Why use idealisation?
It is not economic to fully describe material behaviours over a large range of conditions. Instead, the mathematical expression is idealised such that material behaviour is only represented for a reasonable range of loading conditions. See examples.
How to mathematically idealise/ model a material?
In continuum mechanics, materials are modelled using PDEs. These are equations assembled from (1) physical laws that are universal and (2) constitutive equations that are material specific.
PDEs in continuum mechanics = physical laws + constitutive equations
Examples of PDEs
- Navier equation
- Wave equation as special case of Navier equation
- Heat equation
- Laplace's equation as special case of heat equation
- Navier-Stokes equations