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International Conference Continuum Mechanics Focusing on Singularities

CoMFoS 2016: Mathematical Analysis of Continuum Mechanics and Industrial Applications II pp 115–124Cite as

Brief Introduction to Damage Mechanics and Its Relation to Deformations

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Part of the book series: Mathematics for Industry ((MFI,volume 30))

Abstract

We discuss some principle concepts of damage mechanics and outline a possibility to address the open question of the damage-to-deformation relation by suggesting a parameter identification setting. To this end, we introduce a variable motivated by the physical damage phenomenon and comment on its accessibility through measurements. We give an extensive survey on analytic results and present an isotropic irreversible partial damage model in a dynamic mechanical setting in form of a second-order hyperbolic equation coupled with an ordinary differential equation for the damage evolution. We end with a note on a possible parameter identification setting.

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Notes

  1. 1.

    Especially in the mechanics of solids, there are so-called anisotropic properties, whose response to external influences depends on the direction of those influences. Properties that do not show this directional dependency are referred to as isotropic.

  2. 2.

    This description separates damage mechanics from fracture or crack mechanics, respectively, which focuses on crack propagation rather than initiation.

  3. 3.

    Analytic results of the Kachanov-Rabotnov model are discussed in [1, 38], e.g.

  4. 4.

    Partial damage refers to a damage state \(0\le d\le \delta _0<1\), where \(\delta _0\) denotes the maximal damage possible. Partial damage models are more feasible for mathematical analysis, because they preserve uniform coercivity of the effective elasticity tensor, whereas complete damage leads to a degenerate momentum balance equation.

  5. 5.

    This approach was also continued in some subsequent publications of the author, e.g. in the analysis of a contact problem in an otherwise equal setting in [8] or a dynamic frictional contact problem of a visco-elastic material with damage in [7]. Both these articles further developed the ideas presented in [10].

  6. 6.

    For numeric investigations and results see [9].

  7. 7.

    This is why, as a long-term project, we aim to identify g. The only assumptions on g stem from the regularity needed to perform the analysis (see [18]).

References

  1. Altenbach, H., Deuring, P., Naumenko, K.: A system of ordinary and partial differential equations describing creep behavior of thin-walled shells. Z. Ana. Anw. 18, 1003–1030 (1999)

    Article  MATH  Google Scholar 

  2. Bouchitté, G., Mielke, A., Roubíček, T.: A complete-damage problem at small strains. Z. Angew. Math. Phys 60, 205–236 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bonetti, E., Bonfanti, G.: Well-posedness results for a model of damage in thermoviscoelastic materials. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25, 1187–1208 (2008)

    Google Scholar 

  4. Bonetti, E., Schimperna, G., Segatti, A.: On a doubly nonlinear model for the evolution of damaging in viscoelastic materials. J. Differ. Equ. 218, 91–116 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bonetti, E., Schimperna, G.: Local existence for Frémond’s model of damage in elastic materials. Contin. Mech. Thermodyn. 16, 319–335 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bonetti, E., Frémond, M.: Damage theory: microscopic effects of vanishing macroscopic motions. Comput. Appl. Math. 22, 313–333 (2003)

    MathSciNet  MATH  Google Scholar 

  7. Campo, M.A., Fernández, J.R., Kuttler, K.L.: Analysis of a dynamical frictional contact problem with damage. Fin. El. Anal. Des. 45, 659–674 (2009)

    Article  Google Scholar 

  8. Campo, M.A., Fernández, J.R., Kuttler, K.L.: An elastic-viscoplastic quasistatic contact problem with damage. Comput. Methods Appl. Mech. Engrg 196, 3219–2339 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Campo, M.A., Fernández, J.R., Kuttler, K.L., Shillor, M.: Quasistatic evolution of damage in an elastic body: numerical analysis and computational experiments. Appl. Numer. Math. 57, 976–988 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Campo, M.A., Fernández, J.R., Han, W., Sofonea, M.: A dynamic viscoelastic contact problem with normal compliance and damage. Fin. Elem. Anal. Des. 42, 1–24 (2005)

    Article  MathSciNet  Google Scholar 

  11. Ciarlet, P.: Mathematical Elasticity, vol. I. North-Holland, Amsterdam (1988)

    MATH  Google Scholar 

  12. Farshbaf-Shaker, M.H., Heinemann, C.: A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media. Math. Models Methods Appl. Sci. 25, 2749–2793 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fiaschi, A., Knees, D., Stefanelli, U.: Young-measure quasi-static damage evolution. Arch. Rational Mech. Anal. 203, 415–453 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Francfort, G.A., Garroni, A.: A variational view of partial brittle damage evolution. Arch. Ration. Mech. Anal. 182, 125–152 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Frémond, M., Kuttler, K.L., Shillor, M.: Existence and uniqueness of solutions for a dynamic one-dimensional damage model. J. Math. Anal. Appl. 229, 271–294 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Frémond, M., Nedjar, B.: Damage, gradient of damage and principle of virtual power. Int. J. Sol. Struct. 33, 1083–1103 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Frémond, M., Nedjar, B.: Damage in concrete: the unilateral phenomenon. Nucl. Eng. Des. 156, 323–335 (1996)

    Article  Google Scholar 

  18. Grützner, S.: An approach to parameter identification in damaged continua. Diploma thesis, University of Bremen (2015)

    Google Scholar 

  19. Heinemann, C., Kraus, C.: Complete damage in linear elastic materials: modeling, weak formulation and existence results. Calc. Var. Partial Differ. Equ. 54, 217–250 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Heinemann, C., Kraus, C.: Existence of weak solutions for a hyperbolic-parabolic phase field system with mixed boundary conditions on nonsmooth domains. SIAM J. Math. Anal. 47, 2044–2073 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Heinemann, C., Kraus, C.: Existence of weak solutions for a pde system describing phase separation and damage processes including inertial effects. Discrete Contin. Dynam. Syst. 35, 2565–2590 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Heinemann, C., Kraus, C.: Degenerating Cahn-Hilliard systems coupled with mechanical effects and complete damage processes. Math. Bohem. 139, 315–331 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Heinemann, C., Kraus, C.: Existence results for diffuse interface models describing phase separation and damage. Eur. J. Appl. Math 24, 179–211 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Heinemann, C., Kraus, C.: Existence of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 21, 321–359 (2011)

    MathSciNet  MATH  Google Scholar 

  25. Kogut, P.I., Leugering, G.: Optimal and approximate boundary controls of an elastic body with quasistatic evolution of damage. Math. Methods Appl. Sci. 38, 2739–2760 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kraus, C., Bonetti, E., Heinemann, C., Segatti, A.: Modeling and analysis of a phase field system for damage in phase separation processes in solids. J. Differ. Equ. 258, 3928–3959 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kuttler, K.L., Shillor, M.: Quasistatic evolution of damage in an elastic body. Nonlinear Anal. Real World Appl. 7, 674–699 (2006)

    Google Scholar 

  28. Kuttler, K.L.: Quasistatic evolution of damage in an elastic-viscoplastic material. Elec. J. Differ. Equ. 2005, 1–25 (2005)

    MathSciNet  MATH  Google Scholar 

  29. Lazzaroni, G., Rossi, R., Thomas, M., Toader, R.: Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics. J. Phys. Conf. Ser. 727 (2016)

    Google Scholar 

  30. Lemaître, J., Chaboche, J.-.L: Mechanics of Solid Materials. Cambridge University Press, New York (1990)

    Google Scholar 

  31. Lemaître, J., Dufailly, J.: Damage measurements. Eng. Fract. Mech. 28, 643–661 (1987)

    Article  Google Scholar 

  32. Mainik, A., Mielke, A.: Existence results for energetic models for rate-independent systems. Calc. Var. Partial Differ. Equ. 22, 73–99 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Mielke, A., Roubíček, T.: Rate-independent damage processes in nonlinear elasticity. Math. Mod. Methods Appl. Sci. 16, 177–209 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mielke, A.: Evolution of rate-independent systems. In: Dafermos, C.M., Feireisl, E. (eds.) Handbook of Differential Equations. 461–559. Elsevier, Amsterdam (2005)

    Google Scholar 

  35. Murakami, S.: Continuum Damage Mechanics. Springer, Heidelberg (2012)

    Book  Google Scholar 

  36. Nedjar, B.: Damage and gradient of damage in transient dynamics. In: IUTAM Argoul, P., Frémond, M., Nguyen, Q.S. (eds.) IUTAM Symposium on Variations of Domain and Free-Boundary Problems in Solid Mechanics. Proceedings of the IUTAM Symosium held in Paris, France, April 22-25, 1997; 189–196. Springer, Berlin (1999)

    Google Scholar 

  37. Roubíček, T., Tomassetti, G.: Thermomechanics of damageable materials under diffusion: modelling and analysis. Z. Angew. Math. Phys. 66, 3535–3572 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. Shutov, A.V., Sändig, A.-M.: Mathematical analysis of fully coupled approach to creep damage. arXiv:math-ph/0601052v1 (2006)

  39. Thomas, M.: Quasistatic damage evolution with spatial BV-regularization. Disc. Cont. Dyn. Syst. 6, 235–255 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  40. Thomas, M., Mielke, A.: Damage of nonlinearly elastic materials at small strain - existence and regularity results. Z. Angew. Math. Mech. 90, 88–112 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Voyiadjis, G.Z., Kattan, P.I.: Advances in Damage Mechanics. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

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Acknowledgements

The authors are indebted to Michael Böhm for initiating and supporting this research.

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Authors and Affiliations

  1. ZeTeM - Center for Industrial Mathematics, University of Bremen, Bremen, Germany

    Simon Grützner

  2. Department of Mathematics and Computer Science, Karlstad University, Karlstad, Sweden

    Adrian Muntean

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  1. Simon Grützner
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Correspondence to Simon Grützner .

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Editors and Affiliations

  1. Kanazawa University, Kanazawa, Ishikawa, Japan

    Patrick van Meurs

  2. Kanazawa University, Kanazawa, Ishikawa, Japan

    Masato Kimura

  3. Kanazawa University, Kanazawa, Ishikawa, Japan

    Hirofumi Notsu

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Grützner, S., Muntean, A. (2018). Brief Introduction to Damage Mechanics and Its Relation to Deformations. In: van Meurs, P., Kimura, M., Notsu, H. (eds) Mathematical Analysis of Continuum Mechanics and Industrial Applications II. CoMFoS 2016. Mathematics for Industry, vol 30. Springer, Singapore. https://doi.org/10.1007/978-981-10-6283-4_10

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  • DOI: https://doi.org/10.1007/978-981-10-6283-4_10

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