On applicability of MQ-RPIM and MLPG meshless methods with 3D extended-enriched base functions for estimation of mode I stress intensity factor and fatigue crack growth in cyclic tensile and bending load of an un-notched and notched shaft

Abstract

In this study, a numerical meshless method is used to solve the weak form of the linear elastic equations in solid mechanics. Evaluation and comparison of the numerical meshless methods have been carried out via the radial point interpolation meshless method with multi-quadrics base functions (MQ-RPIM) and meshless local Petrov-Galerkin method (MLPG). Using these two methods, stress intensity factors in an elastic medium containing geometric discontinuities and cracks are estimated based on tensile and bending cyclic loading. The analysis domain has been identified via three-dimensional modeling of the notched and un-notched shafts with an initial surface semi-elliptical crack subjected to tensile or bending cyclic loadings. To enhance the accuracy of calculations, the RPIM meshless method is applied using polynomial and extended-enriched 3D base functions. Shape functions have been developed using standard and optimal parameters and values with Mono-Objective Function in PSO algorithm. In the MLPG meshless method with the extended-enriched functions, discretization is performed via direct and penalty factor methods, to reach more efficient results and meet the boundary conditions. Efficiency comparison of the selected numerical methods with the experimental findings and the numerical analysis of finite elements method indicates that in comparison with the MLPG method, MQ-RPIM enriched meshless method can be utilized with fewer nodes in the analysis domain while reaching the accuracy and convergence with lower stress intensity factors and gentler slope. However, the processing time of the MLPG meshless method is lower than that of the other methods.

Keywords

Radial point interpolations meshless method meshless local Petrov-Galerkin method 3D extended-enriched base functions fatigue crack growth stress intensity factor cyclic load PSO algorithm

Get full access to this article

View all access options for this article.

References

Lucy

. A numerical approach to the testing of the fission hypothesis. Astron J 1977; 82: 1013–1024.

Gingold

Monaghan

. Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon Not R Astron Soc 1977; 181(3): 375–389.

Lancaster

Salkauskas

. Surfaces generated by moving least squares methods. Math Comput 1981; 37(155): 141.

Nayroles

Touzot

Villon

. The diffuse approximation. Compt R Acad Sci Ser II 1991; 313(3): 293–296.

Nayroles

Touzot

Villon

. Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 1992; 10(5): 307–318.

Belytschko

. Element-free Galerkin methods. Int J Numer Methods Eng 1994; 37(2): 229–256.

Atluri

Zhu

. A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput Mech 1998; 22(2): 117–127.

Shen

Han

, et al. Application of meshless local Petrov-Galerkin (MLPG) to problems with singularities and material discontinuities, in 3-D elasticity. Comput Model Eng Sci 2003; 4(5): 571–586.

Han

Atluri

. Truly meshless local Petrov-Galerkin (MLPG) solutions of traction & displacement BIEs. Comput Model Eng Sci 2003; 4(6): 665–678.

10.

Han

Atluri

. Meshless local Petrov-Galerkin (MLPG) approaches for solving 3D problems in elasto-statics. Comput Model Eng Sci 2004; 6: 169–188.

11.

Liu

G-R

. A point interpolation method for two-dimensional solids. Int J Numer Methods Eng 2001; 50(4): 937–951.

12.

Wang

Liu

. A point interpolation meshless method based on radial basis functions. Int J Numer Methods Eng 2002; 54(11): 1623–1648.

13.

Wang

Liu

. On the optimal shape parameters of radial basis functions used for 2-D Meshless methods. Comput Methods Appl Mech Eng 2002; 191(23–24): 2611–2630.

14.

Liu

. A local point interpolation method for stress analysis of two-dimensional solids. Struct Eng Mech 2001; 11(2): 221–236.

15.

Liu

. A local radial point interpolation method (LRPIM) for free vibration analyses of 2-D solids. J Sound Vib 2001; 246(1): 29–46.

16.

Liu

. Vibration analyses of 2-D solids by the Local Point Interpolation Method (LPIM). In: ICSSD 2000: 1st structural conference on structural stability and dynamics, Taipei, Taiwan, 7–9 December 2000.

17.

Liu

. Comparisons of two meshfree local point interpolation methods for structural analyses. Comput Mech 2002; 29(2): 107–121.

18.

Wang

Zhang

, et al. An enriched radial point interpolation method (e-RPIM) for analysis of crack tip fields. Eng Fract Mech 2011; 78(1): 175–190.

19.

Moussaoui

Bouziane

. Numerical study of the shape parameter dependence of the local radial point interpolation method in linear elasticity. MethodsX 2016; 3: 178–187.

20.

Liu

Dai

, et al. A linearly conforming radial point interpolation method for solid mechanics problems. Int J Comput Methods 2006; 3(4): 401–428.

21.

Zhuang

. Meshless methods: theory and application in 3D fracture modelling with level sets. Durham Thesis, Durham University, Durham, 2010.

22.

Teng

Ungor

. Point placement for meshless methods using sphere packing and advancing front methods. In: ICES00: International conference on computational engineering science, Los Angeles, CA, 21–25 August 2000.

23.

Carpinteri

Brighenti

Vantadori

. Surface cracks in notched round bars under cyclic tension and bending. Int J Fatigue 2006; 28: 251–260.

24.

Wang

Liu

. Radial point interpolation method for elastoplastic problems. In: ICSSD: 1st structural conference on structural stability and dynamics, Taipei, Taiwan, 7–9 December 2000.

25.

Liu

. An introduction to meshfree methods and their programming. Berlin, Germany: Springer Science & Business Media, 2005.