Modeling of Nonlinear Hysteresis in Elastomers under Uniaxial Tension

As a fundamental component of an overall program in modeling smart material damping devices, we consider inactive host material models for moderate to highly filled rubbers undergoing uniaxial tensile deformations. Beginning from a neo-Hookean strain energy function formulation for nonlinear extension, we develop general constitutive models for both quasi-static and dynamic deformations of a viscoelastic rod. The constitutive laws are nonlinear and contain hysteresis through a Boltzmann superposition integral term. The resulting integropartial differential equations models are shown to be equivalent to the usual Lagrangian dynamic distributed parameter models coupled with linear ordinary differential equations for internal variables (internal strains). Comprehensive well-posedness results (existence, uniqueness and continuous dependence) are summarized in a discussion of theoretical aspects of the systems. The models are validated with experiments designed and carried out explicitly for this study. In particular, quasi-static Instron experimental data are used in a least squares inverse problem formulation to estimate nonlinear elastic and nonlinear viscoelastic contributions to the general stress-strain constitutive laws proposed. It is shown that the models provide an excellent prediction of nested hysteresis loops manifested in the data. These models are then used as initial estimates in determining the nonlinear hysteretic constitutive laws for the dynamic experiments. It is shown that in cases of more highly filled rubbers, multiple internal variable models lead to best fits to the data.