168, 168 a] SHEARS. 441 



zero stretch along the axis perpendicular to both. The shear just 

 considered is a pure shear, that is, without rotation. It is easily 

 seen that the above shear might have been obtained if all planes 

 parallel to XOZ had been moved parallel to themselves a distance 

 in the X- direction equal to 2g z y, giving the rotational shear 



u' = 2g,y, v' = 0, w' = 



and then rotating about the Z-axis through an angle a z =g z , accord- 

 ing to the equations 



The lines OA f , OB', which before the strain were perpendicular, 

 have respectively the direction- cosines 1, g z , and g z , 1, and the 

 cosine of their included angle is accordingly 2g z . This change of cosine 

 which, as we have just seen, is equal to the amount of sliding of 

 the plane at unit distance from XOZ is commonly called the amount 

 of shear, so that the stretch and squeeze of the axes are each one 

 half of the amount of the shear. 



We may now define the strain (b) as a combination of three 

 simple shears of amounts 2g x , 2g y , 2g z , with planes mutually perpen- 

 dicular and equivalent to stretches of amounts g zj g x and g y along 

 the bisectors of the angles XOT, YOZ, and ZOX respectively, which 

 make angles of 60 with each other, together with squeezes of the 

 same amounts along the bisectors of the other angles. We have 

 thus the final positions of six points on these lines, or just sufficient 

 to determine an ellipsoid whose center is given. This is the strain- 

 ellipsoid. 



The strain (b) will be called a general shear. The quadric cp is 



54) y = x 2 + f + + %g x ys + 2g v xe + 2g,xy = B 2 , 



and the shears are the coefficients of the product terms. If the 

 equation of the quadric is transformed to its principal axes the 

 product terms vanish. Accordingly we may always find three mutually 

 perpendicular axes with respect to which the shear components vanish. 

 These are the axes of the shear. (It may be remarked that the 

 equations of the general rotational shear are obtained from 1) by 

 putting a = & 2 = C B = 1.) 



In order to distinguish between the geometrical term shear and 

 the dynamical shearing stress, to be presently considered, it will be 

 convenient to characterize the coefficients g as the slides (corresponding 

 to the French glissementj German Gleitung). 



168 a. Elongation and Compression Quadric. Since the 

 equations 50) for the shifts, the components of the vector dis- 

 placement q, as a function of r are of precisely the same form as 



