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

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)

What is a Continuum?

In continuum mechanics, a continuum is an idealised mathematical representation of a real material such as solid, liquid or gas.

Kinetics vs. Kinematics

A summary of the differences between a kinematics and kinetics study.

	Kinematics	Kinetics
Study of	Motion (Deformation and flow)	Forces (Surface and body)
Considers	Motion disregarding the forces and moments that cause the motion.	Motion and the forces underlying this motion.

See also

What is Kinetics?

What is Kinematics?

What is Kinetics?

Whilst kinematics deals with the motion and deformation of a body, kinetics studies the forces that produce motion. The two types of forces are:

1. Body forces (per unit mass or volume) are non-contact forces that act on the total mass of a continuum. Examples include gravitational force, electromagnetic force and inertial force.
2. Surface forces (per unit area) are contact forces that act across an internal or external surface of a body. Examples include pressure, contact forces and frictional forces.

Figure. Magnetic (non-contact) vs. frictional (contact) forces. Images adapted from Newtown High School and Joey's Blog.