Outline of MPM strategy

Paper 1 describes a negligible-deformation constitutive equation for the Burgers model with material parameters that are fit based on temperature. The paper makes the statement that this model is equivalent to a two-term Prony series. My understanding is that this means that it is equivalent to a two-component Generalized Maxwell Model. This intuitiion is confirmed in Paper 2 which derives the equivalence and formulates the three-dimensional analogue as a two-component upper-convected maxwell model (though it notes that basically any objective derivative, e.g. lower oldroyd gives a generalization). This latter paper introduces a multiplicative plasticity that is very similar to the one given in Paper 3. Both Paper 2 and Paper 3 derive similar energy functionals for the (hyper-) elastic portion of deformation. This motivates the implementation of Paper 4 which implements MPM-MLS for a multiplicative plasticity flow rule.

Therefore the project is divided into several phases:

• Implement in numpy the vanilla semi-implicit algorithm from Paper 4 for newtonian fluids with spatially varying viscosity (e.g. WLF) following, e.g., this tutorial for temperatures well above the softening point.
• Translate this implementation into jax.
• Implement thermodynamic solver and pressure split from Paper 5
• Implement the oldroyd-b (actually upper-convected maxwell model) for temperatures between glass transition and softening point from Paper 3.
• Use the parameters from [Paper 1] to model solidification.