- Volume 29 Issue 3
This paper presents a nonlinear Moving Least Squares(MLS) difference method for material nonlinearity problem. The MLS difference method, which employs strong formulation involving the fast derivative approximation, discretizes governing partial differential equation based on a node model. However, the conventional MLS difference method cannot explicitly handle constitutive equation since it solves solid mechanics problems by using the Navier's equation that unifies unknowns into one variable, displacement. In this study, a double derivative approximation is devised to treat the constitutive equation of inelastic material in the framework of strong formulation; in fact, it manipulates the first order derivative approximation two times. The equilibrium equation described by the divergence of stress tensor is directly discretized and is linearized by the Newton method; as a result, an iterative procedure is developed to find convergent solution. Stresses and internal variables are calculated and updated by the return mapping algorithm. Effectiveness and stability of the iterative procedure is improved by using algorithmic tangent modulus. The consistency of the double derivative approximation was shown by the reproducing property test. Also, accuracy and stability of the procedure were verified by analyzing inelastic beam under incremental tensile loading.
material nonlinearity;MLS difference method;strong formulation;double derivative approximation;Newton method
- Dai, K.Y., Liu, G.R., Han, X., Li, Y. (2006) Inelastic Analysis of 2D Solids using a Weak-form RPIM based on Deformation Theory, Comput. Methods Appl. Mech. & Eng., 195, pp.4179-4193. https://doi.org/10.1016/j.cma.2005.07.021
- Gu, Y.T., Wang, Q.X., Lam, K.Y., Dai, K.Y. (2007) A Pseudo-elastic Local Meshless Method for Analysis of Material Nonlinear Problems in Solids, Eng. Anal. Bound. Elem., 31, pp.771-782. https://doi.org/10.1016/j.enganabound.2006.12.008
- Pozo, P.L., Perazzo, F., Angulo, A. (2009) A Meshless FPM Model for Solving Nonlinear Material Problems with Proportional Loading based on Deformation Theory, Adv. Eng. Softw., 40, pp.1148-1154. https://doi.org/10.1016/j.advengsoft.2009.05.004
- Lee, S.H., Yoon, Y.C. (2004) Meshfree Point Collocation Method for Elasticity and Crack Problems, Int. J. Numer. Methods Eng., 61, pp.22-48. https://doi.org/10.1002/nme.1053
- Simo, J.C., Taylor, R.L. (1985) Consistent Tangent Operators for Rate-independent Elastoplasticity, Comput. Methods Appl. Mech. & Eng., 48, pp.101-118. https://doi.org/10.1016/0045-7825(85)90070-2
- Simo, J.C., Hughes, T.J.R. (1998) Computational inelasticity, Springer-Verlag, New York.
- Yoon, Y.C., Kim, D.J., Lee, S.H. (2007) A Gridless Finite Difference Method for Elastic Crack Analysis, J. Comput Struct. Eng., 20(3), pp.321-327.
- Yoon, Y.C., Lee, S.H. (2009) Intrinsically Extended Moving Least Squares Finite Difference Method for Potential Problems with Interfacial Boundary, J. Comput Struct. Eng., 22(5), pp.411-420.
- Yoon, Y.C., Kim, K.H., Lee, S.H. (2012) Dynamic Algorithm for Solid Problems using MLS Difference Method, J. Comput. Struct. Eng., 25(2), pp.139-148.
- Yoon, Y.C., Kim, K.H., Lee, S.H. (2014) Analysis of Dynamic Crack Propagation using MLS Difference Method, J. Comput. Struct. Eng., 27(1), pp.17-26.
- Yoon, Y.C., Song, J.-H. (2014a) Extended Particle Difference Method for Weak and Strong Discontinuity Problems: Part I. Derivation of the Extended Particle Derivative Approximation for the Representation of Weak and Strong Discontinuities, Comput. Mech., 53(6), pp.1087-1103. https://doi.org/10.1007/s00466-013-0950-8
- Yoon, Y.C., Song, J.-H. (2014b) Extended Particle Difference Method for Weak and Strong Discontinuity Problems: Part II. Formulations and Applications for Various Interfacial Singularity Problems, Comput. Mech., 53(6). pp.1105-1128. https://doi.org/10.1007/s00466-013-0951-7
- Dynamic Analysis of MLS Difference Method using First Order Differential Approximation vol.31, pp.6, 2018, https://doi.org/10.7734/COSEIK.2018.31.6.331
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