$14.9 Complex Strain and the Elastic Constants 367 From (1), Gy=vOx+VO: :=(ay-va (3) Substituting in(2), 1 (y-vax)-vax-vay=0 0,-0x-v2ox-v2o,=0 6,(1-v)=0x(w+v2) v(1+ 0,=0x(1-2) Oxv =(1-0 and from (3), strain in x direction=- E v2 (14.10) Again Young's modulus E is effectively changed,this time to 14.9.Relationship between the elastic constants E,G,K and v (a）E,G and v Consider a cube of material subjected to the action of the shear and complementary shear forces shown in Fig.14.5 producing the strained shape indicated. Assuming that the strains are small the angle ACB may be taken as 45. Therefore strain on diagonal OA BC.AC cos45°AC1AC a/2a√W2√2"2a$14.9 Complex Strain and the Elastic Constants From (l), .. 6, = vox + voz 6, = (0, - vox) - 1 V Substituting in (2), 1 - (oy -vox) - vox - Yoy = 0 V .. o, - vox - v20x - v20y = O o,(l-v2)= o,(v+v2) v(l +v) 0, = 6,- (1 - v2) fJXV =- (1 - v) and from (3), .. cz=- '[ -- vox vox] =ox[ (1-v) ] (1-v) v (1 -v) 1 - (1 - v) VOX =- ox o o strain in x direction = - - + v--li + v? EEE E (1-V) (1-V) 367 (3) (14.10) Again Young's modulus E is effectively changed, this time to 14.9. Relationship between the elastic constants E, G, K and v (a) E, G and v Consider a cube of material subjected to the action of the shear and complementary shear Assuming that the strains are small the angle ACB may be taken as 45". Therefore strain on diagonal OA forces shown in Fig. 14.5 producing the strained shape indicated. BC ACcos45" AC 1 AC OA- ad2 a~2~~2=2a =-A =-