Showing papers by "Kumbakonam R. Rajagopal published in 2016"

A promising approach for modeling biological fibers

Abstract: 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 infinite stress at a finite strain while storing a finite amount of internal energy. Many biological fibers have a mechanical response that starts out as being compliant and nonlinear, and transitions into one that is stiff and linear. We present a biological fiber model comprised of a strain-limiting fiber (strain being attributed to molecular reconfiguration) loaded in conjunction with a Hookean fiber (strain being attributed to molecular stretch). The model’s parameters are physical, intuitive and readily extracted from a stress/strain curve. Chordae tendineae data are used to demonstrate the efficacy of the model.

An implicit three-dimensional model for describing the inelastic response of solids undergoing finite deformation

Abstract: 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 persists with quasistatic loading. (2) Encompasses a wide range of different types of inelasticity models (such as Mullins effect in rubber, rock and soil mechanics, traditional metal plasticity, hysteretic behavior of shape memory materials) into a simple unified framework that is relatively easy to implement in computational schemes and (3) does not require any a priori particular notion of plastic strain or yield function. The core idea behind the approach is the development of an system of implicit rate equations that allow for the continuity of the response but with different rates along different directions. The theory, which is in purely mechanical setting, subsumes and generalizes many commonly used approaches for hypoelasticity and rate-independent plasticity. We illustrate its capability by modeling the Mullins effect which is the inelastic behavior of certain rubbery materials. We are able to simulate the entire cyclic response without the use of additional internal variables, i.e., the entire response is modeled by using an implicit function of stress and strain measures and their rates.

Finite element modelling of field compaction of hot mix asphalt. Part II: Applications

Abstract: A constitutive model is developed and implemented in the finite element system three-dimensional computer-aided pavement analysis for the simulation of hot mix asphalt field compaction. The details of this model are presented in a companion paper (Masad et al., Finite element modelling of field compaction of hot mix asphalt. Part I: Theory, International Journal of Pavement Engineering, Accepted, 2014). This model is based on nonlinear viscoelasticity theory and can accommodate large deformations that occur during the compaction process. The model was used to study the influence of frequency and amplitude of vibratory compaction rollers on the level of compaction. In addition, it was used to analyse the influence of various methods for compacting longitudinal joints on the percent air voids near these joints. The model was used to simulate the compaction of asphalt pavements with different structures and compacted using various equipment and patterns. The finite element results of the level of compaction ...

A Novel Approach to the Description of Constitutive Relations

Abstract: Recent advances in the development of implicit constitutive relations to describe the response of both solids and fluids have greatly increased the repertoire of the modeler in his ability to describe natural phenomena more faithfully than hitherto possible. It would not be an exaggeration to claim that such constitutive relations have the potential to lead to breakthroughs in mechanics as they provide very promising novel means to study two of the most important and ill-understood problems in mechanics, that of fracturing of solids, and of turbulence in fluids, in addition to providing a means to describe a plethora of phenomena that have eluded explanation in biomechanics, response of colloids and mixtures, etc. In this article we describe these recent developments within the context of both fluid and solid mechanics. A Novel approach to the description of constitutive relations

Finite element modelling of field compaction of hot mix asphalt. Part I: Theory

Abstract: This paper presents the theoretical background for the development of a constitutive model that is used in the simulation of the compaction of asphalt mixtures. The constitutive model is developed to comply with the principles of thermodynamics, and is derived to represent the macroscopic behaviour of an asphalt mixture as a highly compressible viscoelastic material. The paper presents the details of the mathematical formulation and the computational implementation of the model in the finite element package computer-aided pavement analysis 3D. The capabilities of the compaction model and its sensitivity to changes in model's parameters are illustrated using simple numerical applications. In a companion publication, (Masad et al., Finite element modelling of field compaction of hot mix asphalt. Part II: Application, International Journal of Pavement Engineering, Accepted, 2014), the model is verified against field compaction measurements which demonstrate the ability of the model to capture the general tre...

A thermodynamically compatible model for describing asphalt binders: solutions of problems

Abstract: In this sequel to the first paper (Malek et al., 2014. International Journal of Pavement Engineering), in which we identified a generalisation of the model due to Burgers which was corroborated against two sets of experiments, including a challenging one showing distinctly different relaxation times for shear and normal stresses, we solve several time-dependent boundary value problems wherein the boundary of the material is deforming, that have relevance to applications involving asphalt. Problems wherein the boundary is subject to time-varying compressive loads such as those due to moving automobiles and the attendant rutting, and the compaction due to rollers are considered in additions to other problems.

On the anti-plane state of stress near pointed or sharply radiused notches in strain limiting elastic materials: closed form solution and implications for fracture assessements

Abstract: 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 elasticity is used to determine the full stress and strain field close to a pointed or radiused notch with any notch opening angle. An extensive discussion is provided that highlights the main features of stress and strain distributions, and the implications of the new theory for fracture assessments. In particular, it is proved that the obtained stress and strain solution predicts finite strains at the notch tip and allows the intensity of the stress field to be written as a function of the elastic Notch Stress Intensity Factor $$K_{3}$$ , as in the case of conventional linearized elasticity theory. This makes the strain limiting elasticity an excellent vehicle for justifying theoretically a K based-approach to the fracture of brittle elastic solids, within the context of a self consistent theory, unlike the classical linearized theory that predicts singularities for the strain at crack tips.

On the flow of fluids through inhomogeneous porous media due to high pressure gradients

Abstract: Most porous solids are inhomogeneous and anisotropic, and the flows of fluids taking place through such porous solids may show features very different from that of flow through a porous medium with constant porosity and permeability. In this short paper we allow for the possibility that the medium is inhomogeneous and that the viscosity and drag are dependent on the pressure (there is considerable experimental evidence to support the fact that the viscosity of a fluid depends on the pressure). We then investigate the flow through a rectangular slab for two different permeability distributions, considering both the generalized Darcy and Brinkman models. We observe that the solutions using the Darcy and Brinkman models could be drastically different or practically identical, depending on the inhomogeneity, that is, the permeability and hence the Darcy number.

Modelling the nonlinear viscoelastic response of asphalt binders

Abstract: In a recent article, Narayan et al. [Narayan, S.P.A., et al., 2012b. Nonlinear viscoelastic response of asphalt binders: experimental study of relaxation of torque and normal force in torsion. Mechanics Research Communications, 43, 66–74] recorded both torque and normal force in torsional relaxation experiments on asphalt binders. The data are three-dimensional and thus, require interpretation using three-dimensional constitutive relations. In this article, we develop such a three-dimensional nonlinear viscoelastic model for asphalt binder. The predictions of the model fit the experimental data reasonably well. While there are already some three-dimensional constitutive relations available in the literature for asphalt binders which can describe the normal forces that are developed during torsion, they do not capture the relaxation behaviour presented by Narayan et al. (2012b). This new model, however, captures most of the key features of the experimental data including the development of normal forces in...

On the consequences of the constraint of incompressibility with regard to a new class of constitutive relations for elastic bodies: small displacement gradient approximation

Abstract: Recently, there has been an interest in the development of implicit constitutive relations between the stress and the deformation gradient, to describe the response of elastic bodies as such constitutive relations are capable of describing physically observed phenomena, in which classical models within the construct of Cauchy elasticity are unable to explain. In this paper, we study the consequences of the constraint of incompressibility in a subclass of such implicit constitutive relations.

Euler–Bernoulli type beam theory for elastic bodies with nonlinear response in the small strain range

Abstract: The response of many new metallic alloys as well as ordinary materials such as concrete is elastic and nonlinear even in the small strain range. Thus, using the classical linearized theory to determine the response of bodies could lead to a miscalculation of the stresses corresponding to the given strains, even in the small strain regime. As stresses can determine the failure of structural members, such miscalculation could be critical. We investigate the quantitative impact of the material nonlinearity in the Euler–Bernoulli type beam theory. The governing equations for the deflection are found to be nonlinear integro-differential equations, and the equations are solved numerically using a variant of the spectral collocation method. The deflection and the spatial stress distribution in the beam have been computed for two sets of models, namely the standard linearized model and some recent nonlinear models used in the literature to fit experimental data. The predictions concerning the deflection and the spatial stress distribution based on the standard linearized model and the nonlinear models are considerably different.

A viscoelastic model for describing the response of biological fibers

Abstract: 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 compliance version of the FOV memory function is regularized, thereby removing the singularity from the viscoelastic kernel. The ensuing model is used to describe stress relaxation in mitral-valve chordae tendineae. Numerical solutions for the convolution integral are acquired via a midpoint quadrature rule with a Laplace endpoint correction.

On the response of physical systems governed by non-linear ordinary differential equations to step input

Abstract: 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 distributions. Yet the response of non-linear systems is of interest in many applications, and a framework capable of handling such problems is needed. We argue that a suitable framework is provided by Colombeau algebra that gives one the possibility to overcome the limitations of the classical theory of distributions, namely the possibility to simultaneously handle discontinuity, differentiation and non-linearity. Our thesis is documented by means of studying the response of two systems governed by non-linear ordinary differential equations subject to step input. In particular, we show that using the rules of calculus in Colombeau algebra it is possible to obtain an explicit and practically relevant characterisation of the behaviour of the considered systems at the point of the jump discontinuity.

On the stability and uniqueness of the flow of a fluid through a porous medium

Abstract: 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 porous media are inherently unsteady, and thus, at least a part of the inertial term needs to be retained in many situations. We study the stability of the rest state and find it to be asymptotically stable. Next, we study the stability of a base flow and find that the flow is asymptotically stable, provided the base flow is sufficiently slow. Finally, we establish results concerning the uniqueness of the flow under appropriate conditions, and present some corresponding numerical results.

On a possible methodology for identifying the initiation of damage of a class of polymeric materials

Abstract: 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 body was stress free and/or without any strain), we exploit knowledge of the fact that damage is invariably a consequence of the inhomogeneity of the body that makes the body locally 'weak' and the fact that the material properties of a body invariably depend on the density, among other variables that can be defined in the current configuration, of the body. This allows us to use density, for a class of polymeric materials, as a means to identify incipient damage in the body. The calculations that are carried out for the biaxial stretch of an inhomogeneous multi-network polymeric solid bears out the appropriateness of the thesis that the density of the body can be used to forecast the occurrence of damage, with the predictions of the theory agreeing well with experimental results. The study also suggests a meaningful damage criterion for the class of bodies being considered.

Mechanical oscillators with dampers defined by implicit constitutive relations

Abstract: We study the vibrations of lumped parameter systems, the spring being defined by the classical linear constitutive relationship between the spring force and the elongation while the dashpot is described by a general implicit relationship between the damping force and the velocity. We prove global existence of solutions for the governing equations, and discuss conditions that the implicit relation satisfies that are sufficient for the uniqueness of solutions. We also present some counterexamples to the uniqueness when these conditions are not met.

Large deformations of a new class of incompressible elastic bodies

Abstract: 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 stress tensor should be divided into two parts, a constraint stress and a constitutively specified part, as in the classical theory of nonlinear elasticity.

Numerical and approximate analytical solutions for cylindrical and spherical annuli for a new class of elastic materials

Abstract: 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 models to characterize the response of these newly fashioned materials. Here, we study the response of these new classes of elastic bodies within the context of two boundary value problems: the pressurization of a cylindrical annulus and a spherical shell. In the case of the cylindrical annulus, a stress function is introduced that automatically satisfies the equilibrium equation, and the compatibility equation for strain and the nonlinear constitutive equation is used to obtain the nonlinear compatibility equation in terms of the stress function. For the spherical shell, a displacement formulation is used to arrive at a nonlinear equation for the radial stress. The governing equations in both cases cannot be solved exactly, and we use an approximate technique, the variational iteration technique, to solve the problem. We show that this approximate solution agrees very well with the numerical solution of the governing equations, and the solution is different from that obtained in the classical linearized elasticity. In the case of spherical annulus with an internal pressure of 250 MPa, the hoop stress associated with the linearized solution overpredicts the numerical solution by about $$10\,\%$$ at the inner radius and underpredicts by about $$7\,\%$$ at the outer radius.

On stress-based piecewise elasticity for limited strain extensibility materials

Abstract: A one-dimensional stress-based elasticity model with limited strain extensibility is developed in this paper, based on thermodynamics arguments. Such nonlinear elastic models can be used to model certain rubber-like and biological materials with limiting chain extensibility. The derived constitutive function is a non-smooth piecewise expression, which can be regularized for numerical or physical considerations. This non-smooth constitutive expression is derived from a Gibbs potential. A three-dimensional extension of this stress-based model is also proposed in the paper. Some simple structural examples are investigated for a bar composed of this non-smooth elastic body. A homogeneous bar composed of this new class of nonlinear elastic material that is loaded is studied for different tension states, namely for concentrated or distributed axial loading. It is shown that the displacement limit extensibility can be observed at the structural scale, with finite or infinite axial load parameters.

Response of a class of mechanical oscillators described by a novel system of differential-algebraic equations

Abstract: We study the vibration of lumped parameter systems whose constituents are described through novel constitutive relations, namely implicit relations between the forces acting on the system and appropriate kinematical variables such as the displacement and velocity of the constituent. In the classical approach constitutive expressions are provided for the force in terms of appropriate kinematical variables, which when substituted into the balance of linear momentum leads to a single governing ordinary differential equation for the system as a whole. However, in the case considered we obtain a system of equations: the balance of linear momentum, and the implicit constitutive relation for each constituent, that has to be solved simultaneously. From the mathematical perspective, we have to deal with a differential-algebraic system. We study the vibration of several specific systems using standard techniques such as Poincare’s surface of section, bifurcation diagrams, and Lyapunov exponents. We also perform recurrence analysis on the trajectories obtained.

A fully coupled model for diffusion-induced deformation in polymers

Abstract: Polymers that mechanically respond to the presence of a diffusing fluid/solvent have found various applications in drug delivery, tissue scaffolding, sensors and actuators. These applications involve understanding of both the diffusion process and the evolution of the deformation of the polymers during the diffusion process. For example, in a polymeric actuator one might be interested in the extent of deformation one can achieve given a solvent environment and the time in which it can be achieved. There are two key aspects in modeling such behavior. First, the displacement gradients involved are usually large, especially in problems such as “self-assembly.” Second, since the diffusion occurs in a deforming polymeric medium, an appropriate diffusion model that includes the effect of the deformed state of the body as well as the interaction between the polymeric medium and the diffusing fluid has to be considered. In effect, this results in the diffusion and equilibrium equation being fully coupled and nonlinear. In this work, we model diffusion-induced deformation in an elastic material including large deformations based on thermodynamics framework. For the chemical potential, we use the Flory–Huggins potential adapted to include the effect of stress in the polymers. Using the model, we simulate folding and bending of a rectangular polymeric strip by simultaneous solution of the diffusion equation as well as the equilibrium equation using the finite element method. Parametric studies are also conducted in order to examine the effect of material parameters on the diffusion and deformation behaviors. Finally, using the coupled diffusion–deformation model we simulate deformations of composite domains comprising of polymeric constituents with different diffusion–deformation behaviors in order to achieve various interesting “self-assembly” shapes.

On the lubrication approximation for a class of viscoelastic fluids

Abstract: We present a rigorous derivation of a dimensionally reduced Reynolds type equation for thin film flow lubrication of a class of viscoelastic fluids by employing a perturbation analysis on the upper-convected Maxwell model in natural orthogonal coordinates. This approximation accounts for the viscoelastic and curvature corrections to the classical Reynolds lubrication approximation. Comparison of our approximation with the classical Reynolds approximation suggests that viscoelasticity can have a significant influence on the lubrication characteristics, at least for certain values of the film thickness and of the eccentricity ratios of the journal bearing.

On the Flows of Fluids Defined through Implicit Constitutive Relations between the Stress and the Symmetric Part of the Velocity Gradient

Abstract: Though implicit constitutive relations have been in place for a long time, wherein the stress, the strain (or the symmetric part of the velocity gradient), and their time derivatives have been used to describe the response of viscoelastic and inelastic bodies, it is only recently purely algebraic relationships between the stress and the displacement gradient (or the velocity gradient) have been introduced to describe the response of non-linear fluids and solids. Such models can describe phenomena that the classical theory, wherein the stress is expressed explicitly in terms of kinematical variables, is incapable of describing, and they also present a sensible way to approach important practical problems, such as the flows of colloids and suspensions and the turbulent flows of fluids, and that of the fracture of solids. In this paper we review this new class of algebraic implicit constitutive relations that can be used to describe the response of fluids.

Deformations of infinite slabs of non-linear viscoelastic solids containing an elliptic hole

Abstract: 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 solutions as the aspect ratio (in this case the ratio of the minor axis to the major axis) of the ellipse tends to zero. We consider two classes of non-linear viscoelastic bodies, the classical incompressible Kelvin–Voigt solid (Thomson in R Soc Lond 14:289–297, 1865; Voigt in Ann Phys 283(12):671–693, 1892) and a generalization of a compressible model due to Gent (Rubber Chem Technol 69(1):59–61, 1996). We also study for the sake of comparison the case of a nonlinear neo-Hookean elastic solid with an elliptic hole.

Classical Non-Newtonian Fluids

Abstract: The aim of this section is to discuss the mathematical properties of the governing equations of some non-Newtonian fluids introduced in Chapter 2, Section 2.4, namely, the Reiner-Rivlin fluid and in particular, the Bingham fluid.