238 Mr. C. Chree on some Applications of 



that many materials in the state of ease satisfy these con- 

 ditions with at least a very close approach to exactness, so 

 that the results of the mathematical theory when properly 

 restricted are then sufficiently exact for practical purposes. 



From the preceding statements it will be seen that it is of 

 the utmost importance to know what are the limits within 

 which the conditions assumed by the mathematical theory are 

 satisfied with sufficient exactness to justify its application. 

 This question must of course be settled by experiment, but it 

 is beset by various difficulties which ought to be clearly 

 recognized. These arise in part from the serious obstacles 

 in the way of a complete experimental knowledge, and in 

 part from the want of a proper understanding between those 

 interested in the practical and theoretical sides of the subject, 

 and a consequent confusion in the terms used. 



To avoid complication let us begin by supposing the 

 mathematical limit of perfect elasticity to coincide with the 

 physical. Let us consider the simple case of a bar under 

 uniform longitudinal traction. We may suppose the bar 

 isotropic, and in consequence of suitable treatment perfectly 

 elastic for loads not exceeding L,. No mechanical treatment, 

 we shall suppose, can render it perfectly elastic for loads 

 greaier than L 2 . It does not follow that a load L 2 will 

 necessarily rupture the bar either immediately or in course of 

 time, but simply that for any load greater than L 2 the strain 

 is not perfectly elastic. Increasing the load from zero we 

 should reach a load L 3 , probably greater than L 2 , that would 

 in process of lime rupture the bar, or a load L 4 greater than 

 L 3 that produces immediate rupture. All these loads are 

 supposed to refer to unit of area. 



Now in the initial state of the bar we should be entitled to 

 apply the mathematical theory only until the load L^ was 

 reached. When we aim at finding the utmost capability of 

 the material under longitudinal load, we may perhaps apply 

 the theory until the load L 2 is reached, but here we must 

 stop. To apply it until the loads L 3 or L 4 are reached — 

 assuming these greater than L 2 — is clearly inadmissible. 



Results of a similar kind hold for all the comparatively 

 simple forms of stress, — such as pure compression, torsion, or 

 bending — in which practical men are interested. There are 

 limits to the state of perfect elasticity lower than the limits 

 at which rupture takes place, at least immediately. 



The usual aim of the engineer is that no part of the 

 structure he is designing should ever be strained beyond the 

 elastic limit, and this end he of course desires to obtain with 

 the least possible expenditure of material. Thus ideally he 



