Deriving the Nonlinear Viscoelastic Formulations of Polymer Materials

Current material models for both polymeric and composite materials fall short in properly accounting for the nonlinear viscoelastic effects present in the material. Existing models either use outdated viscoelastic formulations, or rely heavily on fitting functions to estimate the time dependent deformation of the material. Developing a continuum mechanics rooted analytical model for a nonlinear viscoelastic polymer will produce a more accurate representation of the material of interest and provide a physics based model for polymer structures. Lifetime elongation or deformation predictions from this model will allow for the expanded use of polymer materials in aerospace industries under higher restrictions, such as commercial aviation aircraft governed by the Federal Aviation Administration (FAA). The push for lightweight materials in the aerospace industry has already led to the proliferation of both polymers and composites in aerospace structures. For designs where material deformation and aging is a major concern, understanding the time dependent material elongation or degradation is essential. In the case of satellite components, material degradation, relaxation, or creep is a top priority. For future designs and existing structures, the proposed nonlinear viscoelastic model will allow for lighter weight components, and extended lifespans for mission critical structures. The proposed model is characterized by the behavior of the strain and stress tensors in the polymer material within a reference configuration. The constitutive equations of the model consist of a complex function that incorporates both the nonlinear elastic and viscous components while also capturing the time dependent behavior of the polymer. Starting from the base laws of thermodynamics, the overall energy inequality for the polymer system is derived. The energy inequality, paired with expressions for the free energy, and external stress of the material are the basis of the analytical model that is solved as a boundary value problem using Green's functions. Material data for Polymethyl methacrylate (PMMA) in the form of dynamic mechanical analysis (DMA), creep, and stress relaxation are introduced to solve for the stress present in a representative structure. A polymer (PMMA) beam in tension was chosen as the initial trial case for the nonlinear viscoelastic model. The resulting nonlinear viscoelastic model produces formulations to predict the external stress of the material. Additional work is still required to properly correlate the derived moduli tensors with the experimental modulus results.