Stress Function Formulation without Body Forces? Solution of the Airy Stress Function8G(1+n)oVT=0Vy+1+K8G(1+n)=y/l) +yhe) = Vryl) =0, Vylp) +8OT=01+: The traction BCs in terms of Airy Stress Functionyydy(dx)aydT=0,n, +tyny=1ay dsdsds( yaxoy y)dyoy(_dx)-_d(ayTn =t,n +on,Jds"ax(dsds axaxoydyds: Integrate over a portion of the boundarydxSaydyTnds+(T'ds+C =-dxdx_dydyaxnXdn dsdsdn
• Solution of the Airy Stress Function 4 2 4 2 8 1 0 1 8 1 0, 0 1 h p h p G T G T • The traction BCs in terms of Airy Stress Function d d d d , d d d d x y x y y x n n n s n s 2 2 2 2 2 2 d d d d d d d d d d d d n x x x xy y n y xy x y y y x T n n y s x y s s y y x T n n x y s x s s x • Integrate over a portion of the boundary d , d 1 2 n n x y C C T s C T s C y x Stress Function Formulation without Body Forces
Stress Function Formulation without Body Forces: Consider the directional derivative of the Airy StressFunction along the boundary normaldyaudxduL.7ds)()Trds)t. Faxdsadn. where t is the unit tangent vector and F is the resultantboundary force.: For many applications, the BCs aresimply expressed in terms of stresses. For the case of zero surface tractions:dy/dn=0 = y=C.dydsFor simply connected regions, a steadydxtemperature distribution with zeroSboundary tractions will not affect thedxdx_dydyXin-plane stress fielddndsdnds
• Consider the directional derivative of the Airy Stress Function along the boundary normal d d d d d d d d n n x y y x C C y x n n T s T s n x y s s n t F • where t is the unit tangent vector and F is the resultant boundary force. • For many applications, the BCs are simply expressed in terms of stresses. • For the case of zero surface tractions: d d 0 . n C • For simply connected regions, a steady temperature distribution with zero boundary tractions will not affect the in-plane stress field. Stress Function Formulation without Body Forces d d d d , d d d d x y x y y x n n n s n s