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Research Papers

Flaw Tolerance in a Viscoelastic Strip

[+] Author and Article Information
Shaohua Chen

e-mail: chenshaohua72@hotmail.com
LNM,
Institute of Mechanics,
Chinese Academy of Sciences,
Beijing 100190,China

Huajian Gao

School of Engineering,
Brown University,
Providence, RI 02912
e-mail: Huajian_Gao@brown.edu

1Corresponding authors.

Manuscript received June 21, 2012; final manuscript received September 25, 2012; accepted manuscript posted October 22, 2012; published online May 16, 2013. Assoc. Editor: Anand Jagota.

J. Appl. Mech 80(4), 041014 (May 16, 2013) (10 pages) Paper No: JAM-12-1252; doi: 10.1115/1.4007864 History: Received June 21, 2012; Revised September 25, 2012; Accepted October 22, 2012

Load-bearing biological materials such as bone, teeth, and nacre have acquired some interesting mechanical properties through evolution, one of which is the tolerance of cracklike flaws incurred during tissue function, growth, repair, and remodeling. While numerous studies in the literature have addressed flaw tolerance in elastic structures, so far there has been little investigation of this issue in time-dependent, viscoelastic systems, in spite of its importance to biological materials. In this paper, we investigate flaw tolerance in a viscoelastic strip under tension and derive the conditions under which a pre-existing center crack, irrespective of its size, will not grow before the material fails under uniform rupture. The analysis is based on the Griffith and cohesive zone models of crack growth in a viscoelastic material, taking into account the effects of the loading rate along with the fracture energy, Young’s modulus, and theoretical strength of material.

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References

Gao, H., Ji, B., Jäger, 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, pp. 5597–5600. [CrossRef] [PubMed]
Persson, B. N. J., 2003, “Nanoadhesion,” Wear, 254, pp. 832–834. [CrossRef]
Glassmaker, N. J., Jagota, A., Hui, C. Y., and Kim, J., 2004, “Design of Biomimetic Fibrillar Interface: 1. Making Contact,” J. R. Soc. Interface, 1, pp. 23–33. [CrossRef] [PubMed]
Hui, C. Y., Glassmaker, N. J., Tang, T., and Jagota, A., 2004, “Design of Biomimetic Fibrillar Interface: 2. Mechanics of Enhanced Adhesion,” J. R. Soc. Interface, 1, pp. 35–48. [CrossRef] [PubMed]
Gao, H., Wang, X., Yao, H., Gorb, S., and Arzt, E., 2005, “Mechanics of Hierarchical Adhesion Structures of Geckos,” Mech. Mater., 37, pp. 275–285. [CrossRef]
Gao, H., and Chen, S., 2005, “Flaw Tolerance in a Thin Strip Under Tension,” ASME J. Appl. Mech., 72, pp. 732–737. [CrossRef]
Northen, M. T., and Turner, K. L., 2005, “A Batch Fabricated Biomimetic Dry Adhesive,” Nanotechnology, 16, pp. 1159–1166. [CrossRef]
Gao, H., 2006, “Application of Fracture Mechanics Concepts to Hierarchical Biomechanics of Bone and Bone-Like Materials,” Int. J. Fract., 138, pp. 101–137. [CrossRef]
Dugdale, D. S., 1960, “Yielding of Steel Sheets Containing Slits,” J. Mech. Phys. Solids, 8(2), pp. 100–104. [CrossRef]
Bilby, B. A., Cottrell, A. H., and Swinden, K. H., 1963, “The Spread of Plastic Yield From a Notch,” Proc. R. Soc. London, Ser. A, 272, pp. 304–314. [CrossRef]
Bazant, Z. P., 1976, “Instability, Ductility and Size Effect in Strain-Softening Concrete,” J. Eng. Mech., 102, pp. 331–344.
Hillerborg, A., Modeer, M., and Petersson, P. E., 1976, “Analysis of Crack Formation and Crack Growth in Concrete by Means of Fracture Mechanics and Finite Elements,” Cem. Concr. Res., 6, pp. 773–782. [CrossRef]
Kendall, K., 1978, “Complexities of Compression Failure,” Proc. R. Soc. London, Ser. A, 361, pp. 245–263. [CrossRef]
Karihaloo, B. L., 1979, “A Note on Complexities of Compression Failure,” Proc. R. Soc. London, Ser. A, 368, pp. 483–493. [CrossRef]
Rice, J. R., 1980, “The Mechanics of Earthquake Rupture,” International School of Physics “E. Fermi,” Course 78, 1979, Italian Physical Society/North Holland, Amsterdam.
Bazant, Z. P., and Cedolin, L., 1983, “Finite Element Modeling of Crack Band Propagation,” J. Struct. Eng., 109, pp. 69–92. [CrossRef]
Barenblatt, G. I., 1985, “The Mathematical Theory of Equilibrium Cracks in Brittle Fracture,” Adv. Appl. Mech., 7, pp. 55–129. [CrossRef]
Bao, G., and Suo, Z., 1992, “Remarks on Crack-Bridging Concepts,” Appl. Mech. Rev., 45, pp. 355–366. [CrossRef]
Cox, B. N., and Marshall, D. B., 1994, “Concepts for Bridged Cracks in Fracture and Fatigue,” Acta Metall. Mater., 42, pp. 341–363. [CrossRef]
Bazant, Z. P., and Planas, J., 1998, Fracture and Size Effect in Concrete and Other Quasibrittle Materials, CRC, Boca Raton, FL.
Massabo, R., and Cox, B. N., 1999, “Concepts for Bridged Mode II Delamination Cracks,” J. Mech. Phys. Solids, 47, pp. 1265–1300. [CrossRef]
Mulmule, S. V., and Dempsey, J. P., 2000, “LEFM Size Requirement for the Fracture Testing of Sea Ice,” Int. J. Fract., 102, pp. 85–98. [CrossRef]
Drugan, W. J., 2001, “Dynamic Fragmentation of Brittle Materials: Analytical Mechanics-Based Models,” J. Mech. Phys. Solids, 49, pp. 1181–1208. [CrossRef]
Gao, H., Ji, B., Buehler, M. J., and Yao, H., 2004, “Flaw Tolerant Bulk and Surface Nanostructures of Biological Systems,” Mech. Chem. Biosyst., 1(1), pp. 37–52. [CrossRef] [PubMed]
Gao, H., and Yao, H., 2004, “Shape Insensitive Optimal Adhesion of Nanoscale Fibrillar Structures,” Proc. Natl. Acad. Sci. U.S.A., 101, pp. 7851–7856. [CrossRef] [PubMed]
Ji, B., and Gao, H., 2004, “Mechanical Properties of Nanostructure of Biological Materials,” J. Mech. Phys. Solids, 52, pp. 1963–1990. [CrossRef]
Ji, B., and Gao, H., 2004, “A Study of Fracture Mechanisms in Biological Nano-Composites via the Virtual Internal Bond Model,” Mater. Sci. Eng. A, 366, pp. 96–103. [CrossRef]
Buehler, M. J., Yao, H., Ji, B., and Gao, H., 2006, “Cracking and Adhesion at Small Scales: Atomistic and Continuum Studies of Flaw Tolerant Nanostructures,” Modell. Simul. Mater. Sci. Eng., 14, pp. 799–816. [CrossRef]
Yao, H., and Gao, H., 2006, “Mechanics of Robust and Releasable Adhesion in Biology: Bottom-Up Designed Hierarchical Structures of Gecko,” J. Mech. Phys. Solids, 54, pp. 1120–1146. [CrossRef]
Chen, S., and Soh, A., 2008, “Tuning the Geometrical Parameters of Biomimetic Fibrillar Structures to Enhance Adhesion,” J. R. Soc. Interface, 5, pp. 373–382. [CrossRef] [PubMed]
Chen, S., Xu., G., and Soh, A., 2010, “Size-Dependent Adhesion Strength of a Single Viscoelastic Fiber,” Tribol. Lett., 37, pp. 375–379. [CrossRef]
Chen, S., and Chen, P., 2010, “Nanoadhesion of a Power-Law Graded Elastic Material,” Chin. Phys. Lett., 27(10), p. 108102. [CrossRef]
Kumar, S., Haque, M. A., and Gao, H., 2009, “Notch Insensitive Fracture in Nanoscale Thin Films,” Appl. Phys. Lett., 94, p. 253103. [CrossRef]
Kumar, S., Li, X. Y., Haque, A., and Gao, H. J., 2011, “Is Stress Concentration Relevant for Nanocrystalline Metals?,” Nano Lett., 11, pp. 2510–2516. [CrossRef] [PubMed]
Giesa, T., Arslan, M., Pugno, N. M., and Buehler, M. J., 2011, “Nanoconfinement of Spider Silk Fibrils Begets Superior Strength, Extensibility, and Toughness,” Nano Lett., 11(11), pp. 5038–5046. [CrossRef] [PubMed]
Abou Neel, E. A., Salih, V., Revell, P. A., and Young, A. M., 2012, “Viscoelastic and Biological Performance of Low-Modulus, Reactive Calcium Phosphate-Filled, Degradable, Polymetric Bone Adhesive,” Acta Biomater., 8, pp. 313–320. [CrossRef] [PubMed]
Deymier-Black, A. C., Yuan, F., Singhal, A., Almer, J. D., Brinson, L. C., and Dunand, D. C., 2012, “Evolution of Load Transfer Between Hydroxyapatite and Collagen During Creep Deformation of Bone,” Acta Biomater., 8, pp. 253–261. [CrossRef] [PubMed]
Mano, J. F., 2005, “Viscoelastic Properties of Bone: Mechanical Spectroscopy Studies on a Chicken Model,” Mater. Sci. Eng., C, 25, pp. 145–152. [CrossRef]
Fung, Y. C., 1983, Biomechanics, Science Press, Beijing (in Chinese).
Nuismer, R. J., 1974, “Governing Equation for Quasi-Static Crack Growth in Linearly Viscoelastic Materials,” ASME J. Appl. Mech., 41, pp. 631–634. [CrossRef]
Gurtin, M. E., and Sternberg, E., 1962, “On the Linear Theory of Viscoelasticity,” Arch. Ration. Mech. Anal., 11, pp. 291–356. [CrossRef]
Christensen, M. E., 1982, Theory of Viscoelasticity, Academic, New York.
Gao, Q., Lin, S., and Yang, X. J., 2007, “Description of Non-Linear Creep Constitutive Relation for Viscoelastic Butyl Rubber,” Chin. J. Appl. Mech., 24, pp. 386–390 (in Chinese).
Graham, G. A. C., 1968, “Correspondence Principle of Linear Viscoelasticity Theory for Mixed Boundary Value Problems Involving Time-Dependent Boundary Regions,” Q. Appl. Math., 26, pp. 167–174.
Graham, G. A. C., and Sabin, G. C. W., 1973, “Correspondence Principle of Linear Viscoelasticity for Problems That Involve Time-Dependent Regions,” Int. J. Eng. Sci., 11, pp. 123–140. [CrossRef]
Guo, X., and Gao, H., 2005, “Bio-Inspired Material Design and Optimization,” IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials—Status and Perspectives, Rungstedgaard, Copenhagen, Denmark, October 26–29.
Kauffmann, F., Ji, B., Dehm, G., Gao, H., and Arzt, E., 2005, “A Quantitative Study of the Hardness in a Superhard Nanocrystalline Titanium Nitride/Silicon Nitride Coating,” Scr. Mater., 52, pp. 1269–1274. [CrossRef]
Lakes, R. S., and Katz, J. L., 1979, “Viscoelastic Properties of Wet Cortical Bone—II: Relaxation Mechanisms,” J. Biomech., 12(9), pp. 679–687. [CrossRef] [PubMed]
Sasaki, N., Nakayama, Y., Yoshikawa, M., and Enyo, A., 1993, “Stress Relaxation Function of Bone Collagen,” J. Biomech., 26(12), pp. 1369–1376. [CrossRef] [PubMed]
Gao, H., and Klein, P., 1998, “Numerical Simulation of Crack Growth in an Isotropic Solid With Randomized Internal Cohesive Bonds,” J. Mech. Phys. Solids, 46, pp. 187–218. [CrossRef]
Smith, B. L., Schaeffer, T. E., Viani, M., Thompson, J. B., Frederick, N. A., Kindt, J. H., Belcher, A., Stucky, G. D., Morse, D. E., and Hansma, P. K., 1999, “Molecular Mechanistic Origin of the Toughness of Natural Adhesive, Fibres and Composites,” Nature (London), 399, pp. 761–763. [CrossRef]
Thompson, J. B., Kindt, J. H., Drake, B., Hansma, H. G., Morse, D. E., and Hansma, P. K., “Bone Indentation Recovery Time Correlates With Bone Reforming Time,” Nature (London), 414, pp. 773–776. [CrossRef]
de Gennes, P. G., 1996, “Soft Adhesives,” Langmuir, 12, pp. 4497–4500. [CrossRef]
Bilby, B. A., and EshelbyJ. D., 1968, Fracture, Academic, New York.

Figures

Grahic Jump Location
Fig. 1

A center-cracked viscoelastic strip of width 2W and crack size 2a under tension

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Fig. 2

The standard viscoelastic model used in the present study

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Fig. 3

Analysis of the viscoelastic strip problem with Griffith’s model of crack growth via the correspondence principle. (a) The auxiliary problem of an elastic strip of width 2W and crack size 2a, and (b) the corresponding viscoelastic problem.

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Fig. 4

A bi-linear loading profile in the viscoelastic problem

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Fig. 5

The stress-strain relationship of a viscoelastic butyl rubber under different loading rates τ0/t0

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Fig. 6

The normalized critical strip width Wcr/lft for crack growth as a function of the normalized crack length β=a/W in the viscoelastic Griffith problem under different loading rates

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Fig. 7

A center-cracked elastic strip of width 2W subject to remote tension σ∞. The length of the cohesive zone is l and the effective crack length is c=a+l.

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Fig. 8

Superposition scheme used to determine the crack opening displacement in the elastic Dugdale problem. (a) A perfect strip without crack under applied stress σ∞; a strip with crack length 2c subjected to (b) a uniformly normal compressive stress σ∞ on the crack face, and (c) a uniformly normal tensile stress σs in the cohesive zone.

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Fig. 9

The normalized critical strip width Wcr/lft for crack growth as a function of the normalized crack length β=a/W in the elastic Dugdale problem

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Fig. 10

The normalized effective crack length as a function of the loading time for the viscoelastic problem with β=0.2

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Fig. 11

The normalized crack opening displacement as a function of the loading time in the viscoelastic Dugdale problem with β=0.33 and W/lft=1.83

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Fig. 12

The nondimensional critical strip width Wcr/lft for crack growth as a function of the normalized crack length β=a/W in the viscoelastic Dugdale problem under different loading rates

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Fig. 13

The nondimensional flaw tolerance width Wft/lft as a function of the total loading time (different loading rates) with β=0.33

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Fig. 14

Dugdale model of a finite crack in an infinite space

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Fig. 15

Dugdale model of a periodic array of cracks in an infinite space. Each crack has a length a and a cohesive zone length c-a. The length of the period is 2W.

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