The eigenbuckling analysis of hexagonal lattices: closed-form solutions

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences - Tập 477 Số 2251 - 2021
Sondipon Adhikari1
1College of Engineering, Swansea University Bay Campus, Fabian Way, Crymlyn Burrows, Swansea SA1 8EN, UK

Tóm tắt

Elastic instability such as the buckling of cellular materials plays a pivotal role in their analysis and design. Despite extensive research, the quantifi- cation of critical stresses leading to elastic instabi- lities remains challenging due to the inherent nonlinearities. We develop an analytical approach considering the spectral decomposition of the elasticity matrix of two-dimensional hexagonal lattice materials. The necessary and sufficient condition for the buckling is established through the zeros of the eigenvalues of the elasticity matrix. Through the analytical solution of the eigenvalues, the conditions involving equivalent elastic properties of the lattice were directly connected to the mathematical requirement of buckling. The equivalent elastic properties are expressed in closed form using geometric properties of the lattice and trigonometric functions of a non-dimensional axial force parameter. The axial force parameter was identified for four different stress cases, namely, compressive stress in the longitudinal and transverse directions separately and together and torsional stress. By solving the resulting nonlinear equations, we derive exact analytical expressions of critical eigenbuckling stresses for these four cases. Crucial parameter combinations leading to minimum buckling stresses are derived analytically. The exact closed-form analytical expressions derived in the paper can be used for quick engineering design calculations and benchmarking related experimental and numerical studies.

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Tài liệu tham khảo

10.1557/mrc.2015.51

10.1126/science.1252291

10.1038/nature21075

10.1038/nature21044

Gibson L, Ashby MF. 1999 Cellular solids structure and properties. Cambridge: UK: Cambridge University Press.

10.1016/j.taml.2018.04.010

Scarpa F, Adhikari S, Gil AJ, Remillat C. 2010 The bending of single layer graphene sheets: lattice versus continuum approach. Nanotechnology 20, 085405. (doi:10.1088/0957-4484/21/12/125702)

10.1088/0957-4484/22/50/505702

Mukhopadhyay T, Mahata T, Adhikari S, Zaeem MA. 2017 Effective mechanical properties of multilayer nano-heterostructures. Nat. Sci. Rep. 7, 158 181-158 18:13. (doi:10.1038/s41598-017-15664-3)

10.1016/j.ijsolstr.2020.10.009

10.1115/1.1425394

10.1016/0010-4361(79)90021-1

10.1016/0020-7403(92)90013-7

10.1016/j.ijengsci.2017.06.004

10.1016/0020-7403(92)90014-8

10.1016/S0022-5096(97)00060-4

10.1016/0020-7403(88)90060-4

10.1115/1.1646165

10.1016/j.msea.2004.03.051

10.1016/j.ijsolstr.2006.12.017

10.1016/j.msea.2014.04.090

10.1016/S0734-743X(02)00056-8

10.1016/j.ijimpeng.2005.05.007

10.1016/j.compstruct.2004.10.014

10.1016/j.matdes.2009.04.034

10.1016/j.euromechsol.2011.12.003

10.1098/rspa.2013.0856

10.1093/qjmam/43.1.15

10.1115/1.2894040

Rivello RM. 1969 Theory and analysis of flight structures, 1st edn. New York: McGraw-Hill.

10.1016/j.mechmat.2021.103796

10.1017/CBO9780511810817

Abramowitz M, Stegun IA. 1965 Handbook of mathematical functions, with formulas, graphs, and mathematical tables. New York, NY: Dover Publications, Inc.

10.1063/1.4709436

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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Tập 477 Số 2251

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