Partial Differential Equations in Continuum Mechanics

Continuum mechanics studies the mechanical behaviour of a material that has been mathematically idealised using partial differential equations (PDEs). Hence, continuum mechanics solves PDEs to understand material behaviours.

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