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XXL Elements of a Mathematical Theory of Elasticity. 
[ By Professor William Thomson, M.A., F.R.S. 
Received April 16, — Read April 24, 1856. 
PART I. ON STRESSES AND STRAINS*. 
Article I . — Initial D^nitions and Explanations. 
Def. A STRESS is an equilibrating application of force to a body. 
Cor. The stress on any part of a body in equilibrium will thus signify the force 
which it experiences from the matter touching that part all round, whether entirely 
homogeneous with itself or only so across a portion of its bounding surface, 
Def A strain is any definite alteration of form or dimensions experienced by a 
solid. 
Examples. Equal and opposite forces acting at the two ends of a wire or rod of any substance 
constitute a stress upon it. A body pressed equally all round, for instance any mass touched by 
air on all sides, experiences a stress. A stone in a building experiences stress if it is pressed upon 
by other stones, or by any parts of the structure, in contact with it. Any part of a continuous 
solid mass simply resting on a fixed base experiences stress from the surrounding parts in conse- 
quence of their weight. The different parts of a ship in a heavy sea experience stresses from which 
they are exempt when the water is smooth. 
If a rod of any substance become either longer or shorter it is said to experience a strain. If a 
body be uniformly condensed in all directions it experiences a strain. If a stone, a beam, or a mass 
of metal, in a building, or in a piece of framework, becomes condensed or dilated, in any direction, 
or bent, or twisted, or distorted in any way, it is said to experience a strain, to become strained, or 
often in common language, simply “ to strain.” A ship is said ‘‘ to strain” if in launching, or when 
working in a heavy sea, the different parts of it experience relative motions. 
Article II . — Homogeneous Stresses and Homogeneous Strains. 
Def. A stress is said to be homogeneous throughout a body when equal and 
similar portions of the body, with corresponding lines parallel, experience equal and 
parallel pressures or tensions on corresponding elements of their surfaces. 
Cor. When a body is subjected to any homogeneous stress, the mutual tension or 
pressure between the parts of it on two sides of any plane amounts to the same per 
* These terms were first definitively introduced into the Theory of Elasticity by Rankine, and I have found 
them very valuable in writing on the subject. It will be seen that I have deviated slightly from Mr. Rankine’s 
definition of the word “ stress,” as I have applied it to the direct action experienced by a body from the matter 
around it, and not, as proposed by him, to the elastic reaction of the body equal and opposite to that action. 
