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

Physical Laws vs. Constitutive Equations

See also

What are Constitutive Equations?

Physical Laws of Continuum Mechanics

What are Constitutive Equations?

Physical laws of continuum mechanics are valid for any continuum. Constitutive equations are mathematical relationships between kinetics and kinematics quantities of a specific continuum (recall that a continuum is just a mathematical idealisation of a material). In other words, constitutive equations describe the behaviour of a material subjected to certain loading conditions.

Examples
Consider two analyses:

1. With same loading conditions but using two different materials i.e. steel vs. aluminium
2. With same material but test at two different temperatures i.e. room temperature vs. 1000ºC

In the above examples, the constitutive equations are responsible for the the followings

1. Differing response for steel and aluminium under the same loading conditions
2. In practice, a material is only described over a range of conditions it is expected to encounter. Therefore, constitutive equations are formulated to describe different response of the same material over an acceptable range i.e. two different sets of equations for 10-30ºC and 900-1200ºC temperature range for the above analysis