applied fracture mechanics

14 Applied Fracture Mechanics12 Applied Fracture MechanicsCorrespondingly, an expansion coefficient K (m) is available via the mth order weight function,h (m)k (x 1 , x 2 )=u ∗(m)k (x 1 , x 2 )/Y (m) , (41)as a functionalK (m) = h ∗(m)k (x 1 , x 2 )F k ds (42)S Tof the surface loading. Thus the mth order weight function differs from the correspondingfundamental field (33) only in a constant geometry factor (40).(II) If S U = 0, all expansion coefficients in the series (20) are defined unambiguously,including K (0) and K (1) . For m = 0 the fundamental field of the mth order is still given by thesolution (33) provided that its bounded energy part is completed by the rigid body motionterms. The coefficients k p in (33) are now chosen to subject it to the boundary conditions⎧⎨ T ∗(m)k= 0 on S T + S N ,(43)⎩u ∗(m)k= 0 on S U .Modifying the reciprocal relation (35) to include S U , one obtainsΓ (m) = − u ∗(m)k F k ds + T ∗(m)k U k ds. (44)S T S TThe remaining calculations are similar to those performed in the case of vanishing S U . Weightfunctions of the mth order are introduced according toh (m)k (x 1 , x 2 )=u ∗(m)k (x 1 , x 2 )/Y (m) , (45)H (m)k (x 1 , x 2 )=T ∗(m)k (x 1 , x 2 )/Y (m) . (46)The coefficients K (m) in the eigenfunction expansion are now expressed via the mth orderweight functions asK (m) =h ∗(m)k (x 1 , x 2 )F k ds −S TS UH ∗(m)k (x 1 , x 2 )U k ds. (47)As it was noted above, in the case of non-vanishing S U the coefficient K (0) needs a specialtreatment. The complementary field for the rigid body translation (21) has a logarithmic ratherthan power-law functional form. Indeed, the auxiliary source generating this complementarysolution is a concentrated force applied at the notch tip. Unlike other eigenfunctions it is notself-equilibrated. The complementary logarithmic solution can be constructed by means ofthe analytical expression for the elastic field of a force at the tip of a notch in an angularlyinhomogeneous plane [16, 18]. Details of further development and the corresponding 0thorder fundamental field for the calculation of the rigid body translation term are availablefrom [28].