A nonlinear Lax-Milgram lemma arising in the modeling of elastomers

Introduction Elastomers have been used by engineers since the mid 1800s in a variety of roles including bearings, springs, and shock absorbers. For example, rubber composites filled with inactive particles such as carbon black and silica are routinely used as passive vibration suppression devices. The advent of smart materials technology has sparked great interest in the development of rubber composites filled with active elements (piezoelectric, magnetic or conductive particles) for use as active vibration suppression devices. The dynamic mechanical behavior of even the inactively filled rubbers is complex, including nonlinear constitutive laws, large deformations even under small loads, loss of kinetic energy (damping), loss of potential energy (hysteresis), dependence on fillers and environment (e.g., temperature). Many current modeling efforts focus on phenomenological formulations involving strain energy function (SEF) theories (see [8], [10], [11]). The other predominan

Nonlinear Elastomers: Modeling and Estimation

We report on our efforts to model nonlinear dynamics in elastomers. Our efforts include the development of computational techniques for simulation studies and for use in inverse or system identification problems. 1 Introduction A problem of fundamental interest and great importance in modern material sciences is the development of both passive and active ("smart") vibration devices constructed from polymer (long molecular chains of covalently bonded atoms often having cross-linking chains) composites such as elastomers filled with carbon black and/or silica or with active elements (i.e., piezoelectric, magnetic or conductive particles). These rubber based products (even without active elements) involve very complex viscoelastic materials that are not at all like metals (where large deformations lead to permanent material changes) and do not satisfy the usual, well-developed linear theory of (infinitesimal) elasticity for deformable bodies. They typically exhibit mechanical properties ..

Modeling the Dynamic Mechanical Behavior of Elastomers

Accurate modeling of the dynamic mechanical behavior of elastomers presents many challenges, including the nonlinear relationship between stress and strain, the loss of kinetic energy (damping), and the loss of potential energy (hysteresis). Currently available software packages for studying the stress-strain laws in rubber-like materials assume a form of the strain energy function (SEF), such as a cubic Mooney-Rivlin form or an Ogden form. While these methods can produce good results, they are only applicable to static behavior, and they ignore hysteresis and damping. We present a dynamic partial differential equation (PDE) formulation, with a Kelvin-Voigt damping term, as an alternative approach to the SEF formulation. Constitutive laws are estimated using data from simple extension experiments, leading to static results which compare favorably with results achieved by estimating a cubic Mooney-Rivlin SEF, and dynamic results which offer new insight. A neo-Hookean model for generaliz..