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Consistent multiscale analysis of heterogeneous thin plates with smoothed quadratic Hermite triangular elements

  • Boya Dong
  • Congying Li
  • Dongdong WangEmail author
  • Cheng-Tang Wu
Article

Abstract

A consistent multiscale formulation is presented for the bending analysis of heterogeneous thin plate structures containing three dimensional reinforcements with in-plane periodicity. A multiscale asymptotic expansion of the displacement field is proposed to represent the in-plane periodicity, in which the microscopic and macroscopic thickness coordinates are set to be identical. This multiscale displacement expansion yields a local three dimensional unit cell problem and a global homogenized thin plate problem. The local unit cell problem is discretized with the tri-linear hexahedral elements to extract the homogenized material properties. The characteristic macroscopic deformation modes corresponding to the in-plane membrane deformations and out of plane bending deformations are discussed in detail. Thereafter the homogenized material properties are employed for the analysis of global homogenized thin plate with a smoothed quadratic Hermite triangular element formulation. The quadratic Hermite triangular element provides a complete C1 approximation that is very desirable for thin plate modeling. Meanwhile, it corresponds to the constant strain triangle element and is able to reproduce a simple piecewise constant curvature field. Thus a unified numerical implementation for thin plate analysis can be conveniently realized using the triangular elements with discretization flexibility. The curvature smoothing operation is further introduced to improve the accuracy of the quadratic Hermite triangular element. The effectiveness of the proposed methodology is demonstrated through numerical examples.

Keywords

Thin plate Multiscale analysis Homogenization Quadratic Hermite triangular element Curvature smoothing 

Notes

Acknowledgments

The support of this work by the National Natural Science Foundation of China (11222221, 11472233), the Natural Science Foundation of Fujian Province of China (2014J06001), and the Fundamental Research Funds for the Central Universities of China (20720150163) is gratefully acknowledged.

References

  1. Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. North-Holland Publishing Company, Philadelphia (1978)zbMATHGoogle Scholar
  2. Cao, L.Q.: Multiscale asymptotic expansion and finite element methods for the mixed boundary value problems of second order elliptic equation in perforated domains. Numer. Math. 103, 11–45 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Chen, J.S., Wu, C.T., Yoon, S., You, Y.: A stabilized conforming nodal integration for Galerkin meshfree methods. Int. J. Numer. Methods Eng. 50, 435–466 (2001)CrossRefzbMATHGoogle Scholar
  4. Chung, P.W., Tamma, K.K., Namburu, R.R.: Asymptotic expansion homogenization for heterogeneous media: computational issues and applications. Compos. Part A Appl. Sci. Manuf. 32, 1291–1301 (2001)CrossRefGoogle Scholar
  5. Dæhlen, M., Lyche, T., Mørken, K., Schneider, R., Seidel, H.P.: Multiresolution analysis over triangles, based on quadratic Hermite interpolation. J. Comput. Appl. Math. 119, 97–114 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Dierckx, P.: On calculating normalized Powell-Sabin B-splines. Comput. Aided Geom. Des. 15, 61–78 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Fischer P.: C 1 Continuous Methods in Computational Gradient Elasticity. Thesis, Universitat Erlangen-Nürnberg (2011)Google Scholar
  8. Fish, J.: Practical Multiscaling. Wiley, NewYork (2013)Google Scholar
  9. Fish, J., Chen, W.: Space-time multiscale model for wave propagation in heterogeneous media. Comput. Methods Appl. Mech. Eng. 193, 4837–4856 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Ghosh, S., Lee, K., Moorthy, S.: Multiple scale analysis of heterogeneous elastic structures using homogenization theory and voronoi cell finite element method. Int. J. Solids Struct. 32, 27–62 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Guedes, J.S., Kikuchi, N.: Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods. Comput. Methods Appl. Mech. Eng. 83, 143–198 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Han, F., Cui, J.Z., Yu, Y.: The statistical two-order and two-scale method for predicting the mechanics parameters of core-shell particle-filled polymer composites. Interact. Multiscale Mech. 1, 231–250 (2008)CrossRefGoogle Scholar
  13. Hassani, B., Hinton, E.: Homogenization and Structural Topology Optimization. Springer, NewYork (1998)zbMATHGoogle Scholar
  14. Lee, C.Y., Yu, W.: Homogenization and dimensional reduction of composite plates with in-plane heterogeneity. Int. J. Solids Struct. 48, 1474–1484 (2011)CrossRefzbMATHGoogle Scholar
  15. Li, S., Wang, G.: Introduction to Micromechanics and Nanomechanics. World Scientific, Singapore (2008)CrossRefzbMATHGoogle Scholar
  16. Liu, G.R., Dai, K.Y., Nguyen, T.T.: A smoothed finite element method for mechanics problems. Comput. Mech. 39, 859–877 (2007)CrossRefzbMATHGoogle Scholar
  17. Nasution, M.R.E., Watanabe, N., Kondo, A., Yudhanto, A.: Thermomechanical properties and stress analysis of 3-D textile composites by asymptotic expansion homogenization method. Compos. Part B Eng. 60, 378–391 (2014)CrossRefGoogle Scholar
  18. Nemat-Nasser, S., Hori, M.: Micromechanis: Overall Properties of Heterogeneous Materials. Elsevier, Amsterdam (1993)zbMATHGoogle Scholar
  19. Ponte Castaneda, P., Suquet, P.: Nonlinear composites. Adv. Appl. Mech. 34, 171–303 (1998)CrossRefzbMATHGoogle Scholar
  20. Powell, M.J.D., Sabin, M.A.: Piecewise quadratic approximations on triangles. ACM Trans. Math. Softw. 3, 316–325 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Sanchez-Palebncia, E., Zaoui, A.: Homogenization Techniques for Composite Media. Springer, NewYork (1987)CrossRefGoogle Scholar
  22. Temizer, I.: On the asymptotic expansion treatment of two-scale finite thermoelasticity. Int. J. Eng. Sci. 53, 74–84 (2012)MathSciNetCrossRefGoogle Scholar
  23. Timoshenko, S., Woinowsky-Krieger, S.: Theory of Plates and Shells. McGraw-Hill, NewYork (1959)zbMATHGoogle Scholar
  24. Wang, D., Chen, J.S.: Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissner plate formulation. Comput. Methods Appl. Mech. Eng. 193, 1065–1083 (2004)CrossRefzbMATHGoogle Scholar
  25. Wang, D., Chen, J.S.: A Hermite reproducing kernel approximation for thin plate analysis with sub-domain stabilized conforming integration. Int. J. Numer. Methods Eng. 74, 368–390 (2008)CrossRefzbMATHGoogle Scholar
  26. Wang, D., Fang, L.: A multiscale method for analysis of heterogeneous thin slabs with irreducible three dimensional microstructures. Interact. Multiscale Mech. 3, 213–234 (2010)CrossRefGoogle Scholar
  27. Wang, D., Fang, L., Xie, P.: Multiscale asymptotic homogenization of heterogeneous slab and column structures with three dimensional microstructures. In: Li, S., Gao, X. (eds.) Handbook of Micromechanics and Nanomechanics, pp. 1067–1109. Pan Stanford Publishing, Singapore (2013)Google Scholar
  28. Wang, D., Lin, Z.: Dispersion and transient analyses of Hermite reproducing kernel Galerkin meshfree method with sub-domain stabilized conforming integration for thin beam and plate structures. Comput. Mech. 48, 47–63 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. Wang, D., Lin, Z.: Free vibration analysis of thin plates using Hermite reproducing kernel Galerkin meshfree method with sub-domain stabilized conforming integration. Comput. Mech. 46, 703–719 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  30. Wang, D., Peng, H.: A Hermite reproducing kernel Galerkin meshfree approach for buckling analysis of thin plates. Comput. Mech. 51, 1013–1029 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Wang, D., Wu, J.: An efficient nesting sub-domain gradient smoothing integration algorithm with quadratic exactness for Galerkin meshfree methods. Comput. Methods Appl. Mech. Eng. 298, 485–519 (2016)MathSciNetCrossRefGoogle Scholar
  32. Wang, D., Xie, P., Fang, L.: Consistent asymptotic expansion multiscale formulation for heterogeneous column structure. J. Eng. Mater. Technol. ASME 134, 031006 (2012)CrossRefGoogle Scholar
  33. Wu, C.T., Guo, Y., Wang, D.: A pure bending exact nodal-averaged shear strain method for finite element plate analysis. Comput. Mech. 53, 877–892 (2014a)MathSciNetCrossRefzbMATHGoogle Scholar
  34. Wu, C.T., Hu, W., Liu, G.R.: Bubble-enhanced smoothed finite element formulation: a variational multi-scale approach for volume-constrained problems in two-dimensional linear elasticity. Int. J. Numer. Methods Eng. 100, 374–398 (2014b)MathSciNetCrossRefGoogle Scholar
  35. Wu, C.T., Wang, H.P.: An enhanced cell-based smoothed finite element method for the analysis of Reissner-Mindlin plate bending problems involving distorted mesh. Int. J. Numer. Methods Eng. 95, 288–312 (2013)MathSciNetCrossRefGoogle Scholar
  36. Xing, Y.F., Chen, L.: Physical interpretation of multiscale asymptotic expansion method. Compos. Struct. 116, 694–702 (2014)CrossRefGoogle Scholar
  37. Zhao, X., Bordas, S.P.A., Qu, J.: A hybrid smoothed extended finite element/level set method for modeling equilibrium shapes of nano-inhomogeneities. Comput. Mech. 52, 1417–1428 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  38. Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, Oxford (2005)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Boya Dong
    • 1
  • Congying Li
    • 1
  • Dongdong Wang
    • 1
    Email author
  • Cheng-Tang Wu
    • 2
  1. 1.Department of Civil EngineeringXiamen UniversityXiamenChina
  2. 2.Livermore Software Technology CorporationLivermoreUSA

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