PSI - Issue 28

L.V. Stepanova et al. / Procedia Structural Integrity 28 (2020) 2277–2282 Author name / Structural Integrity Procedia 00 (2019) 000–000

2279

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the rotational body force the von Mises equivalent stress

e 

2. Steady state creep of a holed plate under uniform tension Here a plane stress problem for steady creep of a holed plate under uniform tension is considered. The quasilinearisation procedure is applied. By quasilinearisation, the nonlinear constitutive relations are replaced by its linearised form. Since the strain-stress, compatibility and equilibrium equations are linear, no further linearization is required. Thus, consider an infinite plate containing a hole of radius a . The geometry is best described by polar coordinate system   , , . r z  Owing to the loading it is assumed that a state of axisymmetric plane stress with the coordinate system being the principal axes of stress   , , . r z     and strain   , , . r z     is realized. For plane stress r are functions of alone, while 0. z   The field equations are   / , / , / / . r r u r du dr d dr r             (1) The equilibrium equation is , , , , 0, 2 0. rr r r rr r r r r r                      (2) with boundary conditions ( ) 0, . r r a at r            (3) The constitutive relations for plane stress are                 2 2 2 / / 2 , , / / 2 , , / , . r e e r r r e e r r z e e r z e r r f F f F f                                                           (4) The problem (1) – (4) is solved using quasilinearization (Boyle and Spence (1983)). A sequence of approximate solutions   ( ) ( ) , , 0,1, 2,.. k k r k     is generated in the following manner. Linearization of the constitutive relations (4) leads to , . r r rr r r r r a b b a b b                      (5) The coefficients , , , , , r r r r r r r r rr r r r r r r F F F F F F F F a F a F b b b b                                                      are evaluated for the current estimate   ( ) ( ) , . k k r    Eqns. (5) can be rearranged and presented in the form     / , / / / . r r r r r r rr r r r b a b a b a b b b b b b b b                                 (6) Equations (6) with compatibility (1) and equilibrium (2) equations give               / 1 / / 1 / . (1 / ) / (1 / ) / 1 / / r r r r r rr r r r a rb r b rb rb d r a a b rb r b b b b r b rb dr                                                         (7) Equations (7) are now in a standard form: a two-point linear boundary value problem in two unknowns   , . r     which must be solved subject to the boundary conditions at two endpoints. The solution of this problem together with (6) gives the next estimate   ( ) ( ) , . k k r    As an example, this problem has been solved for the special case of a power law . n B     The sequence of linear two-point boundary value problems can be solved using a standard routine from a computer scientific subroutine package. Here Waterloo Maple 17 has been used. In Figure 1 is shown the convergence of r  and   starting from the solution for the zeroth approximation. Figure 2 shows the approximate solution for different values of the creep exponent. One can see that the tangential stress reaches its maximum value not at the circular hole but at the internal point of the plate. To validate the approximate solution finite element calculations have been performed. The comparison of the approximate solution obtained by the quasilinearization method and the numerical solution given by finite element approach shows that both solutions are in good agreement