Semi-analytical finite element method applied for characterizing micropolar fibrous composites

doi:10.1007/s10483-024-3195-6

Abstract

Abstract:

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.

Key words: semi-analytical approach, fiber-reinforced composite (FRC), effective property, finite element method (FEM), asymptotic homogenization method (AHM), micropolar elasticity

2010 MSC Number:

J. A. OTERO, Y. ESPINOSA-ALMEYDA, R. RODRÍGUEZ-RAMOS, J. MERODIO. Semi-analytical finite element method applied for characterizing micropolar fibrous composites. Applied Mathematics and Mechanics (English Edition), 2024, 45(12): 2147-2164.

TrendMD

Figures/Tables 8

Fig. 1

Table 1

Table 2

Table 3

Table 4

Table 5

Table 6

Table 7

References 48

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) doi: 10.1007/s10483-009-0508-x
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)

Phase constituent	$ \lambda $/MPa	$ \mu $/MPa	$ \alpha $/MPa	$ \beta $/N	$ \gamma $/N	$ \epsilon $/N
SyF$ ^{\rm a} $	2 097	1 033	114.8	$ -2.91 $	4.364	$ -0.133 $
PUF$ ^{\rm b} $	762.7	104	4.333	$ -26.65 $	39.98	4.504
$ ^{\rm a} $ SyF means hollow glass spheres in epoxy resin; $ ^{\rm b} $ PUF means high dense polyurethane foam

$ V_2 $	$ C^{*}_{1111} $	$ C^{*}_{1122} $	$ C^{*}_{1133} $	$ C^{*}_{3333} $	$ C^{*}_{1212} $	$ C^{*}_{2323} $
0.05	1.024 356	0.789 586	0.782 901	1.123 882	0.116 477	0.117 685
0.15	1.128 963	0.845 813	0.828 104	1.415 613	0.132 825	0.139 042
0.25	1.258 426	0.904 922	0.881 102	1.711 725	0.150 367	0.164 907
0.35	1.421 265	0.966 540	0.944 186	2.013 507	0.170 812	0.197 064
0.45	1.629 217	1.031 368	1.020 851	2.322 923	0.196 913	0.238 676
0.55	1.898 475	1.104 638	1.117 119	2.643 357	0.233 955	0.296 000
0.65	2.252 212	1.206 856	1.245 265	2.981 711	0.295 106	0.383 791

$ V_2 $	$ C^{*}_{1111} $	$ C^{*}_{1122} $	$ C^{*}_{1133} $	$ C^{*}_{3333} $	$ C^{*}_{2323} $
0.05	1.023 879	0.790 063	0.782 901	1.123 881	0.117 685
0.15	1.123 966	0.850 806	0.828 102	1.415 612	0.139 039
0.25	1.242 545	0.920 726	0.881 080	1.711 713	0.164 849
0.35	1.386 318	1.000 944	0.944 033	2.013 421	0.196 683
0.45	1.565 603	1.092 268	1.020 088	2.322 494	0.236 974
0.55	1.795 818	1.195 802	1.113 889	2.641 541	0.289 792
0.65	2.099 323	1.315 442	1.232 814	2.974 712	0.362 676

$ V_2 $	$ C^{*}_{2323} $		$ C^{*}_{3223} $		$ C^{*}_{3232} $
$ V_2 $	SAFEM	AHM	SAFEM	AHM	SAFEM	AHM
0.05	0.133 877	0.133 877	0.107 031	0.107 031	0.117 685	0.117 685
0.15	0.186 612	0.186 612	0.123 849	0.123 849	0.139 042	0.139 042
0.25	0.242 141	0.242 141	0.144 216	0.144 216	0.164 907	0.164 907
0.35	0.301 572	0.301 572	0.169 539	0.169 539	0.197 064	0.197 064
0.45	0.366 866	0.366 866	0.202 306	0.202 306	0.238 676	0.238 676
0.55	0.441 903	0.441 903	0.247 446	0.247 446	0.296 000	0.296 000
0.65	0.535 831	0.535 829	0.316 577	0.316 575	0.383 791	0.383 788

$ V_2 $	$ D^{*}_{2323} $		$ D^{*}_{3223} $		$ D^{*}_{3232} $
$ V_2 $	SAFEM	AHM	SAFEM	AHM	SAFEM	AHM
0.05	41.572 723	41.572 712	32.759 404	32.759 392	40.954 152	40.954 137
0.15	36.210 713	36.210 710	27.924 632	27.924 629	34.672 024	34.672 020
0.25	31.353 252	31.353 251	23.745 453	23.745 451	29.241 749	29.241 747
0.35	26.890 207	26.890 206	20.078 763	20.078 761	24.477 383	24.477 381
0.45	22.725 897	22.725 896	16.800 238	16.800 237	20.217 386	20.217 384
0.55	18.770 549	18.770 549	13.793 231	13.793 232	16.310 189	16.310 190
0.65	14.925 504	14.925 561	10.929 548	10.929 623	12.589 222	12.589 319

$ V_2 $	$ C^{*}_{2323} $		$ C^{*}_{3223} $		$ C^{*}_{3232} $
$ V_2 $	SAFEM	AHM	SAFEM	AHM	SAFEM	AHM
0.05	0.133 88	0.133 877	0.107 03	0.107 031	0.117 69	0.117 685
0.15	0.186 61	0.186 609	0.123 85	0.123 846	0.139 04	0.139 039
0.25	0.242 11	0.242 105	0.144 17	0.144 171	0.164 85	0.164 849
0.35	0.301 34	0.301 336	0.169 24	0.169 239	0.196 68	0.196 683
0.45	0.365 81	0.365 811	0.200 97	0.200 966	0.236 97	0.236 974
0.55	0.438 05	0.438 053	0.242 56	0.242 558	0.289 79	0.289 792
0.65	0.522 74	0.522 738	0.299 95	0.299 951	0.362 68	0.362 676

$ V_2 $	$ D^{*}_{2323} $		$ D^{*}_{3223} $		$ D^{*}_{3232} $
$ V_2 $	SAFEM	AHM	SAFEM	AHM	SAFEM	AHM
0.05	41.572 75	41.572 714	32.759 43	32.759 395	40.954 19	40.954 141
0.15	36.211 26	36.211 253	27.925 34	27.925 335	34.672 95	34.672 937
0.25	31.359 27	31.359 261	23.753 27	23.753 261	29.251 90	29.251 894
0.35	26.918 17	26.918 164	20.115 09	20.115 089	24.524 59	24.524 584
0.45	22.811 47	22.811 462	16.911 42	16.911 419	20.361 86	20.361 850
0.55	18.976 38	18.976 374	14.060 68	14.060 673	16.657 70	16.657 694
0.65	15.356 94	15.356 938	11.490 14	11.490 139	13.317 64	13.317 634