K. Kannan

A thermomechanical framework for the glass transition phenomenon in certain polymers and its application to fiber spinning

A thermodynamic framework is developed to describe a polymer melt undergoing glass transition that takes into account the fact that during such a process the underlying natural configurations (stress-free states) are continually evolving. Such a framework allows one to take into account changes in the symmetry of the material, if such changes take place. Moreover, the framework allows for a seamless transition of a polymeric melt to a mixture of a melt and an elastic solid to the final purely solid state. The efficacy of the model is tested by studying the fiber spinning problem for polyethylene terephthalate and the predictions agree well with the experimental results.

Flow through porous media due to high pressure gradients

Abstract While Darcy’s equations are adequate for studying a large class of flows through porous media, there are several situations wherein it would be inappropriate to use Darcy’s equations. One such example is a flow wherein the range of pressures involved is very large and high pressures and pressure gradients are at play. Here, after developing an approximation for the flow through a porous solid, that is a generalization of an equation developed by Brinkman, we study a simple boundary value problem that clearly delineates the difference between the solution to these equations and those due to the equations that are referred to as “Darcy Law”. We find that the solutions for the equations under consideration exhibit markedly different characteristics from the counterpart for the Brinkman equations or Darcy’s equations (or the Navier–Stokes equation if one neglects the porosity) in that the solutions for the velocity as well as the vorticity lack symmetry and one finds the maximum value of the vorticity occurs at the boundary near which the fluid is less viscous in virtue of the pressure being lower. We also find that for a certain range of values for the non-dimensional parameters describing the flow, boundary layers develop in that the vorticity is confined next to the boundary adjacent to which the viscosity is lower, such boundary layers being absent in the other classical cases.

Simulation of fiber spinning including flow-induced crystallization

Kannan et al. [J. Rheol. 46, 979–999 (2002)] studied the problem of fiber spinning for polymers that are largely atactic in nature, within a thermodynamic framework developed by Rao and Rajagopal [Z. Angew. Math. Phys. 53, 365–406 (2002)], where significant crystallization does not take place. Here, we develop a model within the same framework that can take into account flow-induced crystallization and the anisotropy of the crystalline part of the semicrystalline polymer. We find that the predictions of the model agree very well with experimental data.Kannan et al. [J. Rheol. 46, 979–999 (2002)] studied the problem of fiber spinning for polymers that are largely atactic in nature, within a thermodynamic framework developed by Rao and Rajagopal [Z. Angew. Math. Phys. 53, 365–406 (2002)], where significant crystallization does not take place. Here, we develop a model within the same framework that can take into account flow-induced crystallization and the anisotropy of the crystalline part of the semicrystalline polymer. We find that the predictions of the model agree very well with experimental data.

A Thermodynamic Framework for Describing Solidification of Polymer Melts

A thermodynamic framework is presented that can be used to describe the solidification of polymer melts, both the solidification of atactic polymers into an amorphous elastic solid and the crystallization of other types of polymer melts to semi-crystalline elastic solids. This framework fits into a general structure that has been developed to describe the response of a large class of dissipative bodies. The framework takes into account the fact that the natural configuration of the viscoelastic melt and the solid evolve during the process and that the symmetries of these natural configurations also evolve. Different choices are made as to how the material stores energy, produces entropy, and for its latent heat, latent energy, etc., that lead to models for different classes of materials. The evolution of the natural configuration is dictated by the manner in which entropy is produced, how the energy is stored etc., and it is assumed that the constitutive choices are such that the rate entropy production is maximized, from an allowable class of constitutive models. Such an assumption also determines the crystallization kinetics, i.e., provides equations such as the Avrami equation. Using the framework, a model is developed within which the problem of fiber spinning is studied and we find that the model is able to predict observed experimental results quite well.

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

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.

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

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 than the rest of the solid. In this study, we show that the criterion articulated by Alagappan et al. is still applicable when bodies have multiple localized regions of lower density, thereby lending credence to the notion that the criterion might be reasonable for a large class of bodies with multiple inhomogeneities. As in the previous study, it is found that damage is not initiated at the location where the stresses are the largest but instead at the location where the densities tend to the lowest value. These locations of lower densities coincide with locations in which the deformation gradient is very large, suggesting large changes in the local volume, which is usually the precursor to phenomena such as the bursting of aneurysms.

A damage initiation criterion for a class of viscoelastic solids

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 initiation of damage being decided by parameters that are based on a quantity such as the strain, that requires information concerning a special reference configuration, or the use of ad hoc parameters that have no physically meaningful origins, in this study we use a physically relevant parameter that is completely determined in the current deformed state of the body to predict the initiation of damage. Damage is initiated due to the inhomogeneity of the body wherein certain regions in the body are unable to withstand the stresses, strains, etc. The specific inhomogeneity that is considered is the variation of the density in the body. We consider damage within the context of the deformation of two representative viscoelastic solids, a generalization of a model proposed by Gent (1996 Rubber Chemistry and Technology 69, 59–61. (doi:10.5254/1.3538357)) for polymeric solids and a generalization of the Kelvin–Voigt model. We find that the criterion leads to results that are in keeping with the experiments of Gent & Lindley (1959 Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 249, 195–205. (doi:10.1098/rspa.1959.0016)).