### Elements of a Mathematical Theory of Elasticity

W. Thomson
1856 Philosophical Transactions of the Royal Society of London
PART I. ON STRESSES AND STR A IN S*. A rticle I.-Initial Definitions 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 straiq is any definite alteration of form or dimensions experienced by a solid. Examples. Equal
more » ... . 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 launch working in a heavy sea, the different parts of it experience relative motions. A rticle 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 R a n k in e , and I have found them very valuable in writing on the subject. It will be seen that I have deviated slightly from Mr. R a n k in e '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.