§14.10 Complex Strain and the Elastic Constants 371 Thus the movement of M parallel to OM,which since the strains are small is practically coincident with MM',is (xa cos0+Yxa sin 0)cos0+(E,a sin 0)sin0 Then strain along OM= extension original length =(ex cos0+Yx,sin 0)cos+(Ey sin 0)sin0 Eo=Ex cos20+sin20+Yxy sin 0 cos g=支(ex+e,)+(ex-e,）c0s20+是Yxy sin28 (14.14) This is identical in form with the equation defining the direct stress on any inclined plane with &x and E,replacing ox and a,and ty replacing tx,i.e.the shear stress is replaced by HALF the shear strain. (b)Shear strain To determine the shear strain in the direction OM consider the displacement of point P at the foot of the perpendicular from N to OM(Fig.14.9). Ex QCOs8+y asin日 ey a sin B)cos 8 D (e,a cos B)cos 8 la cos 8)sin8 N E acos Fig.14.9.Enlarged view of part of Fig.14.8. In the strained condition this point moves to p'. Since strain along OM extension of OM=OM·eg extension of OP=OP·eg But OP (acos0)cos0 extension of OP a cos2 0eg$14.10 Complex Strain and the Elastic Constants 37 1 Thus the movement of M parallel to OM, which since the strains are small is practically coincident with MM', is (&,a cos 8 + y,,a sin 0)cos 8 + (&,a sin 6) sin 8 Then extension original length strain along OM = = (E, cos 0 + yxy sin e) cos 8 + (ey sin e) sin 8 .. E, = E, cosz e + E, sinZ e + yxy sin e cos e .. Eo = 3 (E, + E,,) + 3 (E, - E,,) cos 28 + 3 yxp sin 28 (14.14) This is identical in form with the equation defining the direct stress on any inclined plane 8 with E, and cy replacing ox and oy and &J,, replacing T,~, i.e. the shear stress is replaced by HALF the shear strain. (b) Shear strain To determine the shear strain in the direction OM consider the displacement of point P at the foot of the perpendicular from N to OM (Fig. 14.9). w cX acos B+X, asin 9 Fig. 14.9. Enlarged view of part of Fig. 14.8. In the strained condition this point moves to P' Since But .. strain along OM = E, extension of OM = OM extension of OP = OP~E, extension of OP = a cosz eEo OP = (a COS e) COS e