hersey: stiffness of elastic systems 573 



(<0 dE if, (a) bfx { 



= E dd' ' ti de' J " L dd' 



(5) 



It is significant that there are no terms inseparably involving 

 both shape factors and thermal properties; as a consequence of 

 this, the complete expression for the effect of temperature on 

 the stiffness of a body, made of a material whose temperature 

 coefficients, a, /?, and y, are known, can be developed empir- 

 ically without changing the temperature. The dimensionless 

 factors A and B depend only on Poisson's ratio and the gen- 

 eralized shape, while a, /3, and y are familiar thermal properties 

 of the material as such. 



It would appear that a body of any fixed shape could be com- 

 pensated for temperature, provided materials for its construc- 

 tion could be found having such values of a, /S, and y as will 

 make the right hand side of (4) vanish; and, conversely, that a 

 body of any fixed material could be compensated, if its shape 

 could be so modified as to give to A and B values which would 

 make that member vanish. 



Simplified expressions for homogeneous isotropic bodies. When 

 the relation 



- = 2 (1 + «■) (6) 



characterizing homogeneous isotropic bodies is satisfied, the two 

 factors A and B coalesce into one factor, C, giving 



i^ = (l + C)«-C73 + 7 (7) 



o Old 



in which 



where, by (3) 



C=(l + 4 log*M (8) 



d<r 



♦M - ± (0) 



