PSI - Issue 3

Christos F. Markides et al. / Procedia Structural Integrity 3 (2017) 334–345 Christos F. Markides, Stavros K. Kourkoulis / Structural Integrity Procedia 00 (2017) 000–000

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7

The above condition is fully justified by checking the final results. But even then Eqs.(27) and (28) could not provide yet displacements due to the arbitrariness related to terms involving A 1 and B 1 . However, it can be seen that these terms correspond to certain displacements. Namely, consider the parts Φ 1 (0) and Φ 2 (0) of Φ 1 , Φ 2 due to A 1 and B 1 :         0 0 1 1 1 1 2 2 1 2 z A z , z B z     (30) Combining Eqs.(30) and (18), shows that A 1 and B 1 correspond to the constant stresses (Lekhnitskii 1968):         0 0 0 2 2 2 2 x 1 1 1 2 1 1 1 2 y 1 1 1 1 xy 1 1 1 2 1 1 1 2 A B A B , A B A B , A B A B                         (31) or by taking into account Eqs.(21):                   0 0 0 x 1 1 y 1 1 xy 1 1 1 1 b b Ri , a a R , a a Ri b b R             (32) Substituting   0 x  ,   0 y  from Eqs.(32) in the first two of Eqs.(2) and integrating along x, y, respectively, one finds:                     0 0 0 0 0 0 11 x 12 y 1 12 x 22 y 2 u x, y x f y , v x, y y f x                       (33) Then, using Eqs.(33) in satisfying the third of Eqs.(2), it is obtained that:       0 1 2 66 xy f y f x       (34) whence it is implied that:     1 1 1 2 2 2 f y c y d , f x c x d     (35) so that after differentiations in Eq.(34):   0 1 2 66 xy c c     (36) Clearly, by Eqs.(33) and (35) it will hold that:                 0 0 0 0 0 0 11 x 12 y 1 1 12 x 22 y 2 2 u x, y x c y d , v x, y y c x d                         (37) Next, consider the condition of zeroing rigid body rotation of the disc about its center due to the constant stresses of Eqs.(32) or what is the same thing due to u (0) , v (0) of Eqs.(37):     0 0 1 v u 0 2 x y               (38) Then, introducing Eqs.(37) in Eqs.(38), yields:

(39)

2 1 c c 

whence taking also into account Eq.(36), one takes:

   

  66 0

(40)

c c

1

2

xy

2

Substituting in Eq.(37) from Eqs.(40), omitting in the former the constants d 1 and d 2 expressing rigid body translation, it is obtained that:                     0 0 0 0 0 0 0 0 66 66 11 x 12 y xy 12 x 22 y xy u x, y x y, v x, y y x 2 2                           (41)