Continuum Mechanical Description of an Extrinsic and Autonomous Self-Healing Material Based on the Theory of Porous Media

  • Steffen SpechtEmail author
  • Joachim Bluhm
  • Jörg Schröder
Part of the Advances in Polymer Science book series (POLYMER, volume 273)


Polymers and polymeric composites are used in many engineering applications, but these materials can spontaneously lose structural integrity as a result of microdamage caused by stress peaks during service. This internal microdamage is hard to detect and nearly impossible to repair. To extend the lifetime of such materials and save maintenance costs, self-healing mechanisms can be applied that are able to repair internal microdamage during the usual service load. This can be realized, for example, by incorporating microcapsules filled with monomer and dispersed catalysts into the polymeric matrix material. If a crack occurs, the monomer flows into the crack, reacts with the catalysts, and closes the crack.

This contribution focuses on the development of a thermodynamically consistent constitutive model that is able to describe the damage and healing behavior of a microcapsule-based self-healing material. The material under investigation is an epoxy matrix with microencapsulated dicyclopentadiene healing agents and dispersed Grubbs’ catalysts. The simulation of such a multiphase material is numerically very expensive if the microstructure is to be completely resolved. To overcome this, a homogenization technique can be applied to decrease the computational costs of the simulation. Here, the theoretical framework is based on the theory of porous media, which is a macroscopic continuum mechanical homogenization approach. The developed five-phase model consists of solid matrix material with dispersed catalysts, liquid healing agents, solidified healed material, and gas phase. A discontinuous damage model is used for the description of the damage behavior, and healing is simulated by a phase transition between the liquid-like healing agents and the solidified healed material. Applicability of the developed model is shown by means of numerical simulations of the global damage and healing behavior of a tapered double cantilever beam, as well as simulations of the flow behavior of the healing agents at the microscale.


Multiphase system Phase transition Self-healing Theory of porous media 



This work was supported by the German Research Society (DFG) within the Priority Program SPP 1568 “Design and Generic Principles of Self-healing Materials” under the grant number BL 417/7-2.


  1. 1.
    van der Zwaag S (2007) Self healing materials: an alternative approach to 20 centuries of materials science. Springer, DordrechtCrossRefGoogle Scholar
  2. 2.
    Blaiszik BJ, Kramer SLB, Olugebefola SC, Moore JS, Sottos NR, White SR (2010) Self-healing polymers and composites. Annu Rev Mater Res 40:179–211CrossRefGoogle Scholar
  3. 3.
    van der Zwaag S (2010) Routes and mechanisms towards self healing behaviour in engineering materials. Bull Polish Acad Sci 58:227–236Google Scholar
  4. 4.
    Gosh SK (2009) Self-healing materials: fundamentals, design strategies, and applications. In: Gosh SK (ed) Self-healing materials. Wiley-VCH, Weinheim, pp 1–28Google Scholar
  5. 5.
    Hager MD, Greil P, Leyens C, van der Zwaag S, Schubert US (2010) Self-healing materials. Adv Mater 22:5424–5430CrossRefGoogle Scholar
  6. 6.
    Yuan YC, Yin T, Rong MZ, Zhang MQ (2008) Self healing in polymers and polymer composites. concept, realization and outlook: a review. Express Polym Lett 2:238–250CrossRefGoogle Scholar
  7. 7.
    Grigoleit S (2010) Überblick über Selbstheilende Materialien. Technical report, Frauenhofer-Institut für Naturwissenschaftlich-Technische Trendanalysen (INT)Google Scholar
  8. 8.
    White SR, Sottos NR, Geubelle PH, Moore JS, Kessler MR, Sriram SR, Brown EN, Viswanathan S (2001) Autonomic healing of polymer composites. Nature 409:794–797CrossRefGoogle Scholar
  9. 9.
    Beres W, Koul AK, Thamburaj R (1997) A tapered double-cantilever-beam specimen designed for constant-K testing at elevated temperatures. J Test Eval 25:536–542CrossRefGoogle Scholar
  10. 10.
    Brown EN, Sottos NR, White SR (2002) Fracture testing of a self-healing polymer composite. Exp Mech 42:372–379CrossRefGoogle Scholar
  11. 11.
    Brown EN (2011) Use of the tapered double-cantilever beam geometry for fracture toughness measurements and its applictaion to the quantification of self-healing. J Strain Anal Eng Des 46:167–186CrossRefGoogle Scholar
  12. 12.
    Caruso MM, Blaiszik BJ, White SR, Sottos NR, Moore JS (2008) Full recovery of fracture toughness using a nontoxic solvent based self-healing systems. Adv Funct Mater 18:1898–1904CrossRefGoogle Scholar
  13. 13.
    Guadagno L, Raimondo M, Naddeo C, Longo P, Mariconda A, Binder WH (2014) Healing efficiency and dynamic mechanical properties of self-healing epoxy systems. Smart Mater Struct 23:045001CrossRefGoogle Scholar
  14. 14.
    Raimondo M, Guadagno L (2013) Healing efficiency of epoxy-based materials for structural applications. Polym Compos 34:1525–1532CrossRefGoogle Scholar
  15. 15.
    Barbero EJ, Ford KJ (2007) Characterization of self-healing fiber-reinforced polymer-matrix composite with distributed damage. J Adv Mater 39:20–27Google Scholar
  16. 16.
    Mergheim J, Steinmann P (2013) Phenomenological modelling of self-healing polymers based on integrated healing agents. Comput Mech. doi: 10.1007/s00466-013-0840-0 Google Scholar
  17. 17.
    Schimmel EC, Remmers JJC (2006) Development of a constitutive model for self-healing materials. Technical report, Delft Aerospace Computational ScienceGoogle Scholar
  18. 18.
    Voyiadjis GZ, Shojaei A, Li G, Kattan PI (2012) A theory of anisotropic healing and damage mechanics of materials. Proc R Soc Lond A 468:163–183CrossRefGoogle Scholar
  19. 19.
    Voyiadjis GZ, Shojaei A, Li G (2011) A thermodynamic consistent damage and healing model for self healing materials. Int J Plast 27:1025–1044CrossRefGoogle Scholar
  20. 20.
    Henson GM (2012) Continuum modeling of synthetic microvascular materials. In: Proceedings of the 53rd AIAA structures, dynamics and materials conference, Honolulu, Hawaii. doi: 10.2514/6.2012-2001Google Scholar
  21. 21.
    Maiti S, Shankar C, Geubelle PH, Kieffer J (2006) Continuum and molecular-level modeling of fatigue crack retardation in self-healing polymers. J Eng Mater Technol 128:595–602CrossRefGoogle Scholar
  22. 22.
    Sanada K, Itaya N, Shindo Y (2008) Self-healing of interfacial debonding in fiber-reinforced polymers and effect of microstructure on strength recovery. Open Mech Eng J 2:97–103CrossRefGoogle Scholar
  23. 23.
    Zemskov SV, Jonkers HM, Vermolen FJ (2011) Two analytical models for the probability characteristics of a crack hitting encapsulated particles: application to self-healing materials. Comput Mater Sci 50:3323–3333Google Scholar
  24. 24.
    Yagimli B, Lion A (2011) Experimental investigations and material modelling of curing processes under small deformations. Z Angew Math Mech 91:342–359CrossRefGoogle Scholar
  25. 25.
    de Boer R (2000) Theory of porous media. Springer, BerlinCrossRefGoogle Scholar
  26. 26.
    Bluhm J (2002) Modelling of saturated thermo-elastic porous solids with different phase temperatures. In: Ehlers W, Bluhm J (eds) Porous media. Springer, Berlin, pp 87–118CrossRefGoogle Scholar
  27. 27.
    de Boer R, Ehlers W (1986) Theorie der Mehrkomponentenkontinua mit Anwendung auf bodenmechanische Probleme. Technical report, Universität - Gesamthochschule EssenGoogle Scholar
  28. 28.
    Ehlers W (1996) Grundlegende Konzepte in der Theorie Poröser Medien. Tech Mech 16:63–76Google Scholar
  29. 29.
    Ehlers W (2002) Foundations of multiphasic and porous materials. In: Ehlers W, Bluhm J (eds) Porous media. Springer, Berlin, pp 3–86CrossRefGoogle Scholar
  30. 30.
    Ehlers W (2012) Poröse Medien - ein kontinuummechanisches Modell auf der Basis der Mischungstheorie. Nachdruck der Habilitationsschrift aus dem Jahr 1989, Universität - Gesamthochschule EssenGoogle Scholar
  31. 31.
    Acartürk AY (2009) Simulation of charged hydrated porous materials. PhD thesis, Universität StuttgartGoogle Scholar
  32. 32.
    Truesdell C (1984) Rational thermodynamics, 2nd edn. Springer, New YorkCrossRefGoogle Scholar
  33. 33.
    Bowen RM (1980) Incompressible porous media models by use of the theory of mixtures. Int J Eng Sci 18:1129–1148CrossRefGoogle Scholar
  34. 34.
    Bowen RM (1982) Compressible porous media models by use of the theory of mixtures. Int J Eng Sci 20:697–735CrossRefGoogle Scholar
  35. 35.
    Kachanov LM (1958) Time of the rupture process under creep conditions. Izvestija Akademii Nauk Sojuza Sovetskich Socialisticeskich Republiki (SSSR) Otdelenie Techniceskich Nauk (Moskra) 8:26–31Google Scholar
  36. 36.
    Ateshian GA, Ricken T (2010) Multigenerational interstitial growth of biological tissues. Biomech Model Mechanobiol 9:689–702CrossRefGoogle Scholar
  37. 37.
    Humphrey J, Rajagopal K (2002) A constrained mixture model for growth and remodelling of soft tissues. Math Models Methods Appl Sci 12:407–430CrossRefGoogle Scholar
  38. 38.
    Rodriguez E, Hoger A, McCulloch A (1994) Stress-dependent finite growth in soft elastic tissues. J Biomech 27:455–467CrossRefGoogle Scholar
  39. 39.
    Bluhm J, Specht S, Schröder J (2014) Modeling of self-healing effects in polymeric composites. Arch Appl Mech 85:1469–1481. doi: 10.1007/s00419-014-0946-7 CrossRefGoogle Scholar
  40. 40.
    Ehlers W (1989) On the thermodynamics of elasto-plastic porous media. Arch Mech 41:73–93Google Scholar
  41. 41.
    Simo JC, Pister KS (1984) Remarks on rate constitutive equations for finite deformation problems: computational implications. Comput Methods Appl Mech Eng 46:201–215CrossRefGoogle Scholar
  42. 42.
    Miehe C (1988) Zur numerischen behandlung thermomechanischer Prozesse. PhD thesis, Universität HannoverGoogle Scholar
  43. 43.
    Bluhm J, Ricken T, Bloßfeld M (2011) Ice formation in porous media. In: Markert B (ed) Advances in extended & multifield theories for continua, vol 59, Lecture notes in applied and computational mechanics. Springer, Berlin, pp 153–174CrossRefGoogle Scholar
  44. 44.
    Michalowski RL, Zhu M (2006) Frost heave modelling using porosity rate function. Int J Numer Anal Methods Geomech 30:703–722CrossRefGoogle Scholar
  45. 45.
    Taylor RL (2008) FEAP – a finite element analysis program, Version 8.2. Department of Civil and Environmental Engineering, University of California at Berkeley, Berkeley, CAGoogle Scholar
  46. 46.
    Blaiszik BJ, Sottos NR, White SR (2008) Nanocapsules for self healing materials. Compos Sci Technol 68:978–986CrossRefGoogle Scholar
  47. 47.
    Alzari V, Nuvoli D, Sanna D, Ruiu A, Mariani A (2015) Effect of limonene on the frontal ring opening metathesis polymerization of dicyclopentadiene. J Polym Sci A Polym Chem. doi: 10.1002/pola.27776 Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Steffen Specht
    • 1
    Email author
  • Joachim Bluhm
    • 1
  • Jörg Schröder
    • 1
  1. 1.Institute of MechanicsUniversity of Duisburg-EssenEssenGermany

Personalised recommendations