§14.9 Complex Strain and the Elastic Constants 369 (b)E,K and v Consider a cube subjected to three equal stresses a as in Fig.14.7 (i.e.volumetric stress =a). Fig.14.7.Cubical element subjected to uniform stress a on all faces ("volumetric"or "hydrostatic" stress). Total strain along one edge E E-E =E1-2y But volumetric strain=3 x linear strain (see eqn.14.5) 30 °E1-2) (3) By definition: bulk modulus K volumetric stress volumetric strain volumetric strain=K (4) Equating (3)and (4), 03 K=E(1-2v) E=3K(1-2v) (14.12) (c)G,K and v Equations(14.11)and(14.12)can now be combined to give the final relationship as follows: From eqn.(14.11), E v= 2G-1$14.9 (b) E, K and v Complex Strain and the Elastic Constants 369 Consider a cube subjected to three equal stresses 6 as in Fig. 14.7 (Le. volumetric stress = 6). Fig. 14.7. Cubical element subjected to uniform stress u on all faces (“volumetric” or “hydrostatic” stress). 606 Total strain along one edge = - - v- - v￾EEE 6 = -(1 -2v) E But volumetric strain = 3 x linear strain (see eqn. 14.5) 30 E = -(I -2v) By definition: volumetric stress volumetric strain bulk modulus K = Equating (3) and (4), .. 0 volumetric strain = - K 0 30 KE - = -(1-2v) (3) (4) (14.1 2) (c) G, K and v Equations (14.1 1) and (14.12) can now be combined to give the final relationship as follows: From eqn. (14.1 l), E