Biomechanics and Modeling in Mechanobiology

, Volume 18, Issue 1, pp 175–187 | Cite as

Multiscale model of fatigue of collagen gels

  • Rohit Y. DhumeEmail author
  • Elizabeth D. Shih
  • Victor H. Barocas
Original Paper


Fatigue as a mode of failure becomes increasingly relevant with age in tissues that experience repeated fluctuations in loading. While there has been a growing focus on the mechanics of networks of collagen fibers, which are recognized as the predominant mechanical components of soft tissues, the network’s fatigue behavior has received less attention. Specifically, it must be asked (1) how the fatigue of networks differs from that of its component fibers, and (2) whether this difference in fatigue behaviors is affected by changes in the network’s architecture. In the present study, we simulated cyclic uniaxial loading of Voronoi networks to model fatigue experiments performed on reconstituted collagen gels. Collagen gels were cast into dog-bone shape molds and were tested on a uniaxial machine under a tension-tension cyclic loading protocol. Simulations were performed on networks modeled as trusses of, on average, 600 nonlinear elastic fibers connected at freely rotating pin-joints. We also simulated fatigue failure of Delaunay, and Erdős–Rényi networks, in addition to Voronoi networks, to compare fatigue behavior among different architectures. The uneven distribution of stresses within the fibers of the unstructured networks resulted in all three network geometries being more endurant than a single fiber or a regular lattice under cyclic loading. Among the different network geometries, for low to moderate external loads, the Delaunay networks showed the best fatigue behavior, while at higher loads, the Voronoi networks performed better.


Fiber network mechanics Network fatigue Collagen gel fatigue 



The authors gratefully acknowledge financial support from the National Institutes of Health (R01-EB005813). We also thank the Minnesota Supercomputing Institute (MSI) for providing the computing resources used to carry out this work.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Abel J, Luntz J, Brei D (2013) Hierarchical architecture of active knits. Smart Mater Struct 22:125001CrossRefGoogle Scholar
  2. Abhilash AS, Baker BM, Trappmann B, Chen CS, Shenoy VB (2014) Remodeling of fibrous extracellular matrices by contractile cells: predictions from discrete fiber network simulations. Biophys J 107(8):1829–1840. CrossRefGoogle Scholar
  3. Aghvami M, Billiar KL, Sander EA (2016) Fiber network models predict enhanced cell mechanosensing on fibrous gels. J Biomech Eng 138(10):101006–101011CrossRefGoogle Scholar
  4. Andarawis-Puri N, Flatow EL (2011) Tendon fatigue in response to mechanical loading. J Musculoskelet Neuronal Interact 11:106–114Google Scholar
  5. Andrews E, Gibson L, Ashby M (1999) The creep of cellular solids. Acta Mater 47(10):2853–2863. CrossRefGoogle Scholar
  6. Arruda EM, Boyce MC (1993) A three dimensional constitutive model for the large stretch behavior of rubber elastic materials. J Mech Phys Solids 41:389–412CrossRefzbMATHGoogle Scholar
  7. Ashby M, Evans A, Fleck N, Gibson L, Hutchinson J, Wadley H (eds) (2000) Metal foams: a design guide. Butterworth-Heinemann, Burlington. Google Scholar
  8. Ban E, Barocas VH, Shephard MS, Picu RC (2016) Softening in random networks of non-identical beams. J Mech Phys Solids 87:38–50. MathSciNetCrossRefGoogle Scholar
  9. Ban E, Zhang S, Zarei V, Barocas VH, Winkelstein B, Picu CR (2017) Collagen organization in facet capsular ligaments varies with spinal region and with ligament deformation. J Biomech Eng 139:0710091–0710099Google Scholar
  10. Barnes CP, Sell SA, Boland ED, Simpson DG, Bowlin GL (2007) Nanofiber technology: designing the next generation of tissue engineering scaffolds. Adv Drug Deliv Rev 59(14):1413–1433. (intersection of Nanoscience and Modern Surface Analytical Methodology) CrossRefGoogle Scholar
  11. Billiar KL, Sacks M (2000) Biaxial mechanical properties of the native and glutaraldehyde-treated aortic valve cusp: Part II—a structural constitutive model. J Biomech Eng 122:327–335CrossRefGoogle Scholar
  12. Broedersz CP, MacKintosh FC (2014) Modeling semiflexible polymer networks. [cond-mat, physics:physics] arXiv:1404.4332
  13. Chandran PL, Barocas VH (2006) Affine versus non-affine fibril kinematics in collagen networks: theoretical studies of network behavior. J Biomech Eng 128:259–270CrossRefGoogle Scholar
  14. De Vita R, Slaughter WS (2007) A constitutive law for the failure behavior of medial collateral ligaments. Biomech Model Mechanobiol 6(3):189–197. CrossRefGoogle Scholar
  15. Delaunay B (1934) Sur la sphère vide. Bull Acad Sci l’URSS Classe Sci Math Nat 6:793–800zbMATHGoogle Scholar
  16. Deshpande VS, Fleck NA (2000) Isotropic constitutive models for metallic foams. J Mech Phys Solids 48:1253–1283CrossRefzbMATHGoogle Scholar
  17. Dittmore A, Silver J, Sarkar SK, Marmer B, Goldberg GI, Neuman KC (2016) Internal strain drives spontaneous periodic buckling in collagen and regulates remodeling. Proc Natl Acad Sci 113(30):8436–8441. CrossRefGoogle Scholar
  18. Erdős P, Rényi A (1959) On random graphs. Publ Math 6:290–297MathSciNetzbMATHGoogle Scholar
  19. Flory PJ (1953) Principles of polymer chemistry. Cornell University Press, IthacaGoogle Scholar
  20. Flory PJ, Rehner JJ (1943) Statistical mechanics of crosslinked polymer networks I. Rubberlike elasticity. J Chem Phys 11:512CrossRefGoogle Scholar
  21. Freeman MAR (1999) Is collagen fatigue failure a cause of osteoarthrosis and prosthetic component migration? A hypothesis. J Orthop Res 17:3–8CrossRefGoogle Scholar
  22. Koh C, Strange D, Tonsomboon K, Oyen M (2013) Failure mechanisms in fibrous scaffolds. Acta Biomater 9:7326–7334CrossRefGoogle Scholar
  23. Lai VK, Lake SP, Frey CR, Tranquillo RT, Barocas VH (2012) Mechanical behavior of collagen–fibrin co-gels reflects transition from series to parallel interactions with increasing collagen content. J Biomech Eng 134:011004–011009CrossRefGoogle Scholar
  24. Lake SP, Barocas VH (2012) Mechanics and kinematics of soft tissue under indentation are determined by the degree of initial collagen fiber alignment. J Mech Behav Biomed Mater 13:25–35CrossRefGoogle Scholar
  25. Linka K, Hillgrtner M, Itskov M (2018) Fatigue of soft fibrous tissues: multi-scale mechanics and constitutive modeling. Acta Biomater 71:398–410. CrossRefGoogle Scholar
  26. Martin C, Sun W (2013) Modeling of long-term fatigue damage of soft tissue with stress softening and permanent set effects. Biomech Model Mechanobiol 12:645–655. CrossRefGoogle Scholar
  27. Martin C, Sun W (2014a) Simulation of long-term fatigue damage in bioprosthetic heart valves: effects of leaflet and stent elastic properties. Biomech Model Mechanobiol 13:759–770CrossRefGoogle Scholar
  28. Martin C, Sun W (2014b) Simulation of long-term fatigue damage in bioprosthetic heart valves: effects of leaflet and stent elastic properties. Biomech Model Mechanobiol 13(4):759–770. CrossRefGoogle Scholar
  29. MATLAB (2013) version (R2013b). The MathWorks Inc., Natick, MassachusettsGoogle Scholar
  30. Maxwell JC (1864) L. on the calculation of the equilibrium and stiffness of frames. Philos Mag 27:294–299. CrossRefGoogle Scholar
  31. McCullough KYG, Fleck NA, Ashby MF (1999) The stress-life fatigue behaviour of aluminium alloy foams. Fatigue Fract Eng Mater Struct 23:199–208CrossRefGoogle Scholar
  32. Miner MA (1945) Cumulative damage in fatigue. J Appl Mech 67:159–164Google Scholar
  33. Mitchison T, Cramer L (1996) Actin-based cell motility and cell locomotion. Cell 84(3):371–379. CrossRefGoogle Scholar
  34. Nachtrab S, Kapfer SC, Arns CH, Madadi M, Mecke K, Schrder-Turk GE (2011) Morphology and linear-elastic moduli of random network solids. Adv Mater 23(22–23):2633–2637. CrossRefGoogle Scholar
  35. Pena E (2011) Prediction of the softening and damage effects with permanent set in fibrous biological materials. J Mech Phys Solids 59:1808–1822MathSciNetCrossRefzbMATHGoogle Scholar
  36. Picu R (2011) Mechanics of random fiber networks—a review. Soft Matter 7:6768–6785CrossRefGoogle Scholar
  37. Roberts A, Garboczi E (2001) Elastic moduli of model random three-dimensional closed-cell cellular solids. Acta Mater 49(2):189–197. CrossRefGoogle Scholar
  38. Roberts A, Garboczi E (2002) Elastic properties of model random three-dimensional open-cell solids. J Mech Phys Solids 50(1):33–55. CrossRefzbMATHGoogle Scholar
  39. Sa S, Picu CR (2004) Network model for the viscoelastic behavior of polymer nanocomposites. Polymer 45(22):7779–7790. CrossRefGoogle Scholar
  40. Sander E, Stylianopoulos T, Tranquillo R, Barocas V (2009) Image-based multiscale modeling predicts tissue-level and network-level fiber reorganization in stretched cell-compacted collagen gels. Proc Natl Acad Sci USA 106:17675–17680CrossRefGoogle Scholar
  41. Schechtman H, Bader DL (1997) In-vitro fatigue of human tendons. J Biomech 30:829–835CrossRefGoogle Scholar
  42. Shasavari A, Picu R (2012) Model selection for athermal cross-linked fiber networks. Phys Rev E Stat Phys 86:011923CrossRefGoogle Scholar
  43. Stylianopoulos T, Barocas V (2007) Multiscale, structure-based modeling for the elastic mechanical behavior of arterial walls. J Biomech Eng 129:611–618CrossRefGoogle Scholar
  44. Suki B, Ito S, Stamenovic D, Lutchen KR, Ingenito EP (2005) Biomechanics of the lung parenchyma: critical roles of collagen and mechanical forces. J Appl Physiol 98(5):1892–1899. CrossRefGoogle Scholar
  45. Suki B, Jesudason R, Sato S, Parameswaran H, Araujo AD, Majumdar A, Allen PG, Bartolak-Suki E (2012) Mechanical failure, stress redistribution, elastase activity and binding site availability on elastin during the progression of emphysema. Pulm Pharmacol Ther 25(4):268–275. CrossRefGoogle Scholar
  46. Sun W, Sacks M, Fulchiero G, Lovekamp J, Vyavahare N, Scott M (2004) Response of heterograft heart valve biomaterials to moderate cyclic loading. J Biomed Mater Res A 69:658–669CrossRefGoogle Scholar
  47. Veres SP, Harrison JM, Lee JM (2014) Mechanically overloading collagen fibrils uncoils collagen molecules, placing them in a stable, denatured state. Matrix Biol 33:54–59. CrossRefGoogle Scholar
  48. Voronoi G (1908) Nouvelles applications des paramètres continus à la théorie des formes quadratiques. J Reine Angew Math 133:97–102. MathSciNetzbMATHGoogle Scholar
  49. Wang MC, Guth E (1952) Statistical theory of networks of nongaussian flexible chains. J Chem Phys 20:1144MathSciNetCrossRefGoogle Scholar
  50. Weightman B, Chappell DJ, Jenkins EA (1978) A second study of tensile fatigue properties of human articular cartilage. Ann Rheum Dis 37:58–63CrossRefGoogle Scholar
  51. Weisel J, Nagaswami C (1992) Computer modeling of fibrin polymerization kinetics correlated with electron microscope and turbidity observations: clot structure and assembly are kinetically controlled. Biophys J 63(1):111–128 cited By 199CrossRefGoogle Scholar
  52. Witzenburg CM, Dhume RY, Shah SB, Korenczuk CE, Wagner HP, Alford PW, Barocas VH (2016) Failure of the porcine ascending aorta: multidirectional experiments and a unifying microstructural model. J Biomech Eng 139:031005. CrossRefGoogle Scholar
  53. Zhang L, Lake S, Lai V, Picu C, VH B, MS B (2013) A coupled fiber-matrix model demonstrates highly inhomogeneous microstructural interaction in soft tissues under tensile load. J Biomech Eng 135:011008CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of Biomedical EngineeringUniversity of MinnesotaMinneapolisUSA

Personalised recommendations