PSI - Issue 28

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

2282

6

ε

ε r

θ

n=9

n=9

2

-0.002

0.02

converged solution

3

-0.004

6

5

0.015

4

-0.006

converged solution

4

5

0.01

6

-0.008

3

-0.01

0.005

2

r

r

1.4

1.8

2

1.6

1.4

1.8

2

1.6

1

1.2

1

1.2

Fig. 5. Convergence of radial strain r  (left) and circumferential strain   (right) using quasilinearisation for 9 n 

4. Conclusions The approximate solution to the problem for an infinite plate with the circular hole under creep regime is obtained by the quasilinearization method. Five and six approximations of the solution for the nonlinear problem are found. It is shown that with increasing the number of approximations the solution converges to the limit numerical solution. It is worth to note that the tangential stress reaches its maximum value not at the circular hole but at the internal point of the plate. It is shown that quasilinearization method is the effective method for nonlinear problems. Acknowledgements The work was supported by Russian Foundation for Basic Research (project 19-31-90100). References Aznam, S. Mt., Ghani, N.A.C., Chowdhury M.S.H., 2019. A numerical solution for nonlinear heat transfer of fin problems using the Haar wavelet quasilinearization method. Results in Physics 14, 102393. Bellman, R.E., Kalaba, R.E., 1965. “ Quasilinearization and Nonlinear Boundary Value Problems ”, Elsevier, New York. Boyle, J.T., Spence, J., 1983. “ Stress Analysis for Creep ”, Butterworths, London. Lakshmikantham, V., Vatsala, A., 1998. “ Generalized Quasilinearization for Nonlinear Problems ”, Kluwer Academic Publishers, Dordrecht. Magagula, V.M., Motsa, S.S., Sibanda, P. 2020. On the bivariate spectral quasilinearization method for nonlinear boundary layer partial differential equations, in ” Applications of Heat, Mass and Fluid Boundary Layers” . Woodhead Publishing, Sawston, pp. 177-190. Mandelzweig, V.B., Tabakin, F., 2001. Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs. Comput Phys Commun 141, 268–281. Polyanin, A.D., 2019. Construction of exact solutions in implicit form for PDEs: New functional separable solutions of non-linear reaction-diffusion equations with variable coefficients. International Journal of Non-linear Mechanics 111, 95-105. Rani, D., Mishra, V., 2020. Numerical inverse Laplace transform based on Bernoulli polynomials operational matrix for solving nonlinear differential equations. Results in Physics 16. 102836. Saeed, U., Rehman, M.U., 2013. Haar wavelet-quasilinearization technique for fractional nonlinear differential equations. Applied Mathematics and Computation 220, 630-648. Singh, R., Guleria, V., Singh M., 2020. Haar wavelet quasilinearization method for numerical solution of Emden-Fowler type equations Mathematics and Computers in Simulations 174, 123-133. Stepanova, L., Yakovleva, E.M., 2015. Asymptotic stress field in the vicinity of mixed-mode crack under plane stress conditions for power-law hardening material. J Mech Mater Struct 10 (3), 367–393. Stepanova, L., Yakovleva, E., 2016. Stress-strain state near the crack tip under mixed-mode loading: Asymptotic approach and numerical solutions of nonlinear eigenvalue problems. AIP Conference Proceedings 1785, 030030. Stepanova, L., Yakovleva, E., 2016. Asymptotics of eigenvalues of the nonlinear eigenvalue problem arising from the near mixed-mode crack tip stress-strain. Numerical Analysis and Applications 9(2) 159-170. Verma, A.K., Tiwari, D., 2019. Higher resolution methods based on quasilinearization and Haar wavelets on Lane–Emden equations. International Journal of Wavelets. Multiresolution and Information Processing 17(3), 1950005.