Krishna Kannan

Carbon black filled natural and synthetic rubbers used in automobile tires have to sustain a very large hydrostatic tensile stress between the two steel belts, especially near the belt edge. In addition, the rate effect of the volumetric and the distortional response of filled elastomers becomes significant for vehicles running at high speeds. In other words, by limiting the volumetric and distortional response and making it rate dependent, one can emulate the high stresses observed in tires. There are a number of constitutive equations of the rate and integral type for filled elastomers, but none of the available models employ a rate-dependent dilatational limit. In this paper, we propose a new non-linear rate-type compressible viscoelastic incorporating individual limits for dilatation, contraction, and distortion. The rate type thermodynamic framework of Rajagopal and Srinivasa (2000) is extended to include such limits. By establishing the physical meaning of various parameters used in the constitutive equation, we show that they are tightly correlated with the certain type of experiments and can be determined from the available data in the literature. Only a few parameters need to be determined by curve fitting. The proposed model is validated using in-house as well as the other experimental data available in the literature. In Part B of the paper, extensive validation of the proposed constitutive equation for cyclic barreling and cyclic inhomogeneous planar extension will be demonstrated.

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.

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.