Engineering Transactions, 65, 2, pp. 351–369, 2017

A Generalized Hypothesis of Elastic Energy Equivalence in Continuum Damage Mechanics

George Z. VOYIADJIS
Louisiana State University
United States

Peter I. KATTAN
Independent Researcher
Jordan

A new generalized hypothesis of elastic energy equivalence is proposed. The proposed generalized hypothesis is inclusive of all the existing different hypotheses of equivalence in continuum damage mechanics and all are obtained as special cases. Specifically, the hypothesis of elastic strain equivalence and the hypothesis of elastic energy equivalence are obtained as special cases of the generalized hypothesis proposed here. In addition, the generalized hypothesis has some unusual properties when the integer exponent n approaches infinity. In particular, it turns out that the strain energy density function is a vector for even values of the integer exponent. This conclusion is totally unexpected but an attempt is made to explain this result based on geometry.
Keywords: damage mechanics; energy equivalence; generalized hypothesis;thermodynamics; vector exponentiation
Full Text: PDF

References

Basaran C., Nie S., An irreversible thermodynamic theory for damage mechanics of solids, International Journal of Damage Mechanics, 13(3): 205-224, 2004.

Basaran C., Yan C.Y., A thermodynamic framework for damage mechanics of solder joints, Trans. of ASME, Journal of Electronic Packaging, 120, pp. 379-384, 1998.

Bower A. F., Applied mechanics of solids, CRC Press, 2009.

Bronson R., Schaum’s outline of matrix operations, 2nd ed., McGraw-Hill Education, 2011.

Celentano D.J., Tapia P.E., Chaboche J-L., Experimental and numerical characterization of damage evolution in steels, Mecanica Computacional, Vol. XXIII (G. Buscaglia, E. Dari, O. Zamonsky Eds.), Bariloche, Argentina, 2004.

Cullen C.G., Matrices and linear transformations, 2nd ed., Dover Books on Mathematics, 1990.

Doghri I., Mechanics of deformable solids: linear and nonlinear, analytical and computational aspects, Springer-Verlag, 2000.

Golub G.H., Matrix computations, 4th ed, Johns Hopkins University Press, 2012.

Hansen N.R., Schreyer H.L., A thermodynamically consistent framework for theories of elastoplasticity coupled with damage, International Journal of Solids and Structures, 31(3): 359–389, 1994.

Hartman G., Fundamentals of Matrix Algebra, 3rd ed., Createspace Independent Publishing Platform, 2011.

Harville D.A., Matrix algebra: exercises and solutions, Springer, 2013.

Horn R.A., Johnson C.R., Matrix Analysis, 2nd ed., Cambridge University Press, 2012.

Kachanov L., On the Creep Fracture Time [in Russian], Izv Akad Nauk USSR Otd Tech., Vol. 8, pp. 26–31, 1958.

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.

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.

Kattan P.I., Voyiadjis G.Z., Damage Mechanics with Finite Elements: Practical Applications with Computer Tools, Springer-Verlag, Germany, 2001.

Kattan P.I., Voyiadjis G.Z., Decomposition of damage tensor in continuum damage mechanics, Journal of Engineering Mechanics, ASCE, 127(9): 940–944, 2001.

Krajcinovic D., Damage mechanics, North Holland, 1996.

Ladeveze P., Lemaitre J., Damage effective stress in quasi-unilateral conditions, The 16th International Cogress of Theoretical and Applied Mechanics, Lyngby, Denmark, 1984.

Ladeveze P., Poss M., Proslier L., Damage and fracture of tridirectional composites, [in:] Progress in Science and Engineering of Composites. Proceedings of the 4th International Conference on Composite Materials, Japan Society for Composite Materials, Vol. 1, pp. 649-658, 1982.

Laub A.J., Matrix analysis for scientists and engineers, SIAM, Society for Industrial and Applied Mathematics, 2004.

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.

Lubineau G., A Pyramidal modeling scheme for laminates – identification of transverse cracking, International Journal of Damage Mechanics, 19(4): 499–518, 2010.

Lubineau G., Ladeveze P., Construction of a micromechanics-based intralaminar mesomodel, and illustrations in ABAQUS/Standard, Computational Materials Science, 43(1): 137–145, 2008.

Luccioni B., Oller S., A directional damage model, Computer Methods in Applied Mechanics and Engineering, 192: 1119–1145, 2003.

Meyer C.D., Matrix algebra and applied linear algebra, SIAM, Society for Industrial and Applied Mathematics, 2001.

Rabotnov Y., Creep rupture, [in:] Proceedings of 12th International Congress of Applied Mechanics, M. Hetenyi, W. G. Vincenti [Eds.], Springer-Verlag, Berlin, pp. 342-349, 1969.

Rice J.R., Inelastic constitutive relations for solids: an internal variable theory and its application to metal plasticity, Journal of the Mechanics and Physics of Solids, 19: 433-455, 1971.

Schneider H., Barker G.P., Matrices and linear algebra, Dover Books on Mathematics, 1989.

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.

Voyiadjis G.Z., Degradation of elastic modulus in elastoplastic coupling with finite strains, International Journal of Plasticity, 4: 335-353, 1988.

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.

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.

Voyiadjis G.Z., Kattan P.I., A new class of damage variables in continuum damage mechanics, ASME Journal of Materials and Technology, 134(2): 021016, 2012, doi:10.1115/1.4004422.

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.

Voyiadjis G.Z., Kattan P.I., Advances in damage mechanics: metals and metal matrix composites with an introduction to fabric tensors, 2nd ed., Elsevier, 2006.

Voyiadjis G.Z., Kattan P.I., Damage mechanics, Taylor and Francis (CRC Press), 2005.

Voyiadjis G.Z., Kattan P.I., Decomposition of elastic stiffness degradation in continuum damage mechanics, ASME Journal of Material and Technology, Published Online, 50 manuscript pages, 2016.

Voyiadjis G.Z., Kattan P.I., Elasticity of damaged graphene: a damage mechanics approach, International Journal of Damage Mechanics, Published Online, 50 manuscript pages, 2016.

Voyiadjis G.Z., Kattan P.I., Governing differential equations for the mechanics of undamageable materials, Engineering Transactions, 62(3): 241–267, 2014.

Voyiadjis G.Z., Kattan P.I., Healing and super healing in continuum damage mechanics, International Journal of Damage Mechanics, 23(2): 245–260, 2014.

Voyiadjis G.Z., Kattan P.I., Introduction to the mechanics and design of undamageable materials, International Journal of Damage Mechanics, 22(3): 323–335, 2013.

Voyiadjis G.Z., Kattan P.I., Mechanics of damage processes in series and in parallel: a conceptual framework, Acta Mechanica, 223(9): 1863-1878, 2012.

Voyiadjis G.Z., Kattan P.I., Mechanics of damage, healing, damageability, and integrity of materials: A conceptual framework, International Journal of Damage Mechanics, Publish Online, 55 manuscript pages, 2016.

Voyiadjis G.Z., Kattan P.I., On the theory of elastic undamageable materials, ASME Journal of Materials and Technology, 135(2): 021002, 2013, paper no: MATS-12-1107, doi: 10.1115/1.4023770. 2012.

Voyiadjis G.Z., Kattan P.I., Toward a new matrix operation in continuum and damage mechanics: The exponentiation of vectors, Mechanics Research Communications, submitted, 16 manuscript pages, 2017.

Watkins D.S., Fundamentals of Matrix Computations, 3rd ed., Wiley, 2010.




Copyright © 2014 by Institute of Fundamental Technological Research
Polish Academy of Sciences, Warsaw, Poland