On complete and micropolar-based incomplete strain gradient theories for periodic lattice structures

doi:10.1007/s10483-023-3033-9

Abstract

Abstract: The micropolar (MP) and strain gradient (SG) continua have been generally adopted to investigate the relations between the macroscopic elastic constants and the microstructural geometric parameters. Owing to the fact that the microrotation in the MP theory can be expressed in terms of the displacement gradient components, we may regard the MP theory as a particular incomplete SG theory called the MPSG theory, compared with the existing SG theories which are deemed complete since all the SGs are included. Taking the triangular lattice comprising zigzag beams as an example, it is found that as the angle of the zigzag beams increases, the bending of the beams plays a more important role in the total strain energy, and the difference between the results by the two theories gradually decreases. Finally, the models are verified with the pure bending and simple shear of lattices by comparing with the results obtained by the finite element method (FEM)-based structure analyses.

Key words: periodic lattice metamaterial, energy principle, homogenization, micropolar (MP), strain gradient (SG) theory

2010 MSC Number:

74A60

Zeyang CHI, Jinxing LIU, A. K. SOH. On complete and micropolar-based incomplete strain gradient theories for periodic lattice structures. Applied Mathematics and Mechanics (English Edition), 2023, 44(10): 1651-1674.

TrendMD

References

[1] LAKES, R. Foam structures with a negative Poisson's ratio. Science, 235, 1038-1041 (1987)
[2] PLATUS, D. L. Negative-stiffness-mechanism vibration isolation systems. Vibration Control in Microelectronics, Optics, and Metrology, 1619, 44-54 (1992)
[3] NORRIS, A. N. and SHUVALOV, A. L. Elastic cloaking theory. Wave Motion, 48, 525-538 (2011)
[4] CUMMER, S. A. and SCHURIG, D. One path to acoustic cloaking. New Journal of Physics, 9, 45 (2007)
[5] NASSAR, H., CHEN, Y. Y., and HUANG, G. L. Polar metamaterials: a new outlook on resonance for cloaking applications. Physical Review Letters, 124, 084301 (2020)
[6] DESHPANDE, V. S., FLECK, N. A., and ASHBY, M. F. Effective properties of the octet-truss lattice material. Journal of the Mechanics and Physics of Solids, 49, 1747-1769 (2001)
[7] MA, Q., CHENG, H., JANG, K. I., LUAN, H., HWANG, K. C., ROGERS, J. A., HUANG, Y. G., and ZHANG, Y. H. A nonlinear mechanics model of bio-inspired hierarchical lattice materials consisting of horseshoe microstructures. Journal of the Mechanics and Physics of Solids, 90, 179-202 (2016)
[8] RUEGER, Z. and LAKES, R. Strong Cosserat elasticity in a transversely isotropic polymer lattice. Physical Review Letters, 120, 065501 (2018)
[9] KHAKALO, S., BALOBANOV, V., and NIIRANEN, J. Modelling size-dependent bending, buckling and vibrations of 2D triangular lattices by strain gradient elasticity models: applications to sandwich beams and auxetics. International Journal of Engineering Science, 127, 33-52 (2018)
[10] YAN, J., HU, W. B., WANG, Z. H., and DUAN, Z. Y. Size effect of lattice material and minimum weight design. Acta Mechanica Sinica, 30, 191-197 (2014)
[11] WANG, X. and STRONGE, W. Micropolar theory for two-dimensional stresses in elastic honeycomb. Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences, 455, 2091-2116 (1999)
[12] NIU, B. and YAN, J. A new micromechanical approach of micropolar continuum modeling for 2-D periodic cellular material. Acta Mechanica Sinica, 32, 456-468 (2016)
[13] CHI, Z., LIU, J., and SOH, A. K. Micropolar modeling of a typical bending-dominant lattice comprising zigzag beams. Mechanics of Materials, 160, 103922 (2021)
[14] BAŽANT, Z. and CHRISTENSEN, M. Analogy between micropolar continuum and grid frameworks under initial stress. International Journal of Solids and Structures, 8, 327-346 (1972)
[15] LAKES, R. S. and BENEDICT, R. L. Noncentrosymmetry in micropolar elasticity. International Journal of Engineering Science, 20, 1161-1167 (1982)
[16] LIU, X., HUANG, G., and HU, G. Chiral effect in plane isotropic micropolar elasticity and its application to chiral lattices. Journal of the Mechanics and Physics of Solids, 60, 1907-1921 (2012)
[17] SPADONI, A. and RUZZENE, M. Elasto-static micropolar behavior of a chiral auxetic lattice. Journal of the Mechanics and Physics of Solids, 60, 156-171 (2012)
[18] FRENZEL, T., KADIC, M., and WEGENER, M. Three-dimensional mechanical metamaterials with a twist. Science, 358, 1072-1074 (2017)
[19] YUAN, X., TOMITA, Y., and ANDOU, T. A micromechanical approach of nonlocal modeling for media with periodic microstructures. Mechanics Research Communications, 35, 126-133 (2008)
[20] WANG, B., LIU, J., SOH, A., and LIANG, N. On band gaps of nonlocal acoustic lattice metamaterials: a robust strain gradient model. Applied Mathematics and Mechanics (English Edition), 43(1), 1-20 (2022) https://doi.org/10.1007/s10483-021-2795-5
[21] POLYZOS, D. and FOTIADIS, D. Derivation of Mindlin's first and second strain gradient elastic theory via simple lattice and continuum models. International Journal of Solids and Structures, 49, 470-480 (2012)
[22] MONCHIET, V., AUFFRAY, N., and YVONNET, J. Strain-gradient homogenization: a bridge between the asymptotic expansion and quadratic boundary condition methods. Mechanics of Materials, 143, 103309 (2020)
[23] DE DOMENICO, D., ASKES, H., and AIFANTIS, E. C. Discussion of "Derivation of Mindlin's first and second strain gradient elastic theory via simple lattice and continuum models" by Polyzos and Fotiadis. International Journal of Solids and Structures, 191, 646-651 (2020)
[24] WANG, B. and LIU, J. Padé-based strain gradient modeling of bandgaps in two-dimensional acoustic lattice metamaterials. International Journal of Applied Mechanics, 15, 2350006 (2023)
[25] MINDLIN, R. D. and ESHEL, N. On first strain-gradient theories in linear elasticity. International Journal of Solids and Structures, 4, 109-124 (1968)
[26] AIFANTIS, E. C. Strain gradient interpretation of size effects. International Journal of Fracture, 95, 299-314 (1999)
[27] CHI, Z., LIU, J., and SOH, A. K. Overlapping-field modeling (OFM) of periodic lattice metamaterials. International Journal of Solids and Structures, 269, 112201 (2023)
[28] ERINGEN, A. C. Microcontinuum Field Theories: I. Foundations and Solids, Springer Science & Business Media, New York (1999)
[29] MINDLIN, R. D. Microstructure in linear elasticity. Archives for Rational Mechanics and Analysis, 16, 51-78 (1963)
[30] PARK, S. K. and GAO, X. L. Variational formulation of a modified couple stress theory and its application to a simple shear problem. Zeitschrift für Angewandte Mathematik und Physik, 59, 904-917 (2007)