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Crack propagation and renucleation in soft brittle hydrogels

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Crack tip opening displacement (CTOD) and fracture energy are determined from crack geometry and material properties for very slowly propagating cracks, less than 50 \(\upmu \mathrm {m/s}\), in thin brittle hydrogels on the sub-millimeter scale. 2D fluorescent speckle images are captured using confocal microscopy during propagation, and 3D volumetric images are captured both before propagation begins and after the crack arrests. Fracture energy builds up until a critical value is reached and then remains constant as the crack propagates and eventually arrests when the energy is no longer sufficient for propagation. Once a crack arrests, more energy is needed for renucleation, suggesting that local toughening effects are at play. Based on observations of renucleation events and analysis of 3D crack shapes, this local toughening points to a mechanism for fracture surface roughening observed in the literature for slowly propagating cracks. Additionally, through-thickness variation in fracture energy, while expected from linear elastic fracture mechanics (LEFM) theory, suggests local toughening in the process zone which contributes to this roughening of crack surfaces.

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Computational details:

The Digital Image Correlation (DIC) analyses in this chapter were obtained using the following Matlab packages: FIDIC (Fast Iterative DIC): https://github.com/FranckLab/FIDIC. Ncorr v1.2: http://www.ncorr.com/index.php/downloads.


  1. Abe H, Mura T, Keer LM (1976) Growth rate of a penny-shaped crack in hydraulic fracturing of rocks. J Geophys Res (1896–1977) 81(29):5335–5340. https://doi.org/10.1029/JB081i029p05335

  2. Agnelli S, Baldi F, Bignotti F, Salvadori A, Peroni I (2018) Fracture characterization of hyperelastic polyacrylamide hydrogels. Eng Fract Mech. https://doi.org/10.1016/j.engfracmech.2018.06.004

  3. Anderson TL (2005) Fracture mechanics: fundamentals and applications, 3rd edn. CRC Press, Boca Raton

  4. Annabi N, Nichol JW, Zhong X, Ji C, Koshy S, Khademhosseini A, Dehghani F (2010) Controlling the porosity and microarchitecture of hydrogels for tissue engineering. Tissue Eng Part B Rev 16(4):371–383. https://doi.org/10.1089/ten.teb.2009.0639

  5. Baldi F, Bignotti F, Peroni I, Agnelli S, Riccò T (2012) On the measurement of the fracture resistance of polyacrylamide hydrogels by wire cutting tests. Polym Test 31(3):455–465. https://doi.org/10.1016/j.polymertesting.2012.01.009

  6. Bar-Kochba E, Toyjanova J, Andrews E, Kim KS, Franck C (2015) A fast iterative digital volume correlation algorithm for large deformations. Exp Mech 55(1):261–274. https://doi.org/10.1007/s11340-014-9874-2

  7. Baumberger T, Caroli C, Martina D, Ronsin O (2008) Magic angles and cross-hatching instability in hydrogel fracture. Phys Rev Lett 100(17):178303. https://doi.org/10.1103/PhysRevLett.100.178303

  8. Blaber J, Adair B, Antoniou A (2015) Ncorr: Open-source 2D digital image correlation matlab software. Exp Mech 55(6):1105–1122. https://doi.org/10.1007/s11340-015-0009-1

  9. Bostwick JB, Daniels KE (2013) Capillary fracture of soft gels. Phys Rev E 88(4):042410. https://doi.org/10.1103/PhysRevE.88.042410

  10. Bouchbinder E, Livne A, Fineberg J (2010) Weakly nonlinear fracture mechanics: experiments and theory. Int J Fract 162:3–20. https://doi.org/10.1007/s10704-009-9427-3

  11. Bouklas N, Landis CM, Huang R (2015) Effect of solvent diffusion on crack-tip fields and driving force for fracture of hydrogels. J Appl Mech 82(8):081007. https://doi.org/10.1115/1.4030587

  12. Broek D (1982) Elementary engineering fracture mechanics, 1st edn. Springer, Haarlem. https://doi.org/10.1007/978-94-009-4333-9

  13. Coyle S, Majidi C, LeDuc P, Hsia KJ (2018) Bio-inspired soft robotics: material selection, actuation, and design. Extrem Mech Lett 22:51–59. https://doi.org/10.1016/j.eml.2018.05.003

  14. de Gennes PG (2005) Soft matter: more than words. Soft Matter 1(1):16. https://doi.org/10.1039/B419223K

  15. Denisin AK, Pruitt BL (2016) Tuning the range of polyacrylamide gel stiffness for mechanobiology applications. ACS Appl Mater Interfaces 8(34):21893–21902. https://doi.org/10.1021/acsami.5b09344

  16. Engler AJ, Sen S, Sweeney HL, Discher DE (2006) Matrix elasticity directs stem cell lineage specification. Cell 126(4):677–689. https://doi.org/10.1016/j.cell.2006.06.044

  17. Franck C, Hong S, Maskarinec SA, Tirrell DA, Ravichandran G (2007) Three-dimensional full-field measurements of large deformations in soft materials using confocal microscopy and digital volume correlation. Exp Mech 47(3):427–438. https://doi.org/10.1007/s11340-007-9037-9

  18. Geubelle PH, Knauss WG (1994) Finite strains at the tip of a crack in a sheet of hyperelastic material: I. Homogeneous case. J Elast 35:61–98. https://doi.org/10.1007/BF00115539

  19. Gong JP (2010) Why are double network hydrogels so tough? Soft Matter 6(12):2583–2590. https://doi.org/10.1039/B924290B

  20. Gong JP, Osada Y (2010) Soft and wet materials: from hydrogels to biotissues. In: Cloitre M (ed) High solid dispersions. Springer, Berlin, pp 203–246. https://doi.org/10.1007/12_2010_91

  21. Hui CY, Long R, Ning J (2013) Stress relaxation near the tip of a stationary mode I crack in a poroelastic solid. J Appl Mech 80(2):021014. https://doi.org/10.1115/1.4007228

  22. Krishnan VR, Hui CY, Long R (2008) Finite strain crack tip fields in soft incompressible elastic solids. Langmuir 24:14245–14253. https://doi.org/10.1021/la802795e

  23. Long R, Hui CY (2016) Fracture toughness of hydrogels: measurement and interpretation. Soft Matter 12(39):8069–8086. https://doi.org/10.1039/C6SM01694D

  24. Moshayedi P, Ng G, Kwok JCF, Yeo GSH, Bryant CE, Fawcett JW, Franze K, Guck J (2014) The relationship between glial cell mechanosensitivity and foreign body reactions in the central nervous system. Biomaterials 35(13):3919–3925. https://doi.org/10.1016/j.biomaterials.2014.01.038

  25. Naficy S, Brown HR, Razal JM, Spinks GM, Whitten PG (2011) Progress toward robust polymer hydrogels. Aust J Chem 64(8):1007–1025

  26. Nilsson F (2005) A tentative method for determination of cohesive zone properties in soft materials. Int J Fract 136(1):133–142. https://doi.org/10.1007/s10704-005-5125-y

  27. Noselli G, Lucantonio A, McMeeking RM (2018) Poroelastic toughening in polymer gels: a theoretical and numerical study. J Mech Phys Solids 94:33–46. https://doi.org/10.1016/j.jmps.2016.04.017

  28. Notbohm J, Kim JH, Asthagiri AR, Ravichandran G (2012) Three-dimensional analysis of the effect of epidermal growth factor on cell-cell adhesion in epithelial cell clusters. Biophys J 102(6):1323–1330. https://doi.org/10.1016/j.bpj.2012.02.016

  29. Notbohm J, Lesman A, Tirrell DA, Ravichandran G (2015) Quantifying cell-induced matrix deformation in three dimensions based on imaging matrix fibers. Integr Biol 7(10):1186–1195. https://doi.org/10.1039/C5IB00013K

  30. Rice JR (1968) A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 35(2):379–386. https://doi.org/10.1115/1.3601206

  31. Shih CF (1981) Relationships between the J-integral and the crack opening displacement for stationary and extending cracks. J Mech Phys Solids 29(4):305–326. https://doi.org/10.1016/0022-5096(81)90003-X

  32. Sun CT, Jin ZH (2012) Chapter 6—Crack tip plasticity. In: Sun CT, Jin ZH (eds) Fracture mechanics. Academic Press, Boston, pp 123–169. https://doi.org/10.1016/B978-0-12-385001-0.00006-7

  33. Sutton J, Schreier H, Orteu MA (2009) Image correlation for shape, motion and deformation measurements: basic concepts, theory and applications, 1st edn. Springer, Berlin. https://doi.org/10.1007/978-0-387-78747-3

  34. Tanaka Y, Fukao K, Miyamoto Y (2000) Fracture energy of gels. Eur Phys J E 3(4):395–401. https://doi.org/10.1007/s101890070010

  35. Tanaka Y, Shimazaki R, Yano S, Yoshida G, Yamaguchi T (2016) Solvent effects on the fracture of chemically crosslinked gels. Soft Matter 12(39):8135–8142. https://doi.org/10.1039/C6SM01645F

  36. Whang M, Kim J (2016) Synthetic hydrogels with stiffness gradients for durotaxis study and tissue engineering scaffolds. Tissue Eng Regener Med 13(2):126–139. https://doi.org/10.1007/s13770-016-0026-x

  37. Yeung T, Georges PC, Flanagan LA, Marg B, Ortiz M, Funaki M, Zahir N, Ming W, Weaver V, Janmey PA (2005) Effects of substrate stiffness on cell morphology, cytoskeletal structure, and adhesion. Cell Motil 60(1):24–34. https://doi.org/10.1002/cm.20041

  38. Yew CH (1997) Chapter 1–Fracturing of a wellbore and 2-D fracture models. Mechanics of hydraulic fracturing. Gulf Professional Publishing, Houston, pp 1–29. https://doi.org/10.1016/B978-088415474-7/50013-0

  39. Yu Y, Landis CM, Huang R (2018a) Steady-state crack growth in polymer gels: a linear poroelastic analysis. J Mech Phys Solids 118:15–39

  40. Yu Y, Bouklas N, Landis CM, Huang R (2018b) A Linear poroelastic analysis of time-dependent crack-tip fields in polymer gels. J Appl Mech 85(11):111011. https://doi.org/10.1115/1.4041040

  41. Zhang R, He LF (2012) Measurement of mixed-mode stress intensity factors using digital image correlation method. Opt Lasers Eng 50(7):1001–1007. https://doi.org/10.1016/j.optlaseng.2012.01.009

  42. Zhu XK, Joyce JA (2012) Review of fracture toughness (G, K, J, CTOD, CTOA) testing and standardization. Eng Fract Mech 85:1–46. https://doi.org/10.1016/j.engfracmech.2012.02.001

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This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1144469 and Award No. DMS-1535083 under the Designing Materials to Revolutionize and Engineer our Future (DMREF) program. Imaging was performed in the Biological Imaging Facility, with the support of the Caltech Beckman Institute and the Arnold and Mabel Beckman Foundation.

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Correspondence to Kimberley Ann Mac Donald.

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Mac Donald, K.A., Ravichandran, G. Crack propagation and renucleation in soft brittle hydrogels. Int J Fract (2020). https://doi.org/10.1007/s10704-020-00430-w

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  • Brittle
  • Confocal microscopy
  • Fracture mechanics
  • Slow cracks
  • Soft gels