# Assessing the Interfacial Dynamic Modulus of Biological Composites

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## Abstract

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## 1. Introduction

## 2. Analytical Relationships for the Interface–Biocomposite Dynamic Moduli

## 3. Assessing the Interfacial Dynamic Modulus from a Far-Field Dynamic Mechanical Analysis

#### 3.1. Methodological Approach

- Step 1: Isolate a testing segment (length ${L}_{c}$) from the biocomposite complex and use microscopy observations to identify its underlying interfacial region (length ${L}_{i}$).
- Step 2: Apply DMA testing on a biocomposite segment and quantify its modulus magnitude and loss coefficient (${E}_{c}$ and $\mathrm{tan}{\delta}_{c}$).
- Step 3: Use nanomechanical testing (or the literature data) to determine the elastic modulus of the reinforcements (${E}_{f}$) outside the interfacial region.
- Step 4: Calculate the interface–biocomposite scaling factors (${k}_{E}$ and ${k}_{\delta}$), and use them to back-calculate the modulus magnitude and the loss coefficient of the interfacial region from the corresponding biocomposite characteristics (${E}_{i}={k}_{E}\cdot {E}_{c}$, and $\mathrm{tan}{\delta}_{i}={k}_{\delta}\cdot \mathrm{tan}{\delta}_{c}$).

#### 3.2. Example: Sutural Interfaces

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Meyers, M.A.; Chen, P.Y.; Lin, A.Y.M.; Seki, Y. Biological materials: Structure and mechanical properties. Prog. Mater. Sci.
**2008**, 53, 1–206. [Google Scholar] [CrossRef] [Green Version] - Dunlop, J.W.; Weinkamer, R.; Fratzl, P. Artful interfaces within biological materials. Mater. Today
**2011**, 14, 70–78. [Google Scholar] [CrossRef] - Bar-On, B.; Wagner, H.D. Structural motifs and elastic properties of hierarchical biological tissues—A review. J. Struct. Biol.
**2013**, 183, 149–164. [Google Scholar] [CrossRef] - Barthelat, F.; Yin, Z.; Buehler, M.J. Structure and mechanics of interfaces in biological materials. Nat. Rev. Mater.
**2016**, 1, 16007. [Google Scholar] [CrossRef] - Shelef, Y.; Bar-On, B. Surface protection in bio-shields via a functional soft skin layer: Lessons from the turtle shell. J. Mech. Behav. Biomed. Mater.
**2017**, 73, 68–75. [Google Scholar] [CrossRef] - Serrano, C.V.; Leemreize, H.; Bar-On, B.; Barth, F.G.; Fratzl, P.; Zolotoyabko, E.; Politi, Y. Ordering of protein and water molecules at their interfaces with chitin nano-crystals. J. Struct. Biol.
**2016**, 193, 124–131. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bar-On, B. On the form and bio-mechanics of venom-injection elements. Acta Biomater.
**2019**, 85, 263–271. [Google Scholar] [CrossRef] - Ampaw, E.; Owoseni, T.A.; Du, F.; Pinilla, N.; Obayemi, J.; Hu, J.; Nigay, P.-M.; Nzihou, A.; Uzonwanne, V.; Kana, Z.; et al. Compressive deformation and failure of trabecular structures in a turtle shell. Acta Biomater.
**2019**, 97, 535–543. [Google Scholar] [CrossRef] - Ji, B.; Gao, H. Mechanical properties of nanostructure of biological materials. J. Mech. Phys. Solids
**2004**, 52, 1963–1990. [Google Scholar] [CrossRef] - Smith, B.L.; Schäffer, T.E.; Viani, M.; Thompson, J.B.; Frederick, N.A.; Kindt, J.; Belcher, A.; Stucky, G.D.; Morse, D.E.; Hansma, P.K. Molecular mechanistic origin of the toughness of natural adhesives, fibres and composites. Nature
**1999**, 399, 761. [Google Scholar] [CrossRef] - Lee, N.; Williams, L.N.; Mun, S.; Rhee, H.; Prabhu, R.; Bhattarai, K.R.; Horstemeyer, M.F. Stress wave mitigation at suture interfaces. Biomed. Phys. Eng. Express
**2017**, 3, 035025. [Google Scholar] [CrossRef] - Erko, M.; Younes-Metzler, O.; Rack, A.; Zaslansky, P.; Young, S.L.; Milliron, G.; Chyasnavichyus, M.; Barth, F.G.; Fratzl, P.; Tsukruk, T.; et al. Micro-and nano-structural details of a spider’s filter for substrate vibrations: Relevance for low-frequency signal transmission. J. R. Soc. Interface
**2015**, 12, 20141111. [Google Scholar] [CrossRef] - Launey, M.E.; Buehler, M.J.; Ritchie, R.O. On the mechanistic origins of toughness in bone. Annu. Rev. Mater. Res.
**2010**, 40, 25–53. [Google Scholar] [CrossRef] [Green Version] - Amini, S.; Tadayon, M.; Idapalapati, S.; Miserez, A. The role of quasi-plasticity in the extreme contact damage tolerance of the stomatopod dactyl club. Nat. Mater.
**2015**, 14, 943. [Google Scholar] [CrossRef] - Huang, W.; Shishehbor, M.; Guarín-Zapata, N.; Kirchhofer, N.D.; Li, J.; Cruz, L.; Wang, T.; Bhowmick, S.; Stauffer, D.; Manimunda, P.; et al. A natural impact-resistant bicontinuous composite nanoparticle coating. Nat. Mater.
**2020**, 19, 1236–1243. [Google Scholar] [CrossRef] - Xu, M.; An, B. Dynamic crack propagation in the turtle carapace. Mech. Mater.
**2020**, 151, 103614. [Google Scholar] [CrossRef] - Jearanaisilawong, P.; Jongpairojcosit, N.; Glunrawd, C. Dynamic behaviors and protection mechanisms of sulcata tortoise carapace. Comput. Methods Biomech. Biomed. Eng.
**2021**, 1–13. [Google Scholar] [CrossRef] [PubMed] - Zlotnikov, I.; Zolotoyabko, E.; Fratzl, P. Nano-scale modulus mapping of biological composite materials: Theory and practice. Prog. Mater. Sci.
**2017**, 87, 292–320. [Google Scholar] [CrossRef] - Fratzl, P.; Weinkamer, R. Nature’s hierarchical materials. Prog. Mater. Sci.
**2007**, 52, 1263–1334. [Google Scholar] [CrossRef] [Green Version] - Meyers, M.A.; McKittrick, J.; Chen, P.Y. Structural biological materials: Critical mechanics-materials connections. Science
**2013**, 339, 773–779. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Naleway, S.E.; Taylor, J.R.; Porter, M.M.; Meyers, M.A.; McKittrick, J. Structure and mechanical properties of selected protective systems in marine organisms. Mater. Sci. Eng. C
**2016**, 59, 1143–1167. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Politi, Y.; Bar-On, B.; Fabritius, H.O. Mechanics of Arthropod Cuticle-Versatility by Structural and Compositional Variation. In Architectured Materials in Nature and Engineering; Springer: Cham, Switzerland, 2019; pp. 287–327. [Google Scholar]
- Shtein, I.; Koyfman, A.; Eshel, A.; Bar-On, B. Autotomy in plants: Organ sacrifice in Oxalis leaves. J. R. Soc. Interface
**2019**, 16, 20180737. [Google Scholar] [CrossRef] [Green Version] - Bentov, S.; Palmer, B.A.; Bar-On, B.; Shelef, Y.; Aflalo, E.D.; Sagi, A. Reinforcement of bio-apatite by zinc substitution in the incisor tooth of a prawn. Acta Biomater.
**2021**, 120, 116–123. [Google Scholar] [CrossRef] - Shtein, I.; Bar-On, B.; Popper, Z.A. Plant and algal structure: From cell walls to biomechanical function. Physiol. Plant
**2018**, 164, 56–66. [Google Scholar] [CrossRef] [PubMed] - Ebenstein, D.M.; Pruitt, L.A. Nanoindentation of biological materials. Nano Today
**2006**, 1, 26–33. [Google Scholar] [CrossRef] - Moshe-Drezner, H.; Shilo, D.; Dorogoy, A.; Zolotoyabko, E. Nanometer-Scale Mapping of Elastic Modules in Biogenic Composites: The Nacre of Mollusk Shells. Adv. Funct. Mater.
**2010**, 20, 2723–2728. [Google Scholar] [CrossRef] - Zhou, X.; Miao, H.; Li, F. Nanoscale structural and functional mapping of nacre by scanning probe microscopy techniques. Nanoscale
**2013**, 5, 11885–11893. [Google Scholar] [CrossRef] - Li, T.; Zeng, K. Nanoscale elasticity mappings of micro-constituents of abalone shell by band excitation-contact resonance force microscopy. Nanoscale
**2014**, 6, 2177–2185. [Google Scholar] [CrossRef] - Zlotnikov, I.; Shilo, D.; Dauphin, Y.; Blumtritt, H.; Werner, P.; Zolotoyabko, E.; Fratzl, P. In situ elastic modulus measurements of ultrathin protein-rich organic layers in biosilica: Towards deeper understanding of superior resistance to fracture of biocomposites. RSC Adv.
**2013**, 3, 5798–5802. [Google Scholar] [CrossRef] [Green Version] - Labonte, D.; Lenz, A.K.; Oyen, M.L. On the relationship between indentation hardness and modulus, and the damage resistance of biological materials. Acta Biomater.
**2017**, 57, 373–383. [Google Scholar] [CrossRef] [PubMed] - Shelef, Y.; Bar-On, B. Interfacial indentations in biological composites. J. Mech. Behav. Biomed. Mater.
**2020**, 114, 104209. [Google Scholar] [CrossRef] [PubMed] - Ji, B.; Gao, H. Mechanical principles of biological nanocomposites. Annu. Rev. Mater. Res.
**2010**, 40, 77–100. [Google Scholar] [CrossRef] - Bar-On, B.; Wagner, H.D. Mechanical model for staggered bio-structure. J. Mech. Phys. Solids
**2011**, 59, 1685–1701. [Google Scholar] [CrossRef] - Bar-On, B.; Wagner, H.D. New insights into the Young’s modulus of staggered biological composites. Mater. Sci. Eng. C
**2013**, 33, 603–607. [Google Scholar] [CrossRef] [PubMed] - Li, Y.; Ortiz, C.; Boyce, M.C. A generalized mechanical model for suture interfaces of arbitrary geometry. J. Mech. Phys. Solids
**2013**, 61, 1144–1167. [Google Scholar] [CrossRef] - Zhang, P.; Heyne, M.A.; To, A.C. Biomimetic staggered composites with highly enhanced energy dissipation: Modeling, 3D printing, and testing. J. Mech. Phys. Solids
**2015**, 83, 285–300. [Google Scholar] [CrossRef] [Green Version] - Qwamizadeh, M.; Zhou, K.; Zhang, Y.W. Damping behavior investigation and optimization of the structural layout of load-bearing biological materials. Int. J. Mech. Sci.
**2017**, 120, 263–275. [Google Scholar] [CrossRef] - Wu, J.; Yuan, H.; Li, L.; Fan, K.; Qian, S.; Li, B. Viscoelastic shear lag model to predict the micromechanical behavior of tendon under dynamic tensile loading. J. Theor. Biol.
**2018**, 437, 202–213. [Google Scholar] [CrossRef] - Liu, J.; Zhu, W.; Yu, Z.; Wei, X. Dynamic shear-lag model for understanding the role of matrix in energy dissipation in fiber-reinforced composites. Acta Biomater.
**2018**, 74, 270–279. [Google Scholar] [CrossRef] - Yu, Z.; Liu, J.; Wei, X. Achieving outstanding damping performance through bio-inspired sutural tessellations. J. Mech. Phys. Solids
**2020**, 142, 104010. [Google Scholar] [CrossRef] - Bar-On, B.; Wagner, H.D. Stiffness of the extrafibrillar phase in staggered biological arrays. Phys. Rev. Lett.
**2012**, 109, 078102. [Google Scholar] [CrossRef] [PubMed] - Prapavesis, A.; Tojaga, V.; Östlund, S.; van Vuure, A.W. Back calculated compressive properties of flax fibers utilizing the Impregnated Fiber Bundle Test (IFBT). Compos. Part A Appl. Sci. Manuf.
**2020**, 135, 105930. [Google Scholar] [CrossRef] - Khodayari, A.; Hirn, U.; Van Vuure, A.W.; Seveno, D. Inverse rule of mixtures at the nanoscale: Prediction of elastic properties of cellulose nanofibrils. Compos. Part A Appl. Sci. Manuf.
**2020**, 138, 106046. [Google Scholar] [CrossRef] - An, B.; Sun, W. A theory of biological composites undergoing plastic deformations. J. Mech. Behav. Biomed. Mater.
**2019**, 93, 204–212. [Google Scholar] [CrossRef] [PubMed] - Xu, M.; An, B. An analysis of fracture in staggered mineralized collagen fibril arrays. Int. J. Solids Struct.
**2020**, 193–194, 535–549. [Google Scholar] [CrossRef] - Yu, L.; Ma, Y.; Zhou, C.; Xu, H. Damping efficiency of the coating structure. Int. J. Solids Struct.
**2005**, 42, 3045–3058. [Google Scholar] [CrossRef] - Gusev, A.A.; Lurie, S.A. Loss amplification effect in multiphase materials with viscoelastic interfaces. Macromolecules
**2009**, 42, 5372–5377. [Google Scholar] [CrossRef] - Lurie, S.; Minhat, M.; Tuchkova, N.; Soliaev, J. On remarkable loss amplification mechanism in fiber reinforced laminated composite materials. Appl. Compos. Mater.
**2014**, 21, 179–196. [Google Scholar] [CrossRef] - Bar-On, B.; Bayerlein, B.; Blumtritt, H.; Zlotnikov, I. Dynamic response of a single interface in a biocomposite structure. Phys. Rev. Lett.
**2015**, 115, 238001. [Google Scholar] [CrossRef] [PubMed] - Bayerlein, B.; Bertinetti, L.; Bar-On, B.; Blumtritt, H.; Fratzl, P.; Zlotnikov, I. Inherent Role of Water in Damage Tolerance of the Prismatic Mineral–Organic Biocomposite in the Shell of Pinna Nobilis. Adv. Funct. Mater.
**2016**, 26, 3663–3669. [Google Scholar] [CrossRef] - Funari, M.F.; Greco, F.; Lonetti, P. Dynamic debonding in layered structures: A coupled ALE-cohesive approach. Frat. Integrita Strutt.
**2017**, 11, 524–535. [Google Scholar] [CrossRef] [Green Version] - Cohen, Y.; Ronen, D.; Ya’akobovitz, A. Mechanical behavior of vertically aligned carbon nanotubes under electrostatic tension. Sens. Actuators A Phys.
**2019**, 292, 105–111. [Google Scholar] [CrossRef] - Cohen, Y.; Ya’akobovitz, A. The influence of thermal loads on the physical properties of carbon nanotubes forests. Microelectron. Eng.
**2021**, 247, 111575. [Google Scholar] [CrossRef] - Gibson, R.F. Principles of Composite Material Mechanics; CRC Press: Boca Raton, FL, USA, 2016. [Google Scholar]
- Seto, J.; Gupta, H.S.; Zaslansky, P.; Wagner, H.D.; Fratzl, P. Tough lessons from bone: Extreme mechanical anisotropy at the mesoscale. Adv. Funct. Mater.
**2008**, 18, 1905–1911. [Google Scholar] [CrossRef] - Krauss, S.; Monsonego-Ornan, E.; Zelzer, E.; Fratzl, P.; Shahar, R. Mechanical function of a complex three-dimensional suture joining the bony elements in the shell of the red-eared slider turtle. Adv. Mater.
**2009**, 21, 407–412. [Google Scholar] [CrossRef] - Yang, W.; Chen, I.H.; Gludovatz, B.; Zimmermann, E.A.; Ritchie, R.O.; Meyers, M.A. Natural flexible dermal armor. Adv. Mater.
**2013**, 25, 31–48. [Google Scholar] [CrossRef] - Lee, N.; Horstemeyer, M.F.; Rhee, H.; Nabors, B.; Liao, J.; Williams, L.N. Hierarchical multiscale structure–property relationships of the red-bellied woodpecker (Melanerpes carolinus) beak. J. R. Soc. Interface
**2014**, 11, 20140274. [Google Scholar] [CrossRef] [Green Version] - Gao, C.; Hasseldine, B.P.; Li, L.; Weaver, J.C.; Li, Y. Amplifying strength, toughness, and auxeticity via wavy sutural tessellation in plant seedcoats. Adv. Mater.
**2018**, 30, 1800579. [Google Scholar] [CrossRef] [PubMed] - Achrai, B.; Bar-On, B.; Wagner, H.D. Biological armors under impact—Effect of keratin coating, and synthetic bio-inspired analogues. Bioinspirat. Biomim.
**2015**, 10, 016009. [Google Scholar] [CrossRef] [PubMed] - Liu, L.; Jiang, Y.; Boyce, M.; Ortiz, C.; Baur, J.; Song, J.; Li, Y. The effects of morphological irregularity on the mechanical behavior of interdigitated biological sutures under tension. J. Biomech.
**2017**, 58, 71–78. [Google Scholar] [CrossRef] - Yin, Z.; Hannard, F.; Barthelat, F. Impact-resistant nacre-like transparent materials. Science
**2019**, 364, 1260–1263. [Google Scholar] [CrossRef] - Serra, M.; Arenal, R.; Tenne, R. An overview of the recent advances in inorganic nanotubes. Nanoscale
**2019**, 11, 8073–8090. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Reddy, S.K.; Ya’akobovitz, A. Electromechanical behavior of graphene foams. Appl. Phys. Lett.
**2019**, 115, 211902. [Google Scholar] [CrossRef] - Ben-Shimon, Y.; Ya’akobovitz, A. Flexible and bio-compatible temperature sensors based on carbon nanotube composites. Measurement
**2021**, 172, 108889. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Schematic examples of the interfacial regions in biocomposites: graded matrix properties, shaped matrix-reinforcement connection, and staggered matrix-reinforcement connection. (

**b**) Mechanical modeling of a biocomposite segment (length ${L}_{c}$ and dynamic modulus ${E}_{c}^{*}$), which includes an effective viscoelastic interface (length ${L}_{i}$ and dynamic modulus ${E}_{i}^{*}$) between adjacent elastic reinforcements (elastic modulus ${E}_{f}$).

**Figure 2.**The correspondence between ${E}_{i}$ and ${E}_{c}$ for various biocomposite configurations. The dashed lines indicate the theoretical results via Equation (4), and the symbols indicate the corresponding finite-element results (Table S1 in Supporting Information). The colors indicate data sets with the same ${k}_{E}$ parameter: black, red, and green correspond to ${k}_{E}=0.1,0.3$, and $0.8$, respectively. The circle and square symbols represent different viscoelastic biocomposite configurations ($\mathrm{tan}{\delta}_{c}\ne 0$), while the plus and asterisk symbols represent the corresponding biocomposite configurations, but with completely elastic properties ($\mathrm{tan}{\delta}_{c}=0$).

**Figure 3.**The correspondence between $\mathrm{tan}{\delta}_{i}$ and $\mathrm{tan}{\delta}_{c}$ for various biocomposite configurations. The dashed lines indicate the theoretical results via Equation (4), and the symbols indicate the corresponding finite-element results (Table S2 in Supporting Information). The colors indicate data sets with the same ${k}_{\delta}$ parameter: black, red, and green correspond to ${k}_{\delta}=1.05,2.5$, and $5$, respectively. The range of $\mathrm{tan}{\delta}_{i}$ spans between completely elastic ($\mathrm{tan}{\delta}_{i}=0$) and predominantly viscoelastic ($\mathrm{tan}{\delta}_{i}=1/2$) interfaces.

**Figure 4.**Schematic descriptions of zigzag-shaped sutural interfaces in biocomposites and their geometrical parameters. (

**a**) A biocomposite segment with an underlying sutural interface. (

**b**) An isolated interfacial region (one period) from the biocomposite. (

**c**) Schematic examples of sutural interfaces with the same matrix length (${L}_{m}$), but different zigzag angles ($\theta =10\xb0,45\xb0$, and $~70\xb0$).

**Figure 5.**Simulation results for the interfacial dynamic modulus of biocomposites with zigzag-shaped sutural interfaces. (

**a**) the modulus magnitude and (

**b**) the loss coefficient of the interfacial region for different zigzag angles ($\theta $), achieved by back-calculations from far-field DMA testing on the biocomposite (${L}_{i}/{L}_{c}=0.1$), and by direct DMA testing on the interfacial region (${L}_{i}/{L}_{c}=1$).

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**MDPI and ACS Style**

Shelef, Y.; Uzan, A.Y.; Braunshtein, O.; Bar-On, B.
Assessing the Interfacial Dynamic Modulus of Biological Composites. *Materials* **2021**, *14*, 3428.
https://doi.org/10.3390/ma14123428

**AMA Style**

Shelef Y, Uzan AY, Braunshtein O, Bar-On B.
Assessing the Interfacial Dynamic Modulus of Biological Composites. *Materials*. 2021; 14(12):3428.
https://doi.org/10.3390/ma14123428

**Chicago/Turabian Style**

Shelef, Yaniv, Avihai Yosef Uzan, Ofer Braunshtein, and Benny Bar-On.
2021. "Assessing the Interfacial Dynamic Modulus of Biological Composites" *Materials* 14, no. 12: 3428.
https://doi.org/10.3390/ma14123428