- 10:02 AM

A Stochastic Model for Describing Glassy Materials Subjected to Complex Thermal and Loading Histories

James M. Caruthers, School of Chemical Engineering, Purdue University, 480 Stadium Mall Drive, West Lafayette, IN 47907-2100 and Grigori Medvedev, School of Chemical Engineering, Purdue University, 480 Stadium Mall Drive, West Lafayette, IN 47907-2100.

In a recent series of papers Caruthers and coworkers (Polymer, 45, 4577, 4599, 2004) developed a thermoviscoelastic constitutive theory, which was capable of capturing via a single set of physically significant parameters a wide range of phenomena observed in glassy polymers. This diverse range of phenomena included yield, stress/volume/enthalpy relaxation, non-linear stress-strain response, and physical aging, where most of the model predictions were in good agreement with the experimental data. The exceptions were the heat capacities during reheating for some thermal histories and the post-yield stress softening, where the agreement was qualitative but not quantitative. The thermoviscoelastic constitutive model also does not predict the “expansion gap” effect observed in the classic set of volume relaxation experiments of Kovacs. It has been suggested that the assumption of thermorheological simplicity of the thermoviscoelastic constitutive model may be responsible for the aforementioned deficiencies. A stochastic model for volume relaxation developed in our group, which naturally is thermorheologically complex, has shown promise in capturing the “expansion gap”. However, this existing stochastic model cannot be directly applied for general loading histories, since the assumption that the deformation is isotropic was built in into the framework of the stochastic model. Moreover, the local fluctuations were only allowed to be isotropic.

We will present a generalization of the volume stochastic model allowing (1) the local fluctuations to be anisotropic and (2) the relaxation time determined via the configurational internal energy to depend on the full strain tensor. The key assumption of the model is that a material near and below its glass formation point is dynamically heterogeneous, where mobility at a given location is determined by local and instantaneous values of entropy and strain, where the observed macroscopic response is an ensemble average of local contributions. Mathematically the model is represented by a set of coupled non-linear stochastic differential equations (SDE) for entropy and six components of a symmetric strain tensor. The magnitude of the noise terms in the SDE is completely determined by the requirement that the equilibrium values of mean square fluctuations in entropy and strain be consistent with those predicted by thermodynamics. Predictions of the stochastic model for the cases of (a) isotropic deformations, (b) uniaxial creep experiment, and (c) yield in both uniaxial extension and compression at a constant strain rate will be discussed.