Skip to main content
SpringerLink
Log in

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

  • Published:
International Journal of Mechanics and Materials in Design Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29

Similar content being viewed by others

References

  • Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. North-Holland Publishing Company, Philadelphia (1978)

    MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MATH  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • Dierckx, P.: On calculating normalized Powell-Sabin B-splines. Comput. Aided Geom. Des. 15, 61–78 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Fischer P.: C 1 Continuous Methods in Computational Gradient Elasticity. Thesis, Universitat Erlangen-Nürnberg (2011)

  • Fish, J.: Practical Multiscaling. Wiley, NewYork (2013)

    Google Scholar 

  • Fish, J., Chen, W.: Space-time multiscale model for wave propagation in heterogeneous media. Comput. Methods Appl. Mech. Eng. 193, 4837–4856 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Hassani, B., Hinton, E.: Homogenization and Structural Topology Optimization. Springer, NewYork (1998)

    MATH  Google Scholar 

  • Lee, C.Y., Yu, W.: Homogenization and dimensional reduction of composite plates with in-plane heterogeneity. Int. J. Solids Struct. 48, 1474–1484 (2011)

    Article  MATH  Google Scholar 

  • Li, S., Wang, G.: Introduction to Micromechanics and Nanomechanics. World Scientific, Singapore (2008)

    Book  MATH  Google Scholar 

  • Liu, G.R., Dai, K.Y., Nguyen, T.T.: A smoothed finite element method for mechanics problems. Comput. Mech. 39, 859–877 (2007)

    Article  MATH  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Nemat-Nasser, S., Hori, M.: Micromechanis: Overall Properties of Heterogeneous Materials. Elsevier, Amsterdam (1993)

    MATH  Google Scholar 

  • Ponte Castaneda, P., Suquet, P.: Nonlinear composites. Adv. Appl. Mech. 34, 171–303 (1998)

    Article  MATH  Google Scholar 

  • Powell, M.J.D., Sabin, M.A.: Piecewise quadratic approximations on triangles. ACM Trans. Math. Softw. 3, 316–325 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  • Sanchez-Palebncia, E., Zaoui, A.: Homogenization Techniques for Composite Media. Springer, NewYork (1987)

    Book  Google Scholar 

  • Temizer, I.: On the asymptotic expansion treatment of two-scale finite thermoelasticity. Int. J. Eng. Sci. 53, 74–84 (2012)

    Article  MathSciNet  Google Scholar 

  • Timoshenko, S., Woinowsky-Krieger, S.: Theory of Plates and Shells. McGraw-Hill, NewYork (1959)

    MATH  Google Scholar 

  • 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)

    Article  MATH  Google Scholar 

  • 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)

    Article  MATH  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, D., Peng, H.: A Hermite reproducing kernel Galerkin meshfree approach for buckling analysis of thin plates. Comput. Mech. 51, 1013–1029 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • Wang, D., Xie, P., Fang, L.: Consistent asymptotic expansion multiscale formulation for heterogeneous column structure. J. Eng. Mater. Technol. ASME 134, 031006 (2012)

    Article  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • Xing, Y.F., Chen, L.: Physical interpretation of multiscale asymptotic expansion method. Compos. Struct. 116, 694–702 (2014)

    Article  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, Oxford (2005)

    MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Xiamen University, Xiamen, 361005, Fujian, China

    Boya Dong, Congying Li & Dongdong Wang

  2. Livermore Software Technology Corporation, 7374 Las Positas Road, Livermore, CA, 94551, USA

    Cheng-Tang Wu

Authors
  1. Boya Dong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Congying Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dongdong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Cheng-Tang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dongdong Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dong, B., Li, C., Wang, D. et al. Consistent multiscale analysis of heterogeneous thin plates with smoothed quadratic Hermite triangular elements. Int J Mech Mater Des 12, 539–562 (2016). https://doi.org/10.1007/s10999-015-9334-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10999-015-9334-x

Keywords

Search

Navigation