2,689 research outputs found
Circularly polarized wave propagation in a class of bodies defined by a new class of implicit constitutive relations
No full textIn 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
Bodies described by non-monotonic strain-stress constitutive equations containing a crack subject to anti-plane shear stress
No full textIn 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 highlight 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 intensity of the complete stress field can be expressed as a function of the Stress Intensity Factor K III , as in the case of conventional linearized elasticity theory, thus promoting a K based-approach to the fracture of elastic solids obeying a constitutive relation wherein the linearized strain is expressed as a non-linear function of the Cauchy stres
On a New Class of Models in Elasticity
No full textRecently, Rajagopal and co-workers have shown (see Rajagopal [1], Rajagopal and Srinivasa [2],[3], Bustamante and Rajagopal[4], Rajagopal and Saccomandi [5]) that if by an elastic body one means a body that is incapable of dissipation, then the class of such bodies is far larger than either Green elastic or for that matter Cauchy elastic bodies as one could model elastic bodies using implicit constitutive relations between the Cauchy stress and the deformation gradient or implicit constitutive relations that are rate equations involving the Piola-Kirchhoff stress and the Green-St.Venant Strain (see Rajagopal and Srinivasa [2]). Such a generalized framework allows one to develop models whose linearization with regard to the smallness of the displacement gradient allows one to obtain models that have limited linearized strains even while the stresses are very large. Such a possibility has important consequences to problems which, within the context of the classical linearized theory, leads to singularities. In this short paper, we illustrate the implications of such models by considering simple problems within the context of a specific model belonging to the general class, wherein the strains remain small as the stresses tend to very large values
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
No full textIn 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 (Formula presented.), 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
Unsteady motions of a new class of elastic solids
No full textIn this short paper we study unsteady motions of a new class of elastic solids, wherein one can justify a non-linear relationship between the linearized strain and the stress, an impossibility
within the classical construct of elasticity. For the class of materials concerned, one has to solve simultaneously the balance of mass, balance of linear momentum and the constitutive relation. In general, one has ten scalar unknowns, i.e., density ρ, the components of Cauchy stress T and displacement u, and ten scalar algebraic–partial differential equations,
the balance of mass (1), the balance of linear momentum (3) and the constitutive equation (6). The stress wave that is generated is quite distinct from what one observes within the context of the classical theory
A note on some new classes of constitutive relations for elastic bodies
No full textArtículo de publicación ISIThe class of elastic bodies, that is bodies incapable of dissipation in whatever motion that they undergo,
has been significantly enlarged recently (see Rajagopal 2003, On implicit constitutive theories. Appl.
Math., 48, 279–319; Rajagopal 2007, The elasticity of elasticity. Z. Angew. Math. Phys. 58, 309–317;
Rajagopal, K. R. & Srinivasa, A. R. 2007, On the response of non-dissipative solids. Proc. R. Soc. Lond.
A, 463, 357–367). The new classes include fully implicit constitutive relations for the stress and the
deformation gradient, and the interesting sub-class wherein the Cauchy–Green tensor or the linearized
strain tensor bears a non-linear relationship to the stress. While a fully thermodynamic treatment of
such elastic bodies, when defined through implicit constitutive relations between the Piola stress and the
Green–St. Venant strain, within a 3D framework has been carried out (see Rajagopal, K. R. & Srinivasa,
A. R. 2007, On the response of non-dissipative solids, Proc. R. Soc. Lond. A, 463, 357–367), other possible
implicit relationships between other stress and kinematic measures have not been investigated. This
paper is devoted to the determination of the consequences of thermodynamics on the new class of elastic
bodies, when they are expressed through implicit relations for different stress and stretch/strain measures
K. Rajagopal on making films for and on the ethnic minority in Singapore
No full textThis interview was motivated by an interest in exploring how Singapore film directors perceive the three major Chinese cinema awards, mainly the Golden Horse Awards (GHA), Hong Kong Film Awards (HKFA) and Golden Rooster Awards (GRA), and what they might signify for Singapore cinema, especially for a nation that is predominantly ethnic Chinese. Compared to the number of Singapore Chinese-language films produced in the last two decades, there have been considerably less Indian-language productions. K. Rajagopal’s A Yellow Bird (2016) alongside two other Tamil films, namely Eric Khoo’s My Magic (2008) and T. T. Dhavamanni’s Gurushetram: 24 Hours of Anger (2010) offer critical takes on the vicissitudes of Singapore Indians struggling with issues such as socioeconomic inequality and racial prejudice in a booming Chinese-majority city-state
Implicit nonlinear elastic bodies with density dependent material moduli and its linearization
No full textWe develop an implicit constitutive relation to describe the response of a compressible elastic solid, based on physical considerations, that captures all the characteristics exhibited by the popular Blatz–Ko model, but in addition presents some interesting novel features. The fact that the Cauchy stress appears linearly in the implicit constitutive relation between the stress and the left Cauchy–Green strain with the material moduli depending nonlinearly on the deformation gradient, allows us to capture several characteristic features of the response of rubber-like elastic solids. Interestingly, in the nonlinear implicit model that we develop, we find that it is possible to have the normal stress components of the stress influence the shearing motion at second order, when considering weakly nonlinear waves, that only occurs at third order within the case of the classical nonlinear Cauchy elasticity theory. Linearization of the constitutive relation under the assumption of small displacement gradient reduces the constitutive relation to one whose material moduli can depend on the trace of the linearized strain and hence the density in virtue of the balance of mass, such a feature is not possible within the context of the Blatz–Ko constitutive relation, or for that matter any Cauchy elastic body, as linearization leads to the classical linearized elastic constitutive relation that has constant material moduli
Flow of fluids with pressure- and shear-dependent viscosity down an inclined plane
No full textIn this paper we consider a fluid whose viscosity depends on both the mean normal stress and the shear rate flowing down an inclined plane. Such flows have relevance to geophysical flows. In order to make the problem amenable to analysis, we consider a generalization of the lubrication approximation for the flows of such fluids based on the development of the generalization of the Reynolds equation for such flows. This allows us to obtain analytical solutions to the problem of propagation of waves in a fluid flowing down an inclined plane. We find that the dependence of the viscosity on the pressure can increase the breaking time by an order of magnitude or more than that for the classical Newtonian fluid. In the viscous regime, we find both upslope and downslope travelling wave solutions, and these solutions are quantitatively and qualitatively different from the classical Newtonian solutions
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