152 BELL SYSTEM TECHNICAL JOURNAL 



relationship between the applied stress and ensuing strain is found to exist. 

 Expressed as an equation 



— = £ in which Xx is the force per unit area, 



Xx is the strain per unit length, and -E is a constant known as Young's 

 Modulus. (Refer to Section 7.7 for further definition of terms). 



In an analogous manner, shearing stresses applied to an elastic solid as 

 shown by Fig. 7.2 produce a shearing strain such that 



— = A, the shear modulus. 



Xy 



In general there will be contributions to a particular strain from any of 

 the stresses which may happen to exist. For example, when an isotropic 



Fig. 7.1 — Bar under tensional stress 



Fig. 7.2 — Bar under shearing stress 



bar is stretched, there will be a contraction along the width which has been 

 produced by a stress along the length. A statement of these relationships 

 (known as Hooke's Law) is given by the equations of Section 7.8. 



It is now of interest to consider the conditions of equilibrium for a very 

 small cube cut out of the elastic medium which in general is stressed and in 

 motion. Reference to Fig. 7.3 will help to visualize the stresses which may 

 exist on the faces of this cube. Since these stresses vary continuously 

 within the medium, a summation of the forces acting on the cube along each 

 of the major axes can be made with the use of differential calculus. From 

 Newton's Law previously cited, it is apparent that any unbalance of these 

 forces will result in an acceleration inversely proportional to the mass of 

 our small cube. Three equations may *hen be derived, one for each major 

 direction. If only simple harmonic n\)tion is considered (i.e. all displace- 



^ Refer to "Theory of Elasticity" by S. Timcshenko or to any standard text on elasticity. 



