Governing Differential Equations for the Mechanics of Undamageable Materials

• Autorzy: Voyiadjis G. Z. , Kattan P. I.
• Języki publikacji: EN

Abstrakty

EN

In this work the mathematical foundations of the mechanics of elastic undamageable materials are presented. In particular the governing differential equations are derived for both the scalar and tensorial cases. In the isotropic case it is found that the resulting scalar differential equations are simple and easy to solve. However, in the anisotropic case the tensorial differential equations are complicated and unsolvable at this time. The current work presents the solution in the form of explicit nonlinear stress-strain relations for the simple one-dimensional case. However, the general solution of the three-dimensional case remains unattainable at the present time. Only the governing tensorial differential equations are derived for this latter case. It is to be noted that the term “undamageable” is reflected in the context of the material stiffness and not the property of indestructibility due to various loading conditions. Thus, the undamageable material reflects that no microcracks or microvoids occur as well as no plastic yielding in the material. To illustrate this concept, a last section is added on applications.

Słowa kluczowe

EN

damage; damage mechanics; undamageable material; differential equation; scalar; tensor; rubber; biological tissue; tissue; soft materials; metallic glass; metal rubber

PL

uszkodzenia; mechanizm uszkodzenia; równanie różniczkowe; napinacz; tensor; tkanka biologiczna; tkanka; szkło metaliczne

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Engineering Transactions
• Rocznik: 2014
• Tom: Vol. 62, iss. 3
• Strony: 241–--267
• Opis fizyczny: Bibliogr. 42 poz., wykr.

Twórcy

autor

Voyiadjis G. Z.

• Department of Civil and Environmental Engineering Louisiana State University Baton Rouge, LA 70803, USA

autor

Kattan P. I.

• Department of Civil and Environmental Engineering Louisiana State University Baton Rouge, LA 70803, USA

Bibliografia

• 1. Arruda E.M., Boyce M.C., A Three-Dimensional Constitutive Model for the Large Stretch Behavior of Rubber Elastic Materials, Journal of the Mechanics and Physics of Solids, Vol. 41, 2, 389–412, 1993.
• 2. Castelvecchi D., Glass-like Metal Performs Better Under Stress, Physical Review Focus, 15, 20, 1103/PhysRevFocus.15.20, 2005.
• 3. Celentano D.J., Tapia P.E., Chaboche J-L., Experimental and Numerical Characterization of Damage Evolution in Steels, Mecanica Computacional, Vol. XXIII, (edited by G. Buscaglia, E. Dari, O. Zamonsky), Bariloche, Argentina, 2004.
• 4. Corless R.M., Gonnet G.H, Hare D.E.G., Jeffrey D.J., Knuth E.E., On the Lambert W Function, Advances in Computational Mathematics, 5, 329–359, 1996.
• 5. Das J., Tang M.B., Kim K.B., Theissmann R., Baier F., Wang W.H., Eckert J., Work-Hardenable” Ductile Bulk Metallic Glass, Physical Review Letters, 94, 205501, DOI: 1103/PhysRevLett.94.205501, 2005.
• 6. Doghri I., Mechanics of Deformable Solids: Linear and Nonlinear, Analytical and Computational Aspects, Springer-Verlag, Germany, 2000.
• 7. Fung Y.C., Elasticity of Soft Tissues in Simple Elongation, American Journal of Physiology, 213, 6, 1432–1544, 1967.
• 8. Fung Y.C., Biomechanics: Mechanical Properties of living Tissues, Second Edition, Springer, 1993.
• 9. Hansen N.R., Schreyer H.L., A Thermodynamically Consistent Framework for Theories of Elastoplasticity Coupled with Damage, International Journal of Solids and Strucutres, 31, 3, 359–389, 1994.
• 10. Kachanov L., On the Creep Fracture Time [in Russian], Izv Akad, Nauk USSR Otd Tech., 8, 26–31, 1958.
• 11. Kattan P.I., Voyiadjis G.Z., A Coupled Theory of Damage Mechanics and Finite Strain Elasto-Plasticity – Part I: Damage and Elastic Deformations, International Journal of Engineering Science, 28, 5, 421–435, 1990.
• 12. Kattan P.I., Voyiadjis G.Z., A Plasticity-Damage Theory for Large Deformation of Solids – Part II: Applications to Finite Simple Shear, International Journal of Engineering Science, 31, 1, 183–199, 1993.
• 13. Kattan P.I., Voyiadjis G.Z., Decomposition of Damage Tensor in Continuum Damage Mechanics, Journal of Engineering Mechanics, ASCE, 127, 9, 940–944, 2001.
• 14. Kattan P.I., Voyiadjis G.Z., Damage Mechanics with Finite Elements: Practical Applications with Computer Tools, Springer-Verlag, Germany, 2001.
• 15. Ladeveze P., Poss M., Proslier L., Damage and Fracture of Tridirectional Composites, [in:] Progress in Science and Engineering of Composites. Proceedings of the Fourth International Conference on Composite Materials, Japan Society for Composite Materials, 1, 649–658, 1982.
• 16. Ladeveze P., Lemaitre J., Damage Effective Stress in Quasi-Unilateral Conditions, The 16th International Cogress of Theoretical and Applied Mechanics, Lyngby, Denmark, 1984.
• 17. Lee H., Peng K., Wang J., An Anisotropic Damage Criterion for Deformation Instability and its Application to Forming Limit Analysis of Metal Plates, Engineering Fracture Mechanics, 21, 1031–1054, 1985.
• 18. Lubineau G., A Pyramidal Modeling Scheme for Laminates – Identification of Transverse Cracking, International Journal of Damage Mechanics,
• 19, 4, 499–518, 2010. 19. Lubineau G., Ladeveze P., Construction of a Micromechanics-based Intralaminar Mesomodel, and Illustrations in ABAQUS/Standard, Computational Materials Science, 43, 1, 137–145, 2008.
• 20. Luccioni B., Oller S., A Directional Damage Model, Computer Methods in Applied Mechanics and Engineering, 192, 1119–1145, 2003.
• 21. Martin R., Rekondo A., Echeberria J., Cabanero G., Grande H.J., Odriozola I., Room Temperature Self-healing Power of Silicone Elastomers Having Silver Nanoparticles as Crosslinkers, Chemical Communications, 48, 66, 8255, DOI: 10.1039/c2cc32030d, 2012.
• 22. Nichols J.M., Abell A.B., Implementing the Degrading Effective Stiffness of Masonry in a Finite Element Model, North American Masonry Conference, Clemson, South Carolina, USA, 2003.
• 23. Nichols J.M., Totoev Y.Z., Experimental Investigation of the Damage Mechanics of Masonry Under Dynamic In-plane Loads, North American Masonry Conference, Austin, Texas, USA, 1999.
• 24. Rabotnov Y., Creep Rupture, [in:] Proceedings, Twelfth International Congress of Applied Mechanics, M. Hetenyi and W.G. Vincenti [Eds.], Stanford, 1968, Springer-Verlag, Berlin, pp. 342–349, 1969.
• 25. Rekondo A., Martin R., de Luzuriaga A.R., Cabanero G., Grande H.J., Odriozola I., Catalyst-free Room Temperature Self-healing Elastomers Based on Aromatic Disulfide Metasthesis, Materials Horizons, DOI: 10.1039/c3mh00061c, 2013.
• 26. Science Daily, Chemists Create Self-assembling Conductive Rubber, ScienceDaily, Published online at http://www.sciencedaily.com/videos/2007/0409-metal rubber.htm, 2007.
• 27. Science Daily, ‘Terminator’ Polymer: Self-healing Polymer That Spontaneously and Indpendently Repairs Itself, ScienceDaily, (appeared on September 13, 2013 online), http://www.sciencedaily.com/releases/2013/09/130913101819.htm, 2013.
• 28. Sidoroff F., Description of Anisotropic Damage Application in Elasticity, [in:] IUTAM Colloqium on Physical Nonlinearities in Structural Analysis, pp. 237–244, Springer-Verlag, Berlin, 1981.
• 29. Templeton G., ‘Terminator’ Polymer Can Spontaneously Self-heal in Just Two Hours, ExtremeTech, Published online at http://www.extremetech.com/extreme/166656- terminator-polymer-can-spontaneously-self-heal-in-just-two-hours, 2013.
• 30. Voyiadjis G.Z., Degradation of Elastic Modulus in Elastoplastic Coupling with Finite Strains, International Journal of Plasticity, 4, 335–353, 1988.
• 31. Voyiadjis G.Z., Kattan P.I., A Coupled Theory of Damage Mechanics and Finite Strain Elasto-Plasticity – Part II: Damage and Finite Strain Plasticity, International Journal of Engineering Science, 28, 6, 505–524, 1990.
• 32. Voyiadjis G.Z., Kattan P.I., A Plasticity-Damage Theory for Large Deformation of Solids – Part I: Theoretical Formulation, International Journal of Engineering Science, 30, 9, 1089–1108, 1992.
• 33. Voyiadjis G.Z., Kattan P.I., Damage Mechanics, Taylor and Francis (CRC Press), 2005.
• 34. Voyiadjis G.Z., Kattan P.I., Advances in Damage Mechanics: Metals and Metal Matrix Composites with an Introduction to Fabric Tensors, Second Edition, Elsevier, 2006.
• 35. Voyiadjis G.Z., Kattan P.I., A New Fabric-Based Damage Tensor, Journal of the Mechanical Behavior of Materials, 17, 1, 31–56, 2006.
• 36. Voyiadjis G.Z., Kattan P.I., Damage Mechanics with Fabric Tensors, Mechanics of Advanced Materials and Structures, 13, 4, 285–301, 2006.
• 37. Voyiadjis G.Z., Kattan P.I., A Comparative Study of Damage Variables in Continuum Damage Mechanics, International Journal of Damage Mechanics, 18, 4, 315–340, 2009.
• 38. Voyiadjis G.Z., Kattan P.I., A New Class of Damage Variables in Continuum Damage Mechanics, Journal of Engineering Materials and Technology, ASME, in press, 2011.
• 39. Voyiadjis G.Z., Kattan P.I., Introduction to the Mechanics and Design of Undamageable Materials, International Journal of Damage Mechanics, 22, 3, 323–335, 2012.
• 40. Voyiadjis G.Z., Kattan P.I., On the Theory of Elastic Undamageable Materials, Journal of Engineering Materials and Technology, ASME, Special Issue on Modeling Material Behavior at Multiple Scales, Xi Chen [Ed.], 135, 2, 021002-1–021002-6, 2013.
• 41. Xu G.Q., Demkowicz M.J., Healing of Nanocracks by Disclinations, Physical Review Letters, 111, 145501, DOI: 10.1103/PhysRevLett.111.145501, 2013.
• 42. Zhang B., Zhao D.Q., Pan M.X., Wang W.H., Greer A.L., Amorphous Metallic Plastic, Physical Review Letters, 94, 205502, DOI: 10.1103/PhysRevLett.94.205502, 2005.