§14.3 Complex Strain and the Elastic Constants 363 1 or 82=Eo2-vo1-va3) (14.2) 1 or 63 =E(03-vo1-va2) 14.3.Principal stresses in terms of strains-two-dimensional stress system For a two-dimensional stress system,i.e.o3 =0,the above equations reduce to 1 81=Eo1-o2) 1 and 82=E(o2-vo) 1 with g=E（-o1-G2) E81=01-o2 E82=02-vo1 Solving these equations simultaneously yields the following values for the principal stresses: E 01= (1-v2) (E1+VE2) E (14.3) and 2=1-v吗e2+ve) 14.4.Bulk modulus K It has been shown previously that Young's modulus E and the shear modulus G are defined as the ratio of stress to strain under direct load and shear respectively.Bulk modulus is similarly defined as a ratio of stress to strain under uniform pressure conditions.Thus if a material is subjected to a uniform pressure (or volumetric stress)a in all directions then volumetric stress bulk modulus volumetric strain 0 i.e. K= (14.4) Ev the volumetric strain being defined below. 14.5.Volumetric strain Consider a rectangular block of sides x,y and z subjected to a system of equal direct stresses a on each face.Let the sides be changed in length by &x,oy and oz respectively under stress (Fig.14.2).514.3 or or Complex Strain and the Elastic Constants 1 E - -(az-val -V63) z- E E3 = - (63 - vu1 - vaz) 1 E 363 (14.2) 14.3. Principal stresses in terms of strains - two-dimensional stress system For a two-dimensional stress system, Le. a3 = 0, the above equations reduce to 1 E 1 E 1 E E1 = - (crl - va2) E2 = - (az - Vbl) E3 = - (- Val- YO2) and with .. = a1 - va2 EEZ = 02 - VO~ Solving these equations simultaneously yields the following values for the principal stresses: and (14.3) 14.4. Bulk modulus K It has been shown previously that Young’s modulus E and the shear modulus G are defined as the ratio of stress to strain under direct load and shear respectively. Bulk modulus is similarly defined as a ratio of stress to strain under uniform pressure conditions. Thus if a material is subjected to a uniform pressure (or volumetric stress) 0 in all directions then volumetric stress volumetric strain bulk modulus = a i.e. K=- E, the volumetric strain being defined below. 14.5. Volumetric strain ( 14.4) Consider a rectangular block of sides x, y and z subjected to a system of equal direct stresses a on each face. Let the sides be changed in length by Sx, 6y and 6z respectively under stress (Fig. 14.2)