# A unified hyperbolic formulation for viscous fluids and elastoplastic solids

## TL;DR

This paper introduces a unified hyperbolic PDE framework for modeling both viscous fluids and elastoplastic solids, incorporating finite particle length scales and particle rearrangements for more accurate continuum descriptions.

## Contribution

It presents a novel hyperbolic formulation that unifies fluid and solid mechanics by explicitly modeling particle deformability and rearrangements, unlike classical models.

## Key findings

- Successfully models fluid and solid flows within a single system
- Captures particle rearrangement processes through strain dissipation time
- Numerical examples validate the approach's reliability

## Abstract

We discuss a unified flow theory which in a single system of hyperbolic partial differential equations (PDEs) can describe the two main branches of continuum mechanics, fluid dynamics, and solid dynamics. The fundamental difference from the classical continuum models, such as the Navier-Stokes for example, is that the finite length scale of the continuum particles is not ignored but kept in the model in order to semi-explicitly describe the essence of any flows, that is the process of continuum particles rearrangements. To allow the continuum particle rearrangements, we admit the deformability of particle which is described by the distortion field. The ability of media to flow is characterized by the strain dissipation time which is a characteristic time necessary for a continuum particle to rearrange with one of its neighboring particles. It is shown that the continuum particle length scale is intimately connected with the dissipation time. The governing equations are represented by a system of first order hyperbolic PDEs with source terms modeling the dissipation due to particle rearrangements. Numerical examples justifying the reliability of the proposed approach are demonstrated.