Handbook of Damage Mechanics pp 1493-1513 | Cite as

# Thermodynamics of Continuum Damage Healing Mechanics

## Abstract

In this chapter, the governing thermodynamic laws on the damage and healing processes are revisited. The solid mechanics thermodynamic framework provides a physically consistent description for the deformation mechanisms in solids, and it has been widely examined for the plasticity and damage processes in the literature (S. Yazdani, H.L. Schreyer, Combined plasticity and damage mechanics model for plain concrete. J. Eng. Mech. **116**(7), 1435–1450 (1990); J.L. Chaboche, On some modifications of kinematic hardening to improve the description of ratchetting effects. Int. J. Plast. **7**(7), 661–678 (1991); J.L. Chaboche, Cyclic viscoplastic constitutive equations, part I: a thermodynamically consistent formulation. J. Appl. Mech. **60**(4), 813–821 (1993); N.R. Hansen, H.L. Schreyer, A thermodynamically consistent framework for theories of elastoplasticity coupled with damage. Int. J. Solids Struct. **31**(3), 359–389 (1994); G. Voyiadjis, I. Basuroychowdhury, A plasticity model for multiaxial cyclic loading and ratchetting. Acta Mech. **126**(1), 19–35 (1998); J.L. Chaboche, A review of some plasticity and viscoplasticity constitutive theories. Int. J. Plast. **24**(10), 1642–1693 (2008)). Introduction of the healing process into the thermodynamic framework was formerly proposed by Voyiadjis et al. (A thermodynamic consistent damage and healing model for self healing materials. Int. J. Plast. **27**(7), 1025–1044 (2011)) where a physically consistent description for the healing process is provided.

Basically, the mathematical foundation of the thermodynamic-based solid mechanics modeling was developed formerly for capturing plasticity and damage in metallic structures and it is not directly applicable to polymeric materials. Polymers usually show strain softening after their initial yield and they show strain hardening at higher strain levels. To overcome the mathematical deficiency associated with the classical thermodynamic framework, Voyiadjis, Shojaei, and Li (A generalized coupled viscoplastic- viscodamage- viscohealing theory for glassy polymers. Int. J. Plast. **28**(1), 21–45 (2012a)) established a generalized formulation within the thermodynamic framework in which the mathematical competency for simulating the most nonlinear viscoplastic, viscodamage, and viscohealing effects in polymers was enhanced. They have successfully shown that the proposed framework is able to accurately capture the viscoplastic and viscodamage responses of polymers and the model has enough flexibility to capture the healing response in polymeric-based self-healing materials.

### Keywords

Entropy Polyethylene Terephthalate### References

- R.J. Asaro, V. Lubarda,
*Mechanics of Solids and Materials*(Cambridge University Press, New York, 2006)CrossRefGoogle Scholar - M.F. Ashby, C. Gandhi, D.M.R. Taplin, Overview No. 3 fracture-mechanism maps and their construction for f.c.c. metals and alloys. Acta Metall.
**27**(5), 699–729 (1979)CrossRefGoogle Scholar - E.J. Barbero, F. Greco, P. Lonetti, Continuum damage-healing mechanics with application to self-healing composites. Int. J. Damage Mech.
**14**(1), 51–81 (2005)CrossRefGoogle Scholar - V.A. Beloshenko, V.N. Varyukhin, Y.V. Voznyak, The shape memory effect in polymers. Russ. Chem. Rev.
**74**(3), 265 (2005)CrossRefGoogle Scholar - J.L. Chaboche, On some modifications of kinematic hardening to improve the description of ratchetting effects. Int. J. Plast.
**7**(7), 661–678 (1991)CrossRefGoogle Scholar - J.L. Chaboche, Cyclic viscoplastic constitutive equations, part I: a thermodynamically consistent formulation. J. Appl. Mech.
**60**(4), 813–821 (1993)CrossRefMATHGoogle Scholar - J.L. Chaboche, Thermodynamic formulation of constitutive equations and application to the viscoplasticity and viscoelasticity of metals and polymers. Int. J. Solids Struct.
**34**(18), 2239–2254 (1997)CrossRefMATHGoogle Scholar - J.L. Chaboche, A review of some plasticity and viscoplasticity constitutive theories. Int. J. Plast.
**24**(10), 1642–1693 (2008)CrossRefMATHGoogle Scholar - C. G’Sell, J.M. Hiver, A. Dahoun, Experimental characterization of deformation damage in solid polymers under tension, and its interrelation with necking. Int. J. Solids Struct.
**39**, 3857–3872 (2002)Google Scholar - N.R. Hansen, H.L. Schreyer, A thermodynamically consistent framework for theories of elastoplasticity coupled with damage. Int. J. Solids Struct.
**31**(3), 359–389 (1994)CrossRefMATHGoogle Scholar - A.S. Khan, M. Baig, Anisotropic responses, constitutive modeling and the effects of strain-rate and temperature on the formability of an aluminum alloy. Int. J. Plast.
**27**(4), 522–538 (2011)CrossRefMATHGoogle Scholar - A.S. Khan, A. Pandey, T. Stoughton, Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part II: a very high work hardening aluminum alloy (annealed 1100 Al). Int. J. Plast.
**26**(10), 1421–1431 (2010a)CrossRefMATHGoogle Scholar - A.S. Khan, A. Pandey, T. Stoughton, Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part III: yield surface in tension-tension stress space (Al 6061-T 6511 and annealed 1100 Al). Int. J. Plast.
**26**(10), 1432–1441 (2010b)CrossRefMATHGoogle Scholar - E.L. Kirkby, V.J. Michaud, J.A.E. Månson, N.R. Sottos, S.R. White, Performance of self-healing epoxy with microencapsulated healing agent and shape memory alloy wires. Polymer
**50**(23), 5533–5538 (2009)CrossRefGoogle Scholar - H. Lee, K. Peng, J. Wang, An anisotropic damage criterion for deformation instability and its application to forming limit analysis of metal plates. Eng. Fract. Mech.
**21**(5), 1031–1054 (1985)CrossRefGoogle Scholar - G. Li, D. Nettles, Thermomechanical characterization of a shape memory polymer based self-repairing syntactic foam. Polymer
**51**(3), 755–762 (2010)CrossRefGoogle Scholar - G. Li, A. Shojaei, A viscoplastic theory of shape memory polymer fibres with application to self-healing materials. Proc. R. Soc. A
**468**(2144), 2319–2346 (2012). doi:10.1098/rspa.2011.0628MathSciNetCrossRefGoogle Scholar - J. Lubliner, On the thermodynamic foundations of non-linear solid mechanics. Int. J. Non-Linear Mech.
**7**(3), 237–254 (1972)CrossRefMATHGoogle Scholar - S. Miao, M.L. Wang, H.L. Schreyer, Constitutive models for healing of materials with application to compaction of crushed rock salt. J. Eng. Mech.
**121**(10), 1122–1129 (1995)CrossRefGoogle Scholar - R.W. Rice, S.W. Freiman, J.J. Mecholsky, The dependence of strength-controlling fracture energy on the flaw-size to grain-size ratio. J. Am. Ceram. Soc.
**63**(3–4), 129–136 (1980)CrossRefGoogle Scholar - A. Shojaei, G. Li, G.Z. Voyiadjis, Cyclic viscoplastic-viscodamage analysis of shape memory polymers fibers with application to self-healing smart materials. J. Appl. Mech.
**80**(1), 011014–011015 (2013a)CrossRefGoogle Scholar - A. Shojaei, G.Z. Voyiadjis, P.J. Tan, Viscoplastic constitutive theory for brittle to ductile damage in polycrystalline materials under dynamic loading. Int. J. Plast. (2013b). doi:10.1016/j.ijplas.2013.02.009Google Scholar
- J.C. Simo, T.J.R. Hughes, Computational inelasticity. New York, Springer (1997)Google Scholar
- M.S. Sivakumar, G.Z. Voyiadjis, A simple implicit scheme for stress response computation in plasticity models. Journal of Computational Mechanics.
**20**(6), 520–529 (1997)Google Scholar - V. Tvergaard, J.W. Hutchinson, The relation between crack growth resistance and fracture process parameters in elastic–plastic solids. J. Mech. Phys. Solids
**40**(6), 1377–1397 (1992)CrossRefMATHGoogle Scholar - G.Z. Voyiadjis, R.K. Abu Al-Rub, Thermodynamic based model for the evolution equation of the backstress in cyclic plasticity. Int. J. Plast.
**19**(12), 2121–2147 (2003)CrossRefMATHGoogle Scholar - G. Voyiadjis, I. Basuroychowdhury, A plasticity model for multiaxial cyclic loading and ratchetting. Acta Mech.
**126**(1), 19–35 (1998)CrossRefMATHGoogle Scholar - G.Z. Voyiadjis, M. Foroozesh, Anisotropic distortional yield model. J. Appl. Mech.
**57**(3), 537–547 (1990)CrossRefGoogle Scholar - Z. Voyiadjis, P.I. Kattan,
*Advances in Damage Mechanics*(Elsevier, London, 2006)Google Scholar - G.Z. Voyiadjis, P.I. Kattan, A comparative study of damage variables in continuum damage mechanics. Int. J. Damage Mech.
**18**(4), 315–340 (2009)CrossRefGoogle Scholar - G.Z. Voyiadjis, G. Pekmezi, B. Deliktas, Nonlocal gradient-dependent modeling of plasticity with anisotropic hardening. Int. J. Plast.
**26**(9), 1335–1356 (2010)CrossRefMATHGoogle Scholar - G.Z. Voyiadjis, A. Shojaei, G. Li, A thermodynamic consistent damage and healing model for self healing materials. Int. J. Plast.
**27**(7), 1025–1044 (2011)CrossRefMATHGoogle Scholar - G.Z. Voyiadjis, A. Shojaei, G. Li, A generalized coupled viscoplastic- viscodamage- viscohealing theory for glassy polymers. Int. J. Plast.
**28**(1), 21–45 (2012a)CrossRefGoogle Scholar - G.Z. Voyiadjis, A. Shojaei, G. Li, P. Kattan, Continuum damage-healing mechanics with introduction to new healing variables. Int. J. Damage Mech.
**21**(3), 391–414 (2012b)CrossRefGoogle Scholar - G.Z. Voyiadjis, A. Shojaei, G. Li, P.I. Kattan, A theory of anisotropic healing and damage mechanics of materials. Proc. R. Soc. A Math. Phys. Eng. Sci.
**468**(2137), 163–183 (2012c). doi:10.1098/rspa.2011.0326MathSciNetCrossRefMATHGoogle Scholar - S.R. White, N.R. Sottos, P.H. Geubelle, J.S. Moore, M.R. Kessler, S.R. Sriram, E.N. Brown, S. Viswanathan, Autonomic healing of polymer composites. Nature
**409**(6822), 794–797 (2001)CrossRefGoogle Scholar - S. Yazdani, H.L. Schreyer, Combined plasticity and damage mechanics model for plain concrete. J. Eng. Mech.
**116**(7), 1435–1450 (1990)CrossRefGoogle Scholar