STRAINED ELASTIC SOLIDS 397 



placed to PQ, and this for all possible positions of Q' in the 

 neighborhood of P'; and this again, if the strain is heterogene- 

 ous, for all possible positions of P' in the body. 



The use of the words "homogeneous" and "heterogeneous" in 

 connection with strain must not lead to confusion with their 

 use as referring to substances. A homogeneous material may- 

 very readily be subjected to a heterogeneous strain, as will 

 appear presently. It is as well also at this point to reahze what 

 is meant by an elastically isotropic material as distinct from 

 one which is elastically anisotropic (or aeolotropic). Thus we 

 suppose that the body is deformed from its state of reference 

 by a completely defined set of external forces acting on each 

 element of volume (gravitational, for example; or definite 

 mechanical pulls applied to definite elements of volume in the 

 periphery of the body). Each element of length P'Q' in the 

 body is subject to a definite change in length and direction. 

 Suppose now that all the external forces remain unchanged in 

 magnitude but all are changed by the same amount in direction, 

 then the strain in the linear element P'Q', i.e., its change in 

 magnitude and direction from the state of reference, will not in 

 general remain as before; but if the body is isotropic a linear 

 element P'R' which bears the same relation of direction to the 

 directionally changed forces as did P'Q' to the external forces 

 formerly, will experience the same strain as that to which P'Q' 

 was subject in the first case. But for an anisotropic (crystal- 

 line) body even this statement is not in general true. These 

 definitions in general terms will be more clearly stated in 

 precise mathematical form presently; but the fact mentioned 

 embodies the essence of the distinction between anisotropy and 

 isotropy. 



Before proceeding to a general mathematical treatment of 

 strain it may be advisable to consider one or two special cases 

 where there are certain simplifying conditions. Imagine for 

 example that all points are displaced in one direction, parallel 

 to the axis OX' say, and that the displacement of the point 

 P'ix', y', z') is a function of x' only. Representing this dis- 

 placement by u{x') (or briefly by u), we have 



X = x' -{■ u{x'), y = y', z = z'y 



