Krishna Kannan

We develop a 3D nonlinear viscoelastic model for filled elastomeric solids that exhibit good predictive capabilities across multiple deformation modes and strain rates using at most 11 parameters. Through the analysis-driven construction of the rate of dissipation within the rate-type thermodynamic framework of Rajagopal and Srinivasa (2000), we reduce the number of parameters and also introduce the mode-of-deformation-rate dependent viscosity (ηm(K3)) into the constitutive relations. The special form of ηm(K3) accounts for higher values of viscosity in tension as compared to that of other modes of deformation. The broad spectrum of relaxation times exhibited by the elastomers are characterized by categorizing it into short, medium, and long relaxations, each assumed to be associated with one of the three natural configurations. The strong mode-dependent response exhibited by HNBR50, where the compression–relaxation is faster than tension–relaxation, is predicted accurately only when all the natural configurations are active. In contrast, the response of NR is predicted by using just two natural configurations because the polymer molecules are restricted to two extremes of the relaxation spectrum as a consequence of the high affinity between carbon black and the polymer molecules. The entire model is implemented in Abaqus/Standard through the user subroutine UMAT that interacts with an external solver, DDASPK, which solves for the internal variables. We show that the analytical form for the consistent Jacobian can be derived, and establish the efficacy of the implementation by simulating non-homogeneous shear on a hockey puck geometry made of HNBR50 with a concave lateral surface. The simulation shows good agreement with experimental data.

This paper presents the implementation of a nonlinear viscoelastic multi-configurational rate type material model within the general framework of the commercial finite element package Abaqus (Implicit). A user-defined subroutine, UMAT for continuum elements, is employed to implement a model similar to the one developed and extensively validated experimentally by Devendiran et al. One of the major challenges of using an implicit finite element (FE) solver is the description of material Jacobian, which may be difficult to obtain in the form of an analytical equation for multi-configurational models. Hence, numerical approximations are used to implement the consistent material Jacobian used in the global Newton iterations. We have used a Jacobian that is formulated by perturbing the deformation gradient, by extending an idea developed by Miehe and widely implemented in hyperelasticity. In order to determine the multiple intermediate configurations that keep evolving with time, this study also examines the computational efficiency of a fully second order integrator in comparison to traditional first order BDF integrator. A highly efficient TR-BDF2 integrator, originally developed for simulating transients of silicon VLSI devices, is utilised to determine the “current relaxed configuration”, which to the author’s knowledge is the first implementation in literature for rate type viscoelastic constitutive equations. Coupling the numerically approximated Jacobian and TR-BDF2 integrator, the UMAT is generic and can easily be modified for various combinations of stored energy potential functions and rate of dissipation equations. The UMAT is validated against material point solution of the constitutive model in MATLAB; and is found to be exact within a tight tolerance. The implemented UMAT is used to determine the rolling resistance of a Grosch wheel which demonstrates the practical application of such a material model focussed towards the workflows for tyre analysis.

We develop new rate-type constitutive relations on a set of orthonormal tensor basis and the corresponding set of Lode invariants, which require only 9 material parameters to predict the mechanical response of the human brain tissue. The mode-dependent response of the tissue is captured by invoking the Hill-stable elastic potential of Prasad and Kannan (2020) and constructing a new form for the rate of dissipation, thus introducing the mode-of-deformation dependent modulus terms and the mode-of-deformation-rate dependent viscosities into the rate-type thermodynamic framework of Rajagopal and Srinivasa (2000). Through the analysis-driven construction of the rate of dissipation, we incorporate maximum change in the viscosities with respect to the mode-of-deformation rates and limit the number of material parameters. Our model satisfactorily predicts the complicated load-unload cycles (pre-conditioned and conditioned) and the stress relaxation data under multiple modes of deformation and multiple rates for the Corona Radiata (CR) region of the brain tissue. It also captures the tension–compression asymmetry in the response and the higher relaxation time in compression loading than in shear loading.

There are many machine components made of polymeric materials, such as gears, which are subjected to cyclic loading conditions. To design such components, it is necessary to arrive at a suitable mathematical model that can describe the mechanical response of polymeric materials. In this paper, we derive a mathematical model for rate-type solids using thermodynamical framework developed by Rajagopal and Srinivasa (K.R. Rajagopal, A.R. Srinivasa, A thermodynamic frame work for rate type fluid model, Journal of Non-Newtonian Fluid Mechanics 88 (2000) 207-227) (also see Section 5 of Kannan and Rajagopal (K. Kannan, K.R. Rajagopal, A thermomechanical framework for the transition of a viscoelastic liquid to a viscoelastic solid, Mathematics and Mechanics of Solids 9 (2004) 37-59)), which was used by Rajagopal and Srinivasa to derive a mathematical model for isotropic, rate-type liquids. Uniaxial cyclic loading and stress relaxation experiments were conducted. The predictions of the model agreed well with the experimental data. © 2009 Elsevier Ltd. All rights reserved.

Viscoelastic liquids exhibit diverse mechanical behavior and offers a tremendous challenge in modeling its nonlinear response. This work is concerned with the development of a class of constitutive equations for viscoelastic liquids, which can capture the nonlinear response, especially, under torsional loading. By extending a universal solution developed by Rivlin (1948) for elastic bodies under torsional loading to viscoelastic bodies, one can arrive at a constitutive equation that can simultaneously predict torque and normal force with a reasonable accuracy. In order to develop such a constitutive equation, a suitable thermodynamical framework developed by Rajagopal and Srinivasa (2000) is chosen because of the fact that it can be used to exploit the mentioned result of Rivlin. Consequently, a suitable rate-type constitutive equation is also derived in Section 5. The efficacy of the developed model is checked by comparing the predictions of the model with that of the experimental data for torsional deformation of asphalt. It is found that the predictions of the model agree reasonably with that of the experimental data. © 2013 Elsevier Ltd. All rights reserved.

In the previous part of this work (see Devendiran et al. (2018)), a new non-linear rate-type compressible viscoelastic model with independent volumetric and distortional limiters was proposed. In the present work, the developed constitutive equation is validated using experimental data generated in-house for two cyclic inhomogeneous deformations which are predominantly two-dimensional: non-uniform planar extension and barreling compression of carbon-black filled elastomeric sheets and circular cylinders, respectively, under different rates of cyclic deformations. In the finite element simulations of these two boundary value problems, the two-dimensional governing equations are formulated in Lagrangian basis and the resulting system of partial differential equations are solved using COMSOL®. A good agreement is observed between the numerical results and the corresponding experimental data, including the cyclic stress softening effect (Mullins effect).