conditions that are viewed as the constitutive relations on the boundary. In the second part of the paper, we develop a robust mathematical theory for activated Euler fluids associated with different types of the boundary conditions ranging from no-slip to freeslip and include Navier's slip as well as stick-slip. Both steady and unsteady flows of such fluids in three-dimensional domains are analyzed.

math.AP↗

arXiv2019

Motion of a finite composite cylindrical annulus comprised of nonlinear elastic solids subject to periodic shear

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, we consider the annulus to be comprised of an isotropic material, namely a generalized neo-Hookean solid and study the effects of annular thickness on the stress distribution within the annulus. We solve the governing partial differential equations and examine the stress response for the strain hardening and strain softening cases of the generalized neo-Hookean model. We also solve the problem of the annular region being infinite in length, that reduces the problem to a partial differential equation in only time and one spatial dimension. When the thickness of the annulus is sufficiently large, the solutions to the problems exhibit very interesting boundary layer structure in that the norm of the strain has a large gradient in a narrow region adjacent to one of the boundaries, with the strain being relatively uniform outside the narrow region. We also find that beyond a distance of two times the annular wall thickness from the ends of the cylinder the solutions for the infinite length cylinder match solutions for the finite length cylinder implying that end effects are not felt in most of the length of a sufficiently long annulus.

physics.class-ph↗

arXiv2018

An adaptive finite element method for the inequality-constrained Reynolds equation

We present a stabilized finite element method for the numerical solution of cavitation in lubrication, modeled as an inequality-constrained Reynolds equation. The cavitation model is written as a variable coefficient saddle-point problem and approximated by a residual-based stabilized method. Based on our recent results on the classical obstacle problem, we present optimal a priori estimates and derive novel a posteriori error estimators. The method is implemented as a Nitsche-type finite element technique and shown in numerical computations to be superior to the usually applied penalty methods.

math.NA↗

arXiv2017

Numerical simulations of an incompressible piezoviscous fluid flowing in a plane slider bearing

We provide numerical simulations of an incompressible pressure-thickening and shear-thinning lubricant flowing in a plane slider bearing. We study the influence of several parameters, namely the ratio of the characteristic lengths $\varepsilon>0$ (with $\varepsilon\searrow0$ representing the Reynolds lubrication approximation); the coefficient of the exponential pressure--viscosity relation $α^*\geq0$; the parameter $G^*\geq0$ related to the Carreau--Yasuda shear-thinning model and the modified Reynolds number $\mathrm{Re}_\varepsilon\geq0$. The finite element approximations to the steady isothermal flows are computed without resorting to the lubrication approximation. We obtain the numerical solutions as long as the variation of the viscous stress $\mathbf{S}=2η(p,\mathrm{tr}\mathbf{D}^2)\mathbf{D}$ with the pressure is limited, say $|\partial\mathbf{S}/\partial p|\leq1$. We show conclusively that the existing practice of avoiding the numerical difficulties by cutting the viscosity off for large pressures leads to results that depend sorely on the artificial cut-off parameter. We observe that the piezoviscous rheology generates pressure differences across the fluid film.

physics.flu-dyn↗

arXiv2017

Wave Patterns in A Nonclassic Nonlinearly-Elastic Bar Under Riemann Data

Recently there has been interest in studying a new class of elastic materials, which is described by implicit constitutive relations. Under some basic assumption for elasticity constants, the system of governing equations of motion for this elastic material is strictly hyperbolic but without the convexity property. In this paper, all wave patterns for the nonclassic nonlinearly elastic materials under Riemann data are established completely by separating the phase plane into twelve disjoint regions and by using a nonnegative dissipation rate assumption and the maximally dissipative kinetics at any stress discontinuity. Depending on the initial data, a variety of wave patterns can arise, and in particular there exist composite waves composed of a rarefaction wave and a shock wave. The solutions for a physically realizable case are presented in detail, which may be used to test whether the material belongs to the class of classical elastic bodies or the one wherein the stretch is expressed as a function of the stress.

math.AP↗