Krishna Kannan

A rigorous, comprehensive, and thermodynamically consistent theory has been developed for the fused deposition modelling (FDM) of semi-crystalline polymers. It is sufficiently general in that it can accommodate multiple phase transition mechanisms (crystallization, glass transition, and melting) during the heating and cooling cycles of the process encountered during FDM. The theory predicts the residual stresses and the resulting warpage in the polymer part due to the temperature-dependent, spatially varying specific volumes of each phase, precipitated by the inhomogeneous distribution of temperature. The theory treats the semi-crystalline polymer as a constrained mixture of multiple phases, where glass is assumed to be a new phase of the polymer. The statistically based Avrami kinetics for crystallization, modified for non-isothermal processes, is recovered as a particular case of our non-equilibrium thermodynamic analysis. Moreover, the theory predicts the temperature corresponding to the local free energy minima as the ideal glass transition temperature analogous to that of Franz and Parisi's mean field theory with a statistical basis.

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.

There are many alloys used in orthopaedic applications that are nonlinear in the elastic regime even when the strains are ‘small’ (see Hao et al., 2005; Saito et al., 2003; Sakaguch et al., 2004). By using conventional theories of elasticity, either Cauchy or Green elasticity, it is impossible to systematically arrive at constitutive equations, which would be applicable in the elastic domain of such metals as such materials exhibit non-linear response for small strains1 where the classical linearized response is supposed to hold in the sense that the norm of the squares of the displacement gradient are much smaller than the displacement gradient. We delineate a new framework for developing constitutive equations for a new class of elastic materials, termed as implicit elastic materials, which can be used to describe the response of such alloys. In addition to a fully implicit constitutive relation, we discuss a non-linear constitutive relation between the linearized strain and the stress that can be properly justified to describe the response of such alloys. By using the example of a rectangular plate with a hole subject to uniform loading, a classical problem, we illustrate the differences in the stress and strain fields when compared to that predicted by the classical linearized relation.

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.

In 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. © 2014 Elsevier B.V.

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 constitutive relation that has been developed, we consider four ribbons of polymeric melt stacked on top of 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 into the previous laid layer 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 preexisting 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.

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)).

A constitutive model is developed for describing the response of rock glaciers that takes into account the effect of the shear rate, pressure and the volume fraction of the rocks and sand grains trapped within the interstices of the rock glacier on the viscosity of the rock glacier. Using the model, we describe the motion of the Murtèl-Corvatsch rock glacier by corroborating the predictions of the model with the experimental data available for the displacement of a vertical borehole, by appealing to a simple semi-inverse solution for the velocity, pressure and volume fraction fields. The predictions of the model agree quite well with the experimental results.© 2012 Published by Elsevier Ltd.

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.

We study the initiation of damage in a polymeric body in which there is a line defect due to the formation of a "weld line" that occurs when two polymer streams join together and then solidify. We show that damage initiates in the region of weakness, namely the "weld line" based on a criterion for damage that was developed earlier. We also show that if there are other stress concentrators also additionally present, such as a hole, then there is a competition between the stresses induced due to the weakness and the stress as a consequence of the stress concentrator (in this instance a hole). This study adds more credence to the criterion for the initiation of damage that is based completely on knowledge of information at the current configuration of the body, that is, the criterion for damage is not based on the value of quantities that also need information based on a reference configuration such as the stress or strain.