×

High-order three-scale computational method for elastic behavior analysis and strength prediction of axisymmetric composite structures with multiple spatial scales. (English) Zbl 07357434

Summary: A novel high-order three-scale (HOTS) computational method for elastic behavior analysis and strength prediction of axisymmetric composite structures with multiple spatial scales is developed in this paper. The multiple heterogeneities of axisymmetric composite structures we investigated are taken into account by periodic distributions of representative unit cells on the mesoscale and microscale. First, the new micro-meso-macro coupled HOTS computational model for elastic problems of axisymmetric composite structures is established based on multiscale asymptotic analysis, which breaks through the traditional multiscale assumptions and includes three main components. Two classes of mesoscopic and microscopic auxiliary cell functions are constructed on the mesoscale and microscale, respectively. The macroscopic homogenization problems are defined on global axisymmetric structures by twice up-scaling procedures from microscale to mesoscale and then from mesoscale to macroscale. Moreover, the asymptotic HOTS solutions are constructed for approximating multiscale elastic problems of axisymmetric structures and the numerical accuracy analysis of the HOTS solutions is given in detail. Then, the strength prediction formulas for axisymmetric composite structures with multiple spatial scales are presented based on the high-accuracy elastic behavior analysis from the proposed HOTS computational model. Furthermore, the corresponding HOTS numerical algorithm based on the finite element method (FEM) is presented for analyzing the mechanical behaviors and predicting the strength of axisymmetric composite structures with multiple spatial scales in detail. Finally, some numerical examples are reported to verify the feasibility and effectiveness of the proposed HOTS computational method. In this study, a unified three-scale computational framework is offered, which enables the simulation of mechanical behaviors of axisymmetric composite structures with multiple spatial scales.

MSC:

74-XX Mechanics of deformable solids

Software:

FreeFem++
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Shen, X, Luisa, Â, Paolo, P. Numerical simulations for the dynamics of flexural shells. Math Mech Solids 2020; 25(4): 887-912. · Zbl 1446.74168
[2] Shen, X, Yang, Q, Li, L, et al. Numerical approximation of the dynamic Koiter’s model for the hyperbolic parabolic shell. Appl Numer Math 2020; 150: 194-205. · Zbl 1437.74027
[3] Nasution, MRE, Watanabe, N, Kondo, A, et al. Thermo-mechanical properties and stress analysis of 3-d textile composites by asymptotic expansion homogenization method. Composites B Eng 2014; 60(1): 378-391.
[4] Savatorova, VL, Talonov, AV, Vlasov, AN. Homogenization of thermoelasticity processes in composite materials with periodic structure of heterogeneities. Z Angew Math Mech 2013; 93(8): 575-596. · Zbl 1274.74095
[5] Yang, ZH, Cui, JZ, Nie, YF, et al. Microstructural modeling and second-order two-scale computation for mechanical properties of 3D 4-directional braided composites. Comput Mater Continua 2013; 38(175): 1-20.
[6] Cioranescu, D, Donato, P. An Introduction to Homogenization. Oxford: Oxford University Press, 1999. · Zbl 0939.35001
[7] Sanchez-Palencia, E. Non-Homogeneous Media and Vibration Theory. Berlin: Springer, 1980. · Zbl 0432.70002
[8] Wu, YT, Nie, YF, Yang, ZH. Comparison of four multiscale methods for elliptic problems. Comput Model Eng Sci 2014; 99(4): 297-325. · Zbl 1357.65270
[9] Dong, H, Nie, Y, Yang, Z, et al. The numerical accuracy analysis of asymptotic homogenization method and multiscale finite element method for periodic composite materials. Comput Model Eng Sci 2016; 111(5): 395-419.
[10] Bensousson, A, Lions, J, Papanicolaou, G. Asymptotic Analysis for Periodic Structure. Amsterdam: North-Holland, 1978.
[11] Hughes, TJR, Feijoo, GR, Mazzei, L, et al. The variational multiscale method - a paradigm for computational mechanics. Comput Meth Appl Mech Eng 1998; 166(1): 3-24. · Zbl 1017.65525
[12] Weinan, E, Ming, P, Zhang, P. Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J Amer Math Soc 2005; 18(1): 121-156. · Zbl 1060.65118
[13] Hou, TY, Wu, X, Cai, Z. Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math Computat 1999; 68(227): 913-943. · Zbl 0922.65071
[14] Yang, ZH, Cui, JZ. The statistical second-order two-scale analysis for dynamic thermo-mechanical performances of the composite structure with consistent random distribution of particles. Computat Mater Sci 2013; 69: 359-373.
[15] Yang, ZQ, Cui, JZ, Sun, Y, et al. Multiscale analysis method for thermo-mechanical performance of periodic porous materials with interior surface radiation. Int J Numer Meth Eng 2016; 105(5): 323-350.
[16] Li, Z, Ma, Q, Cui, J. Second-order two-scale finite element algorithm for dynamic thermo-mechanical coupling problem in symmetric structure. J Computat Phys 2016; 314: 712-748. · Zbl 1349.65459
[17] Dong, H, Cui, J, Nie, Y, et al. Second-order two-scale computational method for nonlinear dynamic thermo-mechanical problems of composites with cylindrical periodicity. Commun Computat Phys 2016; 49(5): 1027-1057.
[18] Feng, Y, Cui, J. Multi-scale analysis and FE computation for the structure of composite materials with small periodic configuration under condition of coupled thermoelasticity. Int J Numer Meth Eng 2004; 60(11): 1879-1910. · Zbl 1060.74626
[19] Wang, X, Cao, L, Wong, Y. Multiscale computation and convergence for coupled thermoelastic system in composite materials. SIAM Multiscale Model Sim 2015; 13: 661-690. · Zbl 1317.65094
[20] Zhang, Y, Cui, JZ, Nie, YF, et al. In: High-order Triple-scale Method for Composite Structures of the Configurations with Small Periodicities of Two-levels. Seoul, Korea, 2016.
[21] Takano, N, Yoshihiro, O. Three-scale finite element analysis of heterogeneous media by asymptotic homogenization and mesh superposition methods. Int J Solids Struct 2004; 41(15): 4121-4135. · Zbl 1079.74653
[22] Mahnken, R, Dammann, C. A three-scale framework for fibre-reinforced-polymer curing part I: Microscopic modeling and mesoscopic effective properties. Int J Solids Struct 2016; 100-101: 341-355.
[23] Dimitrienko, Y, Dimitrienko, I, Sborschikov, S. Multiscale hierarchical modeling of fiber reinforced composites by asymptotic homogenization method. Appl Math Sci 2015; 9(145): 7211-7220.
[24] Cao, L. Iterated two-scale asymptotic method and numerical algorithm for the elastic structures of composite materials. Comput Meth Appl Mech Eng 2005; 194: 2899-2926. · Zbl 1091.74041
[25] Liu, S, Liu, X, Guan, XF, et al. A stochastic multi-scale model for predicting the thermal expansion coefficient of early-age concrete. Comput Model Eng Sci 2013; 92: 173-191.
[26] Guan, X, Liu, X, Jia, X, et al. A stochastic multiscale model for predicting mechanical properties of fiber reinforced concrete. Int J Solids Struct 2015; 56: 280-289.
[27] Rodríguez, EI, Cruz, ME, Bravo-Castillero, J. Reiterated homogenization applied to heat conduction in heterogeneous media with multiple spatial scales and perfect thermal contact between the phases. J Brazil Soc Mech Sci Eng 2016; 38: 1333-1343.
[28] Ramireztorres, A, Penta, R, Rodriguezramos, R, et al. Three scales asymptotic homogenization and its application to layered hierarchical hard tissues. Int J Solids Struct 2018; 130-131: 190-198.
[29] Ramireztorres, A, Penta, R, Rodriguezramos, R, et al. Homogenized out-of-plane shear response of three-scale fiber-reinforced composites. Comput Visualiz Sci 2019; 20: 85-93.
[30] Ramireztorres, A, Penta, R, Rodriguezramos, R, et al. Effective properties of hierarchical fiber-reinforced composites via a three-scale asymptotic homogenization approach. Math Mech Solids 2019; 24(11): 3554-3574. · Zbl 07273382
[31] Allaire, G, Briane, M. Multiscale convergence and reiterated homogenisation. Proc R Soc Edinburgh Sec A Math 1996; 126(2): 297-342. · Zbl 0866.35017
[32] Telega, JJ, Gałka, A, Tokarzewski, S. Application of the reiterated homogenization to determination of effective noduli of a compact bone. J Theoret Appl Mech 1999; 37(3): 687-706. · Zbl 0963.74037
[33] Almqvist, A, Essel, EK, Fabricius, J, et al. Reiterated homogenization applied in hydrodynamic lubrication. Proc IMechE Part J: J Eng Tribol 2008; 222(7): 827-841.
[34] Trucu, D, Chaplain, M, Marciniak-Czochra, A. Three-scale convergence for processes in heterogeneous media. Applicable Anal 2012; 91(7): 1351-1373. · Zbl 1252.35038
[35] Yang, Z, Zhang, Y, Dong, H, et al. High-order three-scale method for mechanical behavior analysis of composite structures with multiple periodic configurations. Composites Sci Technol 2017; 152: 198-210.
[36] Yang, Z, Sun, Y, Guan, T, et al. A high-order three-scale approach for predicting thermo-mechanical properties of porous materials with interior surface radiation. Comput Math Applicat 2019; in press. · Zbl 1433.74042
[37] Chatzigeorgiou, G, Charalambakis, N, Murat, F. Homogenization problems of a hollow cylinder made of elastic materials with discontinuous properties. Int J Solids Struct 2008; 45: 5165-5180. · Zbl 1169.74532
[38] Chatzigeorgiou, G, Efendiev, Y, Charalambakis, N, et al. Effective thermoelastic properties of composites with periodicity in cylindrical coordinates. Int J Solids Struct 2012; 49(5): 2590-2603.
[39] Ma, Q, Cui, JZ, Li, ZH. Second-order two-scale asymptotic analysis for axisymmetric and spherical symmetric structure with periodic configurations. Int J Solids Struct 2016; 78-79(5): 77-100.
[40] Han, F. The Second-Order Two-Scale Method for Predicting Mechanical Performance of Random Composite Materials. PhD Thesis, Northwestern Polytechnical University, 2010.
[41] Dong, H, Cui, JZ, Nie, YF, et al. High-order three-scale computational method for heat conduction problems of axisymmetric composite structures with multiple spatial scales. Adv Eng Softw 2018; 121: 1-12.
[42] Huang, Y, Huang, HR, He, FS. Linear Theory of Elastic Shell. Science Press, 2007.
[43] Cui, J . Multiscale computational method for unified design of structure, components and their materials. In: Proceedings on Computational Mechanics in Science and Engineering (CCCM-2001), Guangzhou, 5-8 December. Beijing: Peking University Press, pp. 33-43.
[44] Dong, H, Cui, J, Nie, Y, et al. Multiscale computational method for thermoelastic problems of composite materials with orthogonal periodic configurations. Appl Math Modell 2018; 60: 634-660. · Zbl 1480.74052
[45] Dong, Q, Cao, L. Multiscale asymptotic expansions methods and numerical algorithms for the wave equations of second order with rapidly oscillating coefficients. Appl Numer Math 2009; 59(12): 3008-3032. · Zbl 1177.65147
[46] Lin, Q, Zhu, Q. The Preprocessing snd Preprocessing for the Finite Element Method. Shanghai: Shanghai Scientific and Technical Publishers, 1994.
[47] Sauer, T. Numerical Analysis (2nd edn). China Machine Press, 2020.
[48] Hecht, F. FreeFem++ (Third Edition), Version 3.60. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 2015.
[49] Kim, C, Jung, M, Yamada, T, et al. Freefem++ code for reaction-diffusion equation-based topology optimization: For high-resolution boundary representation using adaptive mesh refinement. Structural Multidisciplinary Optimiz 2020; 62: 439-455.
[50] Wu, Y. Multiscale Algorithms Research for Performance Analysis of Random Heterogeneous Materials. PhD Thesis, Northwestern Polytechnical University, 2015.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.