Continuum Mechanics

Matter is formed of molecules which in turn consist of atoms and subatomic particles. Thus matter is not continuous. However, there are many aspects of everyday experience regarding the behavior of materials, such as the amount of lengthening of a steel bar under the action of given forces, the rate of discharge of water in a pipe under a given pressure difference or the drag force experienced by a body moving in air etc., which can be described and predicted with theories that pay no attention to the molecular structure of materials. The theory which describes relationships between gross phenomena, neglecting the structure of materials on a smaller scale, is known as the continuum theory. The continuum theory regards matter as indefinitely divisible. Thus, within the theory, one accepts the idea of an infinitesimal volume of material referred to as a particle in the continuum, and in every neighborhood of a particle there are always infinitely many particles present. Whether the continuum theory is justified or not depends upon the given situation. For example, the molecular dimension of water is about $1\ {\rm A}^\circ\ (10^{-8}\ {\rm cm})$; hence, if we are concerned about the liquid water in a problem in which we never have to consider dimensions less than say $10^{-5}$ cm, we are safe to treat water as a continuum. The mean free path of the molecules of air on the surface of the earth at room temperature is about $5\times 10^{-6}$ cm; hence, if we consider the flow of air about an airplane, we may treat air as a continuum. The diameter of a red blood cell in our body is about $8.5\times 10^{-4}$ cm; hence, we can treat our blood as a continuum if we consider the flow in arteries of diameter say 0.5 mm.

Thus the concept of a material continuum as a mathematical idealization of the real world is applicable to problems in which the fine structure of the matter can be ignored. When the consideration of fine structure is important, we should use principles of particle physics, statistical mechanics, or a theory of micropolar continuum.