Showing papers by "Kumbakonam R. Rajagopal published in 2018"

Derivation of the Variants of the Burgers Model Using a Thermodynamic Approach and Appealing to the Concept of Evolving Natural Configurations

Abstract: Viscoelastic rate-type fluid models involving the stress and frame-indifferent time derivatives of second order, like those in Burgers’ model, are used to describe the complicated response of fluid like materials that are endowed with a complex microstructure that allows them to possess two different relaxation mechanisms as well as other non-Newtonian characteristics. Such models are used in geomechanics, biomechanics, chemical engineering and material sciences. We show how to develop such rate-type fluid models that include the classical Burgers’ model as well as variants of Burgers’ model, using a thermodynamic approach based on constitutive assumptions for two scalar quantities (namely, how the material stores energy and how the energy is dissipated) and appealing to the concept of natural configuration associated with the placement of the body that evolves as the body deforms.

On the states of stress and strain adjacent to a crack in a strain-limiting viscoelastic body.

Abstract: The viscoelastic Kelvin–Voigt model is considered within the context of quasi-static deformations and generalized with respect to a nonlinear constitutive response within the framework of limiting

A nonlinear model for describing the mechanical behaviour of rock

Abstract: 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 some experimental data for the compression of a sample of rock, and several boundary value problems are analysed, considering homogeneous distributions of strains, as well as a problem wherein one has a non-homogeneous distribution of strains. Finally, the behaviour of the P- and S-waves is studied for a sample of rock under compression, and it is discovered that the wave speed depends on the compressive load, a result supported by experiments.

Reversal of flow of a non-Newtonian fluid in an expanding channel

Abstract: 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 and Rajagopal (1991) wherein the viscosity could be zero or infinity for certain values of the power-law exponent. We observe the appearance of boundary layers for the non-Newtonian fluid, even in the case of divergent flow. Sharp and pronounced boundary layers develop adjacent to the boundaries, even at zero Reynolds number. Furthermore, for values of the angle beyond a critical value, we detect regions of flow reversal; i.e. different flow regimes are observed wherein there is inflow and outflow. We are also able to assess the consequences of introducing a traction boundary condition at the boundaries of the channel on the behaviour of the fluid. In this case we find the possibility of asymmetric solutions. We also find a new solution in the case of the Navier–Stokes fluid, albeit numerical, by setting the power-law exponent to zero.

Simulation of inextensible elasto-plastic beams based on an implicit rate type model

Abstract: The aim of this paper is to show that it is possible to model the elasto-plastic behavior of an inextensible beam undergoing finite bending by using a novel implicit rate type response relation directly between the bending moment, curvature, and their rates. This is in contrast to conventional approaches that require the integration of elasto-plastic stress–strain relations across the thickness of the beam at each time step. The model proposed here (a) is rate independent and exhibits hysteresis, (b) requires no notion of plastic strain and no integration across the thickness, and (c) does not have a sharp yield point and instead transits smoothly between nominally elastic and inelastic response. The governing equations are solved numerically as a set of first order differential equations with very low order interpolation functions for quasi-static response and are capable of modeling both hardening and softening behavior as well as the formation of plastic hinges, etc. Since there is no yield function, there is no necessity to use a return mapping algorithm that was developed specifically for use in elasto-plasticity with sharp yield functions. This simplifies the numerical algorithm also. The simulations for an elastic perfectly plastic beam are compared with elasto-plastic models in ABAQUS TM , and they show good agreement at a fraction of the computational time. Moreover we have compared simulations of large deformations of an aluminum alloy bar subject to three point bending experiments. The simulations also match the experimental results very well. We demonstrate that it is possible to incorporate frictionless sliding contact (which is needed for the three point bending comparison) in a relatively straightforward manner without the complexity that arises when it is handled in a full three-dimensional (3D) model.

On viscoelastic beams undergoing cyclic loading: Determining the onset of structural instabilities

Abstract: 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 aimed at understanding the consequences of heat generation due to energy dissipation in viscoelastic materials under cyclic loading and its consequences on the overall time-dependent deformations of structures. A relevant practical application would be understanding cyclic failure (fatigue) in viscoelastic structures. As an example, we consider a simple polymeric beam under bending deformation due to cyclic mechanical loading. A viscoelastic constitutive model for thermorheologically simple materials is used to describe the thermo-mechanical response of the polymeric beam. The amount of energy dissipation that is converted into heat is accounted for in formulating the constitutive model. In beams under bending, the regions with the larger stresses generate more heat, which accelerate the stress relaxation in these regions and cause a temperature gradient in the beam. In the analyses we also allow the generated heat to be conducted. The governing equations are implemented in ABAQUS finite element for coupled heat conduction and deformation analyses. The accelerated stress relaxation (or creep strain) due to an increase in temperature from the energy dissipation is found to have significant effect on the time-dependent deformation of the structures. After certain period of cyclic loading, the structure becomes unstable. The heat conduction process within the beam lead to the softening in the overall structural stiffness (due to the accelerated stress relaxation).

Chemo-mechanical coupling in curing and material-interphase evolution in multi-constituent materials

Abstract: Chemical reactions at bimaterial interfaces during manufacturing of fiber–matrix systems result in an interphase that plays a dominant role in the response of the composite when subjected to mechanical loads. An accurate modeling of the degree of cure in the interfacial region, because of its effect on the evolving properties of the interphase material, is critical to determining the coupled chemo-mechanical interphase stresses that influence the structural integrity of the composite and its fatigue life. A mixture model for curing and interphase evolution is presented that is based on a consistent thermodynamic theory for multi-constituent materials. The mixture model is cast in a stabilized finite element method that is developed employing variational multi-scale ideas for edge-based stabilization and consistent tying of the constituents at the domain boundaries. The ensuing computational method accounts for curing and interphase chemical reactions for the evolution of the density and material modulus of the constituents that have a direct effect on the interfacial stiffness and strength. Several test cases are presented to show the range of applicability of the model and the method.

A damage initiation criterion for a class of viscoelastic solids

Abstract: 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: Mat...

Initiation of damage in a class of polymeric materials embedded with multiple localized regions of lower density

Abstract: 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 of compressible polymeric solids, namely that damage will be initiated at the location where the derivative of the norm of the stress with respect to the stretch starts to decrease. This hypothesis led to results that were in keeping with the experimental work of Gent and Lindley(“Internal rupture of bonded rubber cylinders in tension. Proc. R. Soc. Lond. A 1959; 249, 195–205 :10.1098) and agrees qualitatively with the results of Penn (“Volume changes accompanying the extension of rubber”, Trans Soc Rheol 1970; 14(4): 509–517) on compressible polymeric solids. Alagappan et al. considered a body wherein there is a localized region in which the density is less th...

A New Class of Models to Describe the Response of Electrorheological and Other Field Dependent Fluids

Abstract: We propose a new class of models for electrorheological fluids. While the standard constitutive relations for electrorheological fluids are based on the assumption that the stress is a function of the symmetric part of the velocity gradient and the intensity of the electric field, we formulate constitutive relations in an implicit way. The stress, the symmetric part of the velocity gradient and the intensity of the electric field are linked via a tensorial implicit equation. The potential benefit of the new class of models is investigated by the analysis of a simple shear flow in a transverse electric field.

Modelling residual stresses in elastic bodies described by implicit constitutive relations

Abstract: 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 interest in the absence of external traction. Problems within the context of the norm of the gradient of the displacement field being small are considered, with regard to the determination of the residual stresses in an anisotropic cylindrical annulus with two preferred directions, and the nature of residual stresses within an anisotropic slab. The residual stresses in a body that is subject to incremental stresses are also studied.

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

Abstract: 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 $$\varepsilon \searrow 0$$ representing the Reynolds lubrication approximation); the coefficient of the exponential pressure–viscosity relation $$\alpha ^*\ge 0$$ ; the parameter $$G^*\ge 0$$ related to the Carreau–Yasuda shear-thinning model and the modified Reynolds number $${\mathrm {Re}}_\varepsilon \ge 0$$ . 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 $$\varvec{S}=2\eta (p,{{\mathrm{tr}}}\,\varvec{D}^2)\varvec{D}$$ with the pressure is limited, say $$|\partial \varvec{S}/\partial p|\le 1$$ . 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.

Correction to: A promising approach for modeling biological fibers

Abstract: 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.”

Experimental investigation on heat transfer enhancement and pressure drop of double pipe heat exchanger in solar water heating system

Abstract: This paper presents the design and performances of double pipe heat exchanger embedded straight rectangular fins in the annulus are presented. Solar water heating systems use heat exchangers to transfer solar energy absorbed in solar collectors to the working fluid used to heat the water or a space. An experimental investigation is conducted for different set values of mass flow rate and varying the number of rectangular fins. The experimental results are validated with plain double pipe heat exchanger. The results of rectangular fins in the annulus side causes increased rate of heat transfer and pressured drop compared to plain double pipe heat exchanger. The experimental study is performed by varying mass flow rate of 0.01 kg/s, 0.02 kg/s and 0.03 kg/s of cold fluid in the annulus side and the mass flow rate of hot fluid in the inner pipe is kept constant. The performance and increased pressure drop is a function of number of fins and mass flow rate.