viscoelastic treatment

This attempt at this project is founded on the belief that behavior nearer to the glass transition temperature is most important for the simulations we want to perform.

Combined with work like this, this should allow us to implement a rapid prototype in Jax+Fenics, where we can use automatic differentiation of the constitutive update with respect to input parameters to do the Newton update step.

We start by working through the constitutive equation and volume relaxation given in this article and this thesis. This approach splits the glass-like behavior into two components: thermo-rheologically simple (TR) viscoelasticity and structural volume relaxation. TR behavior boils down to a generalized Maxwell model and adequately describes the response to external loads around the working point $T_w$ of the glass. The structural volume relaxation describes the processes in glassblowing that implicitly lead to brittle fracture either on the pipe or in the annealer. Our initial implementation will solely describe the TR behavior, which would be a qualitatively satisfying (though slightly mathematically incorrect) description of glass if we artificially clamp $T > T_g$ ($T_g$ is an empirical "glass transition temperature" used in the ceramic+glass literature).

Definitions:

• Stress tensor: $\sigma_{ij}$.
• $G(t), K(t)$ are the impulse response functions describing shear and bulk modulus for an isotropic material
• $\varepsilon_ii$ is trace of strain tensor
• $e_ij$ are the deviatoric strain tensor elements

$$\sigma_{ij} = 2\int_0^t G(t-t') \der{e_{ij}}{t'}(t') \intd{t'} + \delta_{ij} \int_0^t K(t-t') \der{\varepsilon_{ii}}{t }(t') \intd{t'}$$ (note: a factor of three has been absorbed here somewhere).

Misc explanations