§8.4 Introduction to Advanced Elasticity Theory 225 Pn*resultont stress on ABC n normol to ABC n 只0 Cartesian components of resultont 5tre多s Fig.8.5.Cartesian components of resultant stress on an inclined plane. Fig.8.6.Normal and tangential components of resultant stress on an inclined plane. or p听=p层n+p+p (8.17) these being alternative forms of eqns.(8.6)and(8.4)respectively. From eqn.(8.5)the normal stress on the plane is given by: On=pxn·【+Pw·m+Pn·n But from eqns.(8.13),(8.14)and(8.15) Pxt=Oxx·l十Oxy·m+O·n pm=ox·l+oy·m+ox·n pn=ox1+oy·m+ox·n .Substituting into eqn (8.5)and using the relationships oxy=ox:ox=ox and o=o which will be proved in $8.12 om=aa·l2+ow·m2+az·n2+2ay·lm+2oz·mn+2oz·ln. (8.18)$8.4 Introduction to Advanced Elasticity Theory 225 t' X P, sresultont stress n=normol to ABC on ABC P, Cortesion I Components pv ot resultant p,, 1 stress Y Fig. 8.5. Cartesian components of resultant stress on an inclined plane X Fig. 8.6. Normal and tangential components of resultant stress on an inclined plane. 2 or ~,2 =P:n + P;n + PW these being alternative forms of eqns. (8.6) and (8.4) respectively. From eqn. (8.5) the normal stress on the plane is given by: (8.17) an = Pxn .I + pbII . m + pill . But from eqns. (8.13), (8.14) and (8.15) pxr, = a,, . I + a,\ . m + a,,? . n Pyn = aVx .1 + a), . m + 0): . n pzn = a, . I + a,, . m + az, . n :. Substituting into eqn (8.5) and using the relationships a,, = ayx; ax, = which will be proved in 58.12 and a,: = a,, Un =uxx~12+uyy~m 2 +u,.n 2 +2u,,~lm+2uy~~mn+2ux~~ln. (8.18)