175, 176] STRESSES IN ISOTROPIC BODIES. 461 



176. Physical Meaning of the Constants. Let us consider 

 a few simple cases of equilibrium with homogeneous strain under 

 stress, there being no impressed bodily forces X, Y, Z, and putting 

 P = 0. 



1. A simple dilatation. 



u = ax, v = ay, w = as, 

 s x = a, s y = a, s z = a, g x = g y = g z = 0, 



& = s x + s y -f s z = 3a, 

 146) X x = Y y = Z z 



The surface forces become simply 



X n = pcos(nx), 



147) Y n =pco*(ny), 



Z n =pQos (n0), 



or the surface force is normal to the surface, and 



148) F,, = P 



The ratio of normal traction to cubical dilatation, or of normal 

 pressure to cubical compression, 



149) ^, = n^ = fi 



is called the bulk -modulus of elasticity, the term modulus being 

 applied in general to the ratio of the stress to the strain thereby 

 produced. 



2. A simple shear. 



u = ay, v = w = 0, 



s x = s y = s z = (3 = g x = g y = 0, 



150) </, = , X x = Y y = Z 2 = 0, 



= Z, = Z X = X 3 = 0, 

 ny), Y n =Tcos(nx), 



151) F n = Tycos 2 (nx) + cos 2 (ny) = Tsm(nz}, as in 117). 



