CHAPTER IV 



BARS 



40. Elementary Theory of Elasticity. Strains. 



We require a few elementary notions from the theory of 

 elasticity. As regards the purely geometrical study of de- 

 formations, or "strains," it is usual to begin with the con- 

 sideration of a body in a state of uniform or "homogeneous" 

 strain. This is sufficiently defined by the property that any 

 two lines in the substance which were originally straight and 

 parallel remain straight and parallel, although their direction 

 relative to other lines in the substance is usually altered. 

 A parallelogram therefore remains a parallelogram, and it 

 easily follows that the lengths of all finite parallel straight 

 lines are altered in the same ratio ; this ratio will however 

 usually be different for different directions in the substance. 



It can be shewn that there are three mutually perpendicular 

 directions in the substance which remain mutually perpen- 

 dicular after the deformation; these are called the "principal 

 axes" of the strain. It is unnecessary, for our purposes, to 

 give the formal proof, as the existence of such axes will be 

 in evidence in such simple cases as we shall meet with. 

 It follows from this theorem that any originally spherical 

 portion of the substance is deformed into an ellipsoid whose 

 axes are in the directions of the principal strain-axes. 



If PQ, P'Q' denote any straight line in the substance, 

 before and after the strain, the ratio of the increase of length 

 to the original length, viz. 



PQ 



