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

What is Continuum Mechanics?

After generating a continuum (by mathematically representing a real material), the mechanical behaviour of such continuum can then be studied. This is continuum mechanics - the study of materials' mechanical behaviour using mathematical models.

     Solid mechanics and fluid mechanics are two special cases of continuum mechanics.

Physical Laws vs. Constitutive Equations

See also

What are Constitutive Equations?

Physical Laws of Continuum Mechanics

Reynolds' Transport Theorem

The physical laws of continuum mechanics can be generalised using Reynolds' Transport Theorem as follows:

Physical Laws of Continuum Mechanics

Physical laws are empirical laws derived by repeated observation of physical phenomena. These laws established links between kinematics (of continua) and kinetics (of deformation) without considering the physical properties (such as elasticity, density, viscosity and thermal conductivity) of the continuum (of the material) itself. This means that they are valid for any continuum (any material).

     The most important physical laws that govern the mechanics of continua are:

1. Conservation of Mass - the mass of an isolated system will remain constant over time
2. Conservation of Momentum (Newton's Second Law) - rate of change of (linear/ angular) momentum is equal to the resultant (force/ moment) acting on the system
3. Conservation of Energy (First Law of Thermodynamics) - rate of change of energy is equal to the difference between rate of heat input and rate of work output

     Note that the above laws can be generalised using Reynolds' Transport Theorem.

Figure. An experiment illustrating the law of conservation of mass (adapted from Grade 8 EAP)