Research Papers

A Floquet-Based Bar-Spring Model for the Dynamic Modulus of Bioinspired Composites With Arbitrary Staggered Architectures

[+] Author and Article Information
Wen Xie

Department of Engineering Mechanics,
School of Civil Engineering,
Wuhan University,
Wuhan 430072, China
e-mail: benxwen@whu.edu.cn

Yanan Yuan

Department of Engineering Mechanics,
School of Civil Engineering,
Wuhan University,
Wuhan 430072, China
e-mail: yuanyn@whu.edu.cn

Zuoqi Zhang

Department of Engineering Mechanics,
School of Civil Engineering,
Wuhan University,
Wuhan 430072, China;
Suzhou Institute of Wuhan University,
Suzhou 215123, China
e-mail: zhang_zuoqi@whu.edu.cn

1Corresponding authors.

Contributed by the Applied Mechanics Division of ASME for publication in the Journal of Applied Mechanics. Manuscript received April 24, 2019; final manuscript received May 24, 2019; published online June 27, 2019. Assoc. Editor: Caglar Oskay.

J. Appl. Mech 86(9), 091007 (Jun 27, 2019) (8 pages) Paper No: JAM-19-1200; doi: 10.1115/1.4043888 History: Received April 24, 2019; Accepted May 29, 2019

Staggered architectures widely seen in load-bearing biological materials provide not only excellent supporting functions resisting static loading but also brilliant protecting functions attenuating the dynamic impact. However, there are very few efforts to unveil the relationship between staggered architectures and damping properties within load-bearing biological and bioinspired materials, while its static counterpart has been intensively studied over the past decades. Here, based on the Floquet theory, we developed a new generic method to evaluate the dynamic modulus of the composites with various staggered architectures. Comparisons with the finite element method results showed that the new method can give more accurate predictions than previous methods based on the tension-shear chain model. Moreover, the new method is more generic and applicable for two- and three-dimensional arbitrarily staggered architectures. This method provides a useful tool to understand the relationship between micro-architecture and damping property in natural load-bearing biological materials and to facilitate the architectural design of high-damping bioinspired composites.

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Ashby, M. F., Gibson, L. J., Wegst, U., and Olive, R., 1995, “The Mechanical-Properties of Natural Materials. 1. Material Property Charts,” Proc. R. Soc. Math. Phys. Eng. Sci., 450(1938), pp. 123–140. [CrossRef]
Gao, H. J., Ji, B. H., Jager, I. L., Arzt, E., and Fratzl, P., 2003, “Materials Become Insensitive to Flaws at Nanoscale: Lessons From Nature,” Proc.Natl. Acad. Sci. U.S.A., 100(10), pp. 5597–5600. [CrossRef] [PubMed]
Ji, B. H., and Gao, H. J., 2004, “Mechanical Properties of Nanostructure of Biological Materials,” J. Mech. Phys. Solids, 52(9), pp. 1963–1990. [CrossRef]
Zhang, Z. Q., Liu, B., Huang, Y., Hwang, K. C., and Gao, H., 2010, “Mechanical Properties of Unidirectional Nanocomposites With Non-Uniformly or Randomly Staggered Platelet Distribution,” J. Mech. Phys. Solids, 58(10), pp. 1646–1660. [CrossRef]
Zhang, Z. Q., Zhang, Y. W., and Gao, H. J., 2011, “On Optimal Hierarchy of Load-Bearing Biological Materials,” Proc. R. Soc. Lond. B Biol. Sci., 278(1705), pp. 519–525. [CrossRef]
Ritchie, R. O., 2011, “The Conflicts Between Strength and Toughness,” Nat. Mater., 10(11), pp. 817–822. [CrossRef] [PubMed]
Jackson, A. P., Vincent, J. F. V., and Turner, R. M., 1988, “The Mechanical Design of Nacre,” Proc. R. Soc. Lond. B Biol. Sci., 234(1277), pp. 415–440. [CrossRef]
Kamat, S., Su, X., Ballarini, R., and Heuer, A. H., 2000, “Structural Basis for the Fracture Toughness of the Shell of the Conch Strombus Gigas,” Nature, 405(6790), pp. 1036–1040. [CrossRef] [PubMed]
Menig, R., Meyers, M. H., Meyers, M. A., and Vecchio, K. S., 2000, “Quasi-Static and Dynamic Mechanical Response of Haliotis Rufescens (Abalone) Shells,” Acta Mater., 48(9), pp. 2383–2398. [CrossRef]
Khayer Dastjerdi, A., Rabiei, R., and Barthelat, F., 2013, “The Weak Interfaces Within Tough Natural Composites: Experiments on Three Types of Nacre,” J. Mech. Behav. Biomed. Mater., 19, pp. 50–60. [CrossRef] [PubMed]
Meyers, M. A., Chen, P. Y., Lin, A. Y. M., and Seki, Y., 2008, “Biological Materials: Structure and Mechanical Properties,” Prog. Mater. Sci., 53(1), pp. 1–206. [CrossRef]
Currey, J. D., 1977, “Mechanical Properties of Mother of Pearl in Tension,” Proc. R. Soc. Lond. B Biol. Sci., 196(1125), pp. 443–463. [CrossRef]
Landis, W., 1995, “The Strength of a Calcified Tissue Depends in Part on the Molecular Structure and Organization of Its Constituent Mineral Crystals in Their Organic Matrix,” Bone, 16(5), pp. 533–544. [CrossRef] [PubMed]
Landis, W. J., Hodgens, K. J., Song, M. J., Arena, J., Kiyonaga, S., Marko, M., Owen, C., and McEwen, B. F., 1996, “Mineralization of Collagen May Occur on Fibril Surfaces: Evidence From Conventional and High-Voltage Electron Microscopy and Three-Dimensional Imaging,” J. Struct. Biol., 117(1), pp. 24–35. [CrossRef] [PubMed]
Roschger, P., Grabner, B., Rinnerthaler, S., Tesch, W., Kneissel, M., Berzlanovich, A., Klaushofer, K., and Fratzl, P., 2001, “Structural Development of the Mineralized Tissue in the Human L4 Vertebral Body,” J. Struct. Biol., 136(2), pp. 126–136. [CrossRef] [PubMed]
Tesch, W., Eidelman, N., Roschger, P., Goldenberg, F., Klaushofer, K., and Fratzl, P., 2001, “Graded Microstructure and Mechanical Properties of Human Crown Dentin,” Calcif. Tissue Int., 69(3), pp. 147–157. [CrossRef] [PubMed]
Warshawsky, H., 1989, “Organization of Crystals in Enamel,” Anat. Rec., 224(2), pp. 242–262. [CrossRef] [PubMed]
Lei, H. F., Zhang, Z. Q., and Liu, B., 2012, “Effect of Fiber Arrangement on Mechanical Properties of Short Fiber Reinforced Composites,” Compos. Sci. Technol., 72(4), pp. 506–514. [CrossRef]
Lei, H. J., Zhang, Z. Q., Han, F., Liu, B., Zhang, Y. W., and Gao, H. J., 2013, “Elastic Bounds of Bioinspired Nanocomposites,” ASME J. Appl. Mech., 80(6), p. 061017. [CrossRef]
Chen, B., Wu, P. D., and Gao, H., 2009, “A Characteristic Length for Stress Transfer in the Nanostructure of Biological Composites,” Compos. Sci. Technol., 69(7–8), pp. 1160–1164. [CrossRef]
Zhang, P., and To, A. C., 2014, “Highly Enhanced Damping Figure of Merit in Biomimetic Hierarchical Staggered Composites,” ASME J. Appl. Mech., 81(5), p. 051015. [CrossRef]
Zuo, S. C., and Wei, Y. G., 2007, “Effective Elastic Modulus of Bone-Like Hierarchical Materials,” Acta Mech. Solida Sin., 20(3), pp. 198–205. [CrossRef]
Kotha, S. P., Li, Y., and Guzelsu, N., 2001, “Micromechanical Model of Nacre Tested in Tension,” J. Mater. Sci., 36(8), pp. 2001–2007. [CrossRef]
Tang, Z., Kotov, N. A., Magonov, S., and Ozturk, B., 2003, “Nanostructured Artificial Nacre,” Nat. Mater., 2(6), pp. 413–418. [CrossRef] [PubMed]
Munch, E., Launey, M. E., Alsem, D. H., Saiz, E., Tomsia, A. P., and Ritchie, R. O., 2008, “Tough, Bio-Inspired Hybrid Materials,” Science, 322(5907), pp. 1516–1520. [CrossRef] [PubMed]
Dimas, L. S., Bratzel, G. H., Eylon, I., and Buehler, M. J., 2013, “Tough Composites Inspired by Mineralized Natural Materials: Computation, 3D Printing, and Testing,” Adv. Funct. Mater., 23(36), pp. 4629–4638. [CrossRef]
Espinosa, H. D., Juster, A. L., Latourte, F. J., Loh, O. Y., Gregoire, D., and Zavattieri, P. D., 2011, “Tablet-Level Origin of Toughening in Abalone Shells and Translation to Synthetic Composite Materials,” Nat. Commun., 2, p. 173. [CrossRef] [PubMed]
Lakes, R., 2002, “High Damping Composite Materials: Effect of Structural Hierarchy,” J. Compos. Mater., 36(3), pp. 287–297. [CrossRef]
Taber, K. H., Warden, D. L., and Hurley, R. A., 2006, “Blast-Related Traumatic Brain Injury: What Is Known?,” J. Neuropsychiatry Clin. Neurosci., 18(2), pp. 141–145. [CrossRef] [PubMed]
Viano, D. C., Pellman, E. J., Withnall, C., and Shewchenko, N., 2006, “Concussion in Professional Football: Performance of Newer Helmets in Reconstructed Game Impacts—Part 13,” Neurosurgery, 59(3), pp. 591–606. [CrossRef] [PubMed]
Zhang, L., Yang, K. H., and King, A. I., 2004, “A Proposed Injury Threshold for Mild Traumatic Brain Injury,” ASME J. Biomech. Eng., 126(2), pp. 226–236. [CrossRef]
Puxkandl, R., Zizak, I., Paris, O., Keckes, J., Tesch, W., Bernstorff, S., Purslow, P., and Fratzl, P., 2002, “Viscoelastic Properties of Collagen: Synchrotron Radiation Investigations and Structural Model,” Philos. Trans. R. Soc. B, 357(1418), pp. 191–197. [CrossRef]
Sanjeevi, R., Somanathan, N., and Ramaswamy, D., 1982, “A Viscoelastic Model for Collagen Fibres,” J. Biomech., 15(3), pp. 181–183. [CrossRef] [PubMed]
Shen, Z. L., Kahn, H., Ballarini, R., and Eppell, S. J., 2011, “Viscoelastic Properties of Isolated Collagen Fibrils,” Biophys. J., 100(12), pp. 3008–3015. [CrossRef] [PubMed]
Yi, Y.-M., Park, S.-H., and Youn, S.-K., 1998, “Asymptotic Homogenization of Viscoelastic Composites With Periodic Microstructures,” Int. J. Solids Struct., 35(17), pp. 2039–2055. [CrossRef]
Hu, R., and Oskay, C., 2017, “Nonlocal Homogenization Model for Wave Dispersion and Attenuation in Elastic and Viscoelastic Periodic Layered Media,” ASME J. Appl. Mech., 84(3), p. 031003. [CrossRef]
Hu, R., and Oskay, C., 2018, “Spatial–Temporal Nonlocal Homogenization Model for Transient Anti-Plane Shear Wave Propagation in Periodic Viscoelastic Composites,” Comput. Methods Appl. Mech. Eng., 342, pp. 1–31. [CrossRef]
Tran, A., Yvonnet, J., He, Q.-C., Toulemonde, C., and Sanahuja, J., 2011, “A Simple Computational Homogenization Method for Structures Made of Linear Heterogeneous Viscoelastic Materials,” Comput. Methods Appl. Mech. Eng., 200(45–46), pp. 2956–2970. [CrossRef]
Zhang, P., Heyne, M. A., and To, A. C., 2015, “Biomimetic Staggered Composites With Highly Enhanced Energy Dissipation: Modeling, 3D Printing, and Testing,” J. Mech. Phys. Solids, 83, pp. 285–300. [CrossRef]
Hashin, Z., 1970, “Complex Moduli of Viscoelastic Composites—I. General Theory and Application to Particulate Composites,” Int. J. Solids Struct., 6(5), pp. 539–552. [CrossRef]
Qwamizadeh, M., Zhang, Z., Zhou, K., and Zhang, Y. W., 2015, “On the Relationship Between the Dynamic Behavior and Nanoscale Staggered Structure of the Bone,” J. Mech. Phys. Solids, 78, pp. 17–31. [CrossRef]
Qwamizadeh, M., Zhang, Z., Zhou, K., and Zhang, Y. W., 2016, “Protein Viscosity, Mineral Fraction and Staggered Architecture Cooperatively Enable the Fastest Stress Wave Decay in Load-Bearing Biological Materials,” J. Mech. Behav. Biomed. Mater., 60, pp. 339–355. [CrossRef] [PubMed]
Qwamizadeh, M., Lin, M., Zhang, Z. Q., Zhou, K., and Zhang, Y. W., 2017, “Bounds for the Dynamic Modulus of Unidirectional Composites With Bioinspired Staggered Distributions of Platelets,” Compos. Struct., 167, pp. 152–165. [CrossRef]
Qwamizadeh, M., Liu, P., Zhang, Z. Q., Zhou, K., and Zhang, Y. W., 2016, “Hierarchical Structure Enhances and Tunes the Damping Behavior of Load-Bearing Biological Materials,” ASME J. Appl. Mech., 83(5), p. 051009. [CrossRef]
Liu, J. J., Hai, X. S., Zhu, W. Q., and Wei, X. D., 2018, “Optimization of Damping Properties of Staggered Composites Through Microstructure Design,” ASME J. Appl. Mech., 85(10), p. 101002. [CrossRef]
Liu, J. J., Zhu, W. Q., Yu, Z. L., and Wei, X. D., 2018, “Dynamic Shear-Lag Model for Understanding the Role of Matrix in Energy Dissipation in Fiber-Reinforced Composites,” Acta Biomater., 74, pp. 270–279. [CrossRef] [PubMed]
Manca, F., Palla, P. L., Cleri, F., and Giordano, S., 2016, “Characteristic Lengths in Natural Bundle Assemblies Arising From Fiber-Matrix Energy Competition: A Floquet-Based Homogenization Theory,” Eur. J. Mech. A Solid, 60, pp. 145–165. [CrossRef]
Pontryagin, L. S., 1962, Ordinary Differential Equations, Addison-Wesley, New York.
Cox, H. L., 1952, “The Elasticity and Strength of Paper and Other Fibrous Materials,” Br. J. Appl. Phys., 3(3), pp. 72–79. [CrossRef]


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Fig. 1

Diverse staggered microarchitectures found in the load-bearing biological composites in nature: (a) the regular staggered microstructure in nacre [7,9,12], (b) the stairwise staggered pattern in the mineralized collagen of bone [1315], and (c) randomly staggered alignment of enamel rods in tooth [16,17]

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Fig. 2

(a) A two-dimensional schematic of composites with unidirectional reinforcements, (b) a representative unit cell with nonuniform reinforcements nonuniformly aligned in matrices in a staggered fashion, (c) a bar-spring model converted from the representative unit cell, and (d) the axial forces and shear stress on an infinitesimal bar segment

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Fig. 3

A schematic of the regular staggered composites, also well known as “brick-and-mortar” structure

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Fig. 4

The Floquet-based bar-spring (FBBS) model predicted (a) storage modulus and (b) loss modulus varying against the reinforcement aspect ratio, with comparison to the results of FEM simulations, the tension-shear chain (TSC) model, and its refined version

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Fig. 5

The Floquet-based bar-spring model predicted (a) storage modulus and (b) loss modulus varying against the angular frequency for different reinforcement aspect ratio, and (c) and (d) for different modulus ratio, with comparison to the corresponding FEM results for verification purposes

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Fig. 6

Staggered structure with hybrid long and short fibers

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Fig. 7

Variation of (a) storage modulus and (b) loss modulus with frequency

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Fig. 8

(a) Representative elementary volume, (b) longitudinal section view, and (c) cross-sectional view of the 3D model

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Fig. 9

(a) Storage modulus and (b) loss modulus of the 3D model



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