PSI - Issue 37

L.V. Stepanova et al. / Procedia Structural Integrity 37 (2022) 920–925 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

921

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studies (Polyanin, (2019), Polyanin and Sorokin (2021), Stepanova and Yakovleva (2015), Singh et al. (2020), Magagula, et al. (2020), Stepanova and Yakovleva (2016), Verma and Tiwari (2019), Rani and Mishra (2020), Malyk. (2020)). Undoubtely, the combination of computer technology and algorithms and approximate solutions provides a promising way to solve complex problems of natural sciences. The quasilinearisation method is effective in this respect. Based on Newton-Rapson quasilinearisation approach was first proposed by Bellman and Kalaba (1965) to solve the nonlinear ordinary and partial differential equations. In the (Boyle and Spence (1983)) quasilinearisation technique has been widely used for steady state creep analysis to since the most obvious route for nonlinear problems is linearize the equations around an initial estimate and then solve the linear problem iteratively until the process converges. To solving partial differential equations quasilinearisation scheme is insufficient, so some of its modifications are developed. As example in (Pandey and Saurabh (2021)) an effective analytical method has been introduced. Based on combination of Newton's quasilinearisation and Picard iteration method for a solving a class of two-point nonlinear doubly singular boundary value problems has been used. In (Su et al. (2020)) a class of nonlinear Riemann-Liouville fractional-order two-point boundary problem solutions are obtained by using quasilinearisation technique. Haar wavelet quasilinearisation method for numerical solution of Emden-Fowler type boundary value problems with Dirichlet and Robin-Neumann boundary conditions is introduced (Singh et al. (2020)). Presented method based on the Haar-wavelets and quasilinearisation technique.To linearize the nonlinear singular problem the quasilinearisation method has been used and Haar wavelets has been used to obtain numerical solution of linear singular problem. Based on Haar wavelets and quasilinearisation technique is also used in (Aznam et al. (2019)) to solve three different types of nonlinear problems and noted that method was proven to be stable, convergent and easily coded. The two-grid quasilinearisation for solving high-order nonlinear differential equations are presented in (Koleva and Vulkov (2010)). Numerical experiments show that a large class of NODEs, including the Fisher – Kolmogorov, Blasius and Emden – Fowler equations solving with two-grid algorithm will not be much more difficult than solving the corresponding linearized equations and at the same time with significant economy of the computations. Exact solutions of partial differential equations have always played a crucial role in shaping the correct understanding of the qualitative features of many phenomena (Polyanin and Sorokin (2021)). Exact solutions of nonlinear differential equations clearly demonstrate and allow us to better understand the physical phenomena described by the considered partial differential equations. In this contribution the aim is to construct an approximate analytical solution to the nonlinear boundary value problem of determining the stress-strain state in an infinite plate with a central circular hole under creep regime for the Norton constitutive equations by the quasilinearisiation method.

Nomenclature ij 

stress tensor components , , r z  cylindrical coordinates i x cartesian coordinates ij  creep strain rates , ij ijkl a b coefficients in the linearization method n creep exponent e  the von Mises equivalent stress a the hole radius B

material constant of the Bailey-Norton power law

the remote stress

 

2. Steady state creep of a holed plate under uniaxial tension A plane stress problem for steady creep of a holed plate under uniaxial tension is considered. Consider an infinite plate containing a hole of radius a (Fig. 1). The geometry is described by cylindrical coordinate system ( ) , , . r z 