THEORY OF ELASTICITY 2. For a full account of the methods and results of mathemati cal enquiry the special article ELASTICITY should be consulted. Much of the theory is concerned with problems of physical rather than engineering interest, and methods of more restricted range :an be employed to obtain those few and relatively simple solu tions which form the basis of practical design. The essentials of the theory are : (I) a quantitative analysis of stress, or internal action; (2) a quantitative analysis of strain, or distortion; (3) postulates (based on experiment) regarding the relation of stress to strain. These, in conjunction with the accepted principles of mechanics, it employs to derive equations which govern the displacement at every part of a loaded body. The solution of those equations is a purely mathematical problem, which may present considerable difficulties.
contemplates a material having the nature of a structureless jelly, which would have exactly the same appearance when viewed through a microscope of any imaginable magnification.
The assumption must be recognized and its implications weighed, for it:does not accord with our knowledge of real mate rials. These in all cases exhibit a definite structure, even when examined under low magnifications : brass, for example, is revealed as a conglomerate of small crystals of copper and of zinc. Small pieces cut from different parts would therefore not exhibit similar properties ; and if we were to subdivide them further, down to molecular or atomic dimensions, we should ultimately reach a stage at which all resemblance to a continuous substance had dis appeared.
We shall have to consider later how far it is permissible to apply to real materials results which are based on the assumption of continuity. For the moment we are concerned to develop, in accordance with this assumption, precise notions of stress and strain.