A Geometrically Non-linear Stochastic Analysis of Two-dimensional Structures made of Neo-hookean Hyperelastic Materials Uusing MLPG Method: Considering Uncertainty in Mechanical Properties

Document Type : Research Article

Authors

1 Engineering Department, Quchan University of Technology, Quchan, Iran

2 Civil Engineering Department, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

3 Industrial Engineering Department, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

Abstract

In recent decades, analysis of structures considering variability of some parameters for more reliable design has attracted the attention of researchers. In this paper, the stochastic analysis of a cantilever deep beam made of large deformable neo-hookean material is carried out. For this purpose, the meshless local Petrov-Galerkin (MLPG) method is developed to obtain the geometrically non-linear equilibrium equations. The radial point interpolation method is used for generating the shape functions. The incremental iterative Newton-Raphson method with suitable load steps is used to solve the non-linear governing equations. The results of deterministic analysis obtained with proposed method are compared with the finite element results and good agreement is achieved. The initial elasticity modulus of neo-hookean material is considered to be uncertain variable. To generate random field of uncertain variable with normal, lognormal and uniform probability density functions (PDFs), the Monte Carlo Simulation (MCS) technique was employed. The sufficient number of simulations for convergence the results was determined experimentally. The effect of elasticity modulus, PDF and coefficients of variation (COV) on maximum vertical displacement, PDF and COV of results are studied in details. Comparing the stochastic and deterministic results shows that the uncertainty in mechanical properties has significant effect on results.

Keywords

Main Subjects

References

[1] S. Chakraborty, B. Bhattacharyya, An efficient 3D stochastic finite element method, International Journal of Solids and Structures, 39 (9) (2002) 2465-2475.
[2] Y. Xu, Y. Qian, G. Song, Stochastic finite element method for free vibration characteristics of random FGM beams, Applied Mathematical Modelling, 40 (23) (2016) 10238-10253.
[3] B.N. Singh, D. Yadav, N.G. Iyengar, AC° element for free vibration of composite plates with uncertain material properties, advanced composite materials, 11 (4) (2002) 331-350.
[4] S. Sakata, F. Ashida, K. Ohsumimoto, Stochastic homogenization analysis of a porous material with the perturbation method considering a microscopic geometrical random variation, International Journal of Mechanical Sciences, 77 (2013) 145-154.
[5] S. Naskar, T. Mukhopadhyay, S. Sriramula, S. Adhikari, Stochastic natural frequency analysis of damaged thin-walled laminated composite beams with uncertainty in micromechanical properties, Composite Structures, 160 (2017) 312-334.
[6] A.K. Onkar, C.S. Upadhyay, D. Yadav, Generalized buckling analysis of laminated plates with random material properties using stochastic finite elements, International journal of mechanical sciences, 48 (7) (2006) 780-798.
[7] B.S. de Lima, N.F. Ebecken, A comparison of models for uncertainty analysis by the finite element method, Finite Elements in Analysis and Design, 34 (2) (2000) 211-232.
[8] S.M. Hosseini, F. Shahabian, Reliability of stress field in Al–Al2O3 functionally graded thick hollow cylinder subjected to sudden unloading, considering uncertain mechanical properties, Materials & Design, 31 (8) (2010) 3748-3760.
[9] S.M. Hosseini, F. Shahabian, Stochastic hybrid numerical method for transient analysis of stress field in functionally graded thick hollow cylinders subjected to shock loading, Journal of Mechanical Science and Technology, 27 (5) (2013) 1373-1384.
[10] B.N. Rao, S. Rahman, Stochastic meshless analysis of elastic–plastic cracked structures, Computational mechanics, 32 (3) (2003) 199-213.
[11] S.M. Hosseini, F. Shahabian, J. Sladek, V. Sladek, Stochastic Meshless Local Petrov-Galerkin(MLPG) Method for Thermo-Elastic Wave Propagation Analysis in Functionally Graded Thick Hollow Cylinders, Computer Modeling in Engineering & Sciences(CMES), 71 (1) (2011) 39-66.
[12] M. Dehghan, M. Shirzadi, Meshless simulation of stochastic advection–diffusion equations based on radial basis functions, Engineering Analysis with Boundary Elements, 53 (2015) 18-26.
[13] C. Su, Z. Qin, X. Fan, Stochastic spline fictitious boundary element method for modal analysis of plane elastic problems with random fields, Engineering Analysis with Boundary Elements , 66 (2016) 66-76.
[14] B. Kim, S.B. Lee, J. Lee, S. Cho, H. Park, S. Yeom, S.H. Park, A comparison among Neo-Hookean model, Mooney-Rivlin model, and Ogden model for chloroprene rubber, International Journal of Precision Engineering and Manufacturing, 13 (5) (2012) 759-764.
[15] Y. Anani, G.H. Rahimi, Stress analysis of thick pressure vessel composed of functionally graded incompressible hyperelastic materials, International journal of mechanical sciences, 104 (2015) 1-7.
[16] R.M. Soares, P.B. Gonçalves, Large-amplitude nonlinear vibrations of a Mooney–Rivlin rectangular membrane, Journal of Sound and Vibration, 333 (13) (2014) 2920-2935.
[17] Z.Q. Feng, F. Peyraut, Q.C. He, Finite deformations of Ogden’s materials under impact loading, International Journal of Non-Linear Mechanics, 41 (4) (2006) 575-585.
[18] J. Sladek, P. Stanak, Z.D. Han, V. Sladek, S.N. Atluri, Applications of the MLPG method in engineering & sciences: a review, Comput. Model. Eng. Sci, 92 (5) (2013) 423-475.
[19] Y. Gu, Q.X. Wang, K.Y. Lam, A meshless local Kriging method for large deformation analyses, Computer Methods in Applied Mechanics and Engineering, 196 (9-12) (2007) 1673-1684.
[20] M.H.G. Rad, F. Shahabian, S.M Hosseini, A meshless local Petrov–Galerkin method for nonlinear dynamic analyses of hyper-elastic FG thick hollow cylinder with Rayleigh damping, Acta Mechanica, 226 (5) (2015) 1497-1513.
[21] M.H.G. Rad, F. Shahabian, S.M Hosseini, Geometrically nonlinear elastodynamic analysis of hyper-elastic neo-Hooken FG cylinder subjected to shock loading using MLPG method, Engineering Analysis with Boundary Elements, 50 (2015) 83-96.
[22] A.S. Nowak, K.R. Collins, Reliability of structures, CRC Press, 2012.
[23] M.H.G. Rad, F. Shahabian, S.M Hosseini, Large Deformation Hyper-Elastic Modeling for Nonlinear Dynamic Analysis of Two Dimensional Functionally Graded Domains Using the Meshless Local Petrov-Galerkin (MLPG) Method, CMES: Computer Modeling in Engineering & Sciences, 108 (3) (2015) 135-157.
[24] E. Larsson, B. Fornberg, A numerical study of some radial basis function based solution methods for elliptic PDEs, Computers & Mathematics with Applications, 46 (5-6) (2003) 891-902.
AUT Journal of Civil Engineering
Volume 2, Issue 2
December 2018
Pages 209-218

Files

Share

How to cite

Statistics