Applied Mathematics and Mechanics >
Semi-analytical finite element method applied for characterizing micropolar fibrous composites
Received date: 2024-07-19
Online published: 2024-11-30
Supported by
the National Council of Humanities, Sciences, and Technologies of Mexico(CF-2023-G-792);the National Council of Humanities, Sciences, and Technologies of Mexico(CF-2023-G-1458);the National Council for Scientific and Technological Development of Brazil(09/2023);the Research on Productivity of Brazil(307188/2023-0);Project supported by the National Council of Humanities, Sciences, and Technologies of Mexico (Nos. CF-2023-G-792 and CF-2023-G-1458), the National Council for Scientific and Technological Development of Brazil (No. 09/2023), and the Research on Productivity of Brazil (No. 307188/2023-0)
Copyright
A semi-analytical finite element method (SAFEM), based on the two-scale asymptotic homogenization method (AHM) and the finite element method (FEM), is implemented to obtain the effective properties of two-phase fiber-reinforced composites (FRCs). The fibers are periodically distributed and unidirectionally aligned in a homogeneous matrix. This framework addresses the static linear elastic micropolar problem through partial differential equations, subject to boundary conditions and perfect interface contact conditions. The mathematical formulation of the local problems and the effective coefficients are presented by the AHM. The local problems obtained from the AHM are solved by the FEM, which is denoted as the SAFEM. The numerical results are provided, and the accuracy of the solutions is analyzed, indicating that the formulas and results obtained with the SAFEM may serve as the reference points for validating the outcomes of experimental and numerical computations.
J. A. OTERO, Y. ESPINOSA-ALMEYDA, R. RODRÍGUEZ-RAMOS, J. MERODIO . Semi-analytical finite element method applied for characterizing micropolar fibrous composites[J]. Applied Mathematics and Mechanics, 2024 , 45(12) : 2147 -2164 . DOI: 10.1007/s10483-024-3195-6
| 1 | KHLOPOTIN, A., OLSSON, P., and LARSSON, F. A. Transformational cloaking from seismic surface waves by micropolar metamaterials with finite couple stiffness. Wave Motion, 58, 53- 67 (2015) |
| 2 | REDA, H., RAHALI, Y., GANGHOFFER, J. F., and LAKISS, H. Wave propagation in 3D viscoelastic auxetic and textile materials by homogenized continuum micropolar models. Composite Structures, 141, 328- 345 (2016) |
| 3 | YANG, J. F. C., and LAKES, R. S. Experimental study of micropolar and couple stress elasticity in compact bone in bending. Journal of Biomechanics, 15 (2), 91- 98 (1982) |
| 4 | KAMRIN, K., and HENANN, D. L. Nonlocal modeling of granular flows down inclines. Soft Matter, 11 (1), 179- 185 (2015) |
| 5 | WANG, K., SUN, W., SALAGER, S., NA, S., and KHADDOUR, G. Identifying material parameters for a micro-polar plasticity model via X-ray micro-computed tomographic (CT) images: lessons learned from the curve-fitting exercises. International Journal for Multiscale Computational Engineering, 14 (4), 389- 413 (2016) |
| 6 | ERINGEN, A. C. Microcontinuum Field Theories: $ \rm Ⅱ. $ Fluent Media, Springer, New York (2001) |
| 7 | ERINGEN, A. C. Microcontinuum Field Theories: $ \rm I. $ Foundations and Solids, Springer, New York (2012) |
| 8 | MEKHEIMER, K. S., and EL-KOT, M. A. The micropolar fluid model for blood flow through a tapered artery with a stenosis. Acta Mechanica Sinica, 24 (6), 637- 644 (2008) |
| 9 | FRENZEL, T., KADIC, M., and WEGENER, M. Three-dimensional mechanical metamaterials with a twist. Science, 358, 1072- 1074 (1999) |
| 10 | DIEBELS, S. A micropolar theory of porous media: constitutive modelling. Transport in Porous Media, 34 (1), 193- 208 (1999) |
| 11 | HUANG, L., and ZHAO, C. G. A micropolar mixture theory of multi-component porous media. Applied Mathematics and Mechanics (English Edition), 30 (5), 617- 630 (2009) |
| 12 | DE BORST, R., and SLUYS, L. J. Localisation in a Cosserat continuum under static and dynamic loading conditions. Computer Methods in Applied Mechanics and Engineering, 90, 805- 827 (1991) |
| 13 | BAUER, S., SCHÄFER, M., GRAMMENOUDIS, P., and TSAKMAKIS, C. Three-dimensional finite elements for large deformation micropolar elasticity. Computer Methods in Applied Mechanics and Engineering, 199 (41-44), 2643- 2654 (2010) |
| 14 | ERDELJ, S. G., JELENIĆ, G., and IBRAHIMBEGOVIĆ, A. Geometrically non-linear 3D finite-element analysis of micropolar continuum. International Journal of Solids and Structures, 202, 745- 764 (2020) |
| 15 | BELYTSCHKO, T., YUN, Y. L., and GU, L. Element-free Galerkin methods. International Journal for Numerical Methods in Engineering, 37 (2), 229- 256 (1994) |
| 16 | LIU, W. K., JUN, S., and ZHANG, Y. F. Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids, 20 (8-9), 1081- 1106 (1995) |
| 17 | SULSKY, D., CHEN, Z., and SCHREYER, H. L. A particle method for history-dependent materials. Computer Methods in Applied Mechanics and Engineering, 118 (1-2), 179- 196 (1994) |
| 18 | FOREST, S., DENDIEVE, R., and CANOVA, G. R. Estimating the overall properties of heterogeneous Cosserat materials. Modelling and Simulation in Materials Science and Engineering, 7 (5), 829- 840 (1999) |
| 19 | FOREST, S., BARBE, F., and CAILLETAUD, G. Cosserat modeling of size effects in the mechanical behavior of polycrystals and multi-phase materials. International Journal of Solids and Structures, 37, 7105- 7126 (2000) |
| 20 | CHENG, Z. Q., and HE, L. H. Micropolar elastic fields due to a spherical inclusion. International Journal of Engineering Science, 33 (3), 389- 397 (1995) |
| 21 | CHENG, Z. Q., and HE, L. H. Micropolar elastic fields due to a circular cylindrical inclusion. International Journal of Engineering Science, 35 (7), 659- 668 (1997) |
| 22 | SHARMA, P., and DASGUPTA, A. Average elastic field and scale-dependent overall properties of heterogeneous micropolar materials containing spherical and cylindrical inhomogeneities. Physical Review B, 66, 224110 (2002) |
| 23 | LIU, X. N., and HU, G. K. A continue micromechanical theory of overall plasticity for particulate composites including particle size effect. International Journal of Plasticity, 21 (4), 777- 799 (2005) |
| 24 | KOTIN, Y. V., POLILOV, A. N., and VLASOV, D. D. "Van Fo Fy method'' in the micromechanics of fibrous composites. Mechanics of Composite Materials, 59, 847- 868 (2023) |
| 25 | FRANCIS, N. M., POURAHMADIAN, F., LEBENSOHN, R. A., and DINGREVILLE, R. A fast Fourier transform-based solver for elastic micropolar composites. Computer Methods in Applied Mechanics and Engineering, 418, 116510 (2024) |
| 26 | OSTOJA-STARZEWSKI, M., and JASIUK, I. Stress invariance in planar Cosserat elasticity. Proceedings of the Royal Society of London: Series A, 451, 453- 470 (1995) |
| 27 | YUAN, X., and TOMITA, Y. Effective properties of Cosserat composites with periodic microstructure. Mechanics Research Communications, 28 (3), 265- 270 (2001) |
| 28 | SACK, K. L., SKATULLA, S., and SANSOUR, C. Biological tissue mechanics with fibres modelled as one-dimensional Cosserat continua. applications to cardiac tissue. International Journal of Solids and Structures, 81, 84- 94 (2016) |
| 29 | EREMEYEV, V. A., SKRZAT, A., and STACHOWICZ, F. Linear micropolar elasticity analysis of stresses in bones under static loads. Strength of Materials, 49, 575- 585 (2017) |
| 30 | RODRÍGUEZ-RAMOS, R., ESPINOSA-ALMEYDA, Y., GUINOVART-SANJUÁN, D., CAMACHO-MONTES, H., RODRÍGUEZ-BERMÚDEZ, P., BRITO-SANTANA, H., OTERO, J. A., and SABINA, F. J. Analysis of micropolar elastic multi-laminated composite and its application to bioceramic materials for bone reconstruction. Interface Focus, 14, 20230064 (2024) |
| 31 | RODRÍGUEZ-RAMOS, R., YANES, V., ESPINOSA-ALMEYDA, Y., OTERO, J. A., SABINA, F. J., SÁNCHEZ-VALDÉS, C. F., and LEBON, F. Micro-macro asymptotic approach applied to heterogeneous elastic micropolar media: analysis of some examples. International Journal of Solids and Structures, 239-240, 111444 (2022) |
| 32 | YANES, V., SABINA, F. J., ESPINOSA-ALMEYDA, Y., OTERO, J. A., and RODRÍGUEZ-RAMOS, R. Asymptotic homogenization approach applied to Cosserat heterogeneous media. Mechanics and Physics of Structured Media, Academic Press, New York (2022) |
| 33 | SANCHEZ-PALENCIA, E. Non-Homogeneous Media and Vibration Theory, Springer-Verlag, Moscow (1980) |
| 34 | SANCHEZ-PALENCIA, E. Homogenization Techniques for Composite Media, Springer-Verlag, Moscow (1985) |
| 35 | BAKHVALOV, N. and PANASENKO, G. Homogenisation: Averaging Processes in Periodic Media, Kluwer Academic Publisher, the Netherlands (1989) |
| 36 | GORBACHEV, V. I., and EMEL'YANOV, A. N. Homogenization of the equations of the Cosserat theory of elasticity of inhomogeneous bodies. Mechanics of Solids, 49 (1), 73- 82 (2014) |
| 37 | NOWACKI, W. The linear theory of micropolar eelasticity. Micropolar Elasticity. International Centre for Mechanical Sciences, Springer, Vienna (1974) |
| 38 | NOWACKI, W. Theory of Asymmetric Elasticity, Pergamon-Press, Moscow (1986) |
| 39 | ERINGEN, A. C. Microcontinuum Field Theories $ \rm I $: Foundations and Solids, Springer-Verlag, New York (1999) |
| 40 | PARTON, V. Z., and KUDRYAVTSEV, B. A. Engineering Mechanics of Composite Structures, CRC Press, Moscow (1993) |
| 41 | AVELLANEDA, R., RODRIGUEZ-ALEMAN, S., and OTERO, J. A. Semi-analytical method for computing effective thermoelastic properties in fiber-reinforced composite materials. Applied Science, 11, 5354 (2021) |
| 42 | ZIENKIEWICZ, O. C., TAYLOR, R. L., and ZHU, J. Z. The Finite Element Method: Its Basis and Fundamentals, Butterworth-Heinemann, Oxford (2013) |
| 43 | EREMEYEV, V., and PIETRASZKIEWICZ, W. Material symmetry group of the non-linear polar-elastic continuum. International Journal of Solids and Structures, 49 (14), 1993- 2005 (2012) |
| 44 | KWON, Y., and BANG, H. The Finite Element Method Using MATLAB, CRC Press, Washington D. C. (2000) |
| 45 | CHANDRUPATLA, T., and BELEGUNDU, A. Introduction to Finite Elements in Engineering, Pearson Education, Harlow (2012) |
| 46 | HASSANPOUR, S., and HEPPLER, G. R. Micropolar elasticity theory: a survey of linear isotropic equations, representative notations, and experimental investigations. Mathematics and Mechanics of Solids, 22, 224- 242 (2015) |
| 47 | ROYER, D., and DIEULESAINT, E. Elastic Waves in Solids $ \rm I $: Free and Guided Propagation, Springer Science & Business Media, Berlin (1993) |
| 48 | RODRÍGUEZ-RAMOS, R., SABINA, F. J., GUINOVART-DÍAZ, R., and BRAVO-CASTILLERO, J. Closed-form expressions for the effective coefficients of a fiber-reinforced composite with transversely isotropic constituents-I: elastic and square symmetry. Mechanics of Materials, 33 (4), 223- 235 (2001) |
/
| 〈 |
|
〉 |
Email Alert
RSS