Consistent multiscale analysis of heterogeneous thin plates with smoothed quadratic Hermite triangular elements
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Consistent multiscale analysis of heterogeneous thin plates with smoothed quadratic Hermite triangular elements Boya Dong . Congying Li . Dongdong Wang . Cheng-Tang Wu
Received: 6 September 2015 / Accepted: 14 December 2015 Ó Springer Science+Business Media Dordrecht 2015
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 B. Dong C. Li D. Wang (&) Department of Civil Engineering, Xiamen University, Xiamen 361005, Fujian, China e-mail: [email protected] C.-T. Wu Livermore Software Technology Corporation, 7374 Las Positas Road, Livermore, CA 94551, USA
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
1 Introduction The thin plate structures are often enhanced by inplane reinforcements to improve their membrane and bending stiffness. One interesting observation for this type of structures is that the local and global dimension difference, i.e., the local microscopic reinforcements usually with in-plane periodicity are completely three dimensional, while the global thin plates are considered as a two dimensional generalized plane stress state. A three dimensional modeling of these structures is highly computationally demanding, which makes it not feasible for large scale practical problems. Alternatively, various homogenization methods (SanchezPalebncia and Zaoui 1987; Nemat-Nasser and Hori
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B. Dong et al.
1993; Li and Wang 2008) are commonly adopted to analyze these heterogeneous stru
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