167, 168J TYPICAL SIMPLE STRAINS. 439 



The general small strain is accordingly completely defined by 

 the nine small coefficients, 



s x , s y , s s , g xj g y , g,, a*, & y , G> Z . 



168. Simple Strains. Stretches and Shears. The pure 

 strain 50) may be resolved into two parts 



(a) (b) 



52) vl = s y y, <=^-f + g x 0, 



wj = s z s, Wi" = gyX + g x y + 0. 



A strain whose equations contain but a single constant is called a 

 simple strain. Thus we may resolve the strain (a) into three simple 

 strains of which the first is given by 



u = s x x } v = 0, w = 0. 



This represents a displacement in which each point is shifted parallel 

 to the x -axis through a distance proportional to its x coordinate. 

 Such a displacement is called a stretch. The constant s x represents 

 the distance moved by a plane at unit distance from the YZ- plane 

 and measures the magnitude of the stretch or the linear expansion 

 per unit length. If s is negative the stretch becomes a squeeze. 



The strain (a) accordingly represents the resultant of three 

 simple strains, namely stretches, of different amounts in the directions 

 of the coordinate axes, which are evidently the axes of the strain. 

 The semi- axes of the strain -ellipsoid are 1 + s x> 1 + s y , 1 -f s z and 

 its equation 



? " 7i i N2 === 'J 



or neglecting squares of small quantities, 



(1 - 2s*) x* + (1 - 2s y ) ^ -f (1 - 2s,) = 1. 

 The dilatation is by 42) 

 53) 6 = s x + s y + s z . 



Obviously we can have 6 = if at least one of the stretches is 

 replaced by a squeeze. If the three s's are equal we have a simple 



introducing the into the slides is thus obvious", and we have therefore so 



introduced it, although to them "it seemed too great an interference with the 

 nearly general custom.' 1 We have also introduced a single suffix, g x , instead 

 of the more usual double suffix notation, g z , feeling that the brevity and 

 analogy with a> x thus gained justifies the change. 



