§14.7 Complex Strain and the Elastic Constants 365 It will be shown later that the following relationship applies between the elastic constants E,v and K, E=3K(1-2y) Thus the volumetric strain may be written in terms of the bulk modulus as follows: 6,t* (14.7) 3K This equation applies to solid bodies only and cannot be used for the determination of internal volume (or capacity)changes ofhollow vessels.It may be used,however,for changes in cylinder wall volume. 14.7.Change in volume of circular bar A simple application of egn.(14.6)is to the determination of volume changes of circular bars under direct load. Consider,therefore,a circular bar subjected to a direct stress a applied axially as shown in Fig.14.3. Direct stress Fig.14.3.Circular bar subjected to direct axial stress o. Here 0y=0,0x=0andg,=0 Therefore from eqn.(14.6) 8V 8.=E1-2)= change of volume=(1-2v) (14.8) This formula could have been obtained from eqn.(14.5)with e=E and 6s=6=6o=-VE oV then e,=6x+8,+8:= 6v=v(但-2)=Y4-2刘$14.7 Complex Strain and the Elastic Constants 365 It will be shown later that the following relationship applies between the elastic constants E, v and K, Thus the volumetric strain may be written in terms of the bulk modulus as follows: E=3K(1-2~) (14.7) This equation applies to solid bodies only and cannot be used for the determination of internal volume (or capacity) changes of hollow vessels. It may be used, however, for changes in cylinder wall volume. 14.7. Change in volume of circular bar A simple application of eqn. (14.6) is to the determination of volume changes of circular Consider, therefore, a circular bar subjected to a direct stress 6 applied axially as shown in bars under direct load. Fig. 14.3. EY * % c Direct stress U Fig. 14.3. Circular bar subjected to direct axial stress u. cy= 6, cx = 0 and 0, = 0 Here Therefore from eqn. (14.6) 6V E, = -(1-2v) = - E V 6 UV E .. This formula could have been obtained from eqn. (14.5) with change of volume = 6 V = - (1 - 2v) 6 d E, = - and E, = E, = E,, = - v- E E 6V V then E,=E,+E,+E,=- (14.8)