Isotropic Elastoplasticity Fully Coupled with Non-Local Damage

M. Almansba, K. Saanouni, N. E. Hannachi

.

DOI: 10.4236/eng.2010.26055                

Abstract

This paper presents a simple damage-gradient based elastoplastic model with non linear isotropic hardening in order to regularize the associated initial and boundary value problem (IBVP). Using the total energy equivalence hypothesis, fully coupled constitutive equations are used to describe the non local damage induced softening leading to a mesh independent solution. An additional partial differential equation governing the evolution of the non local isotropic damage is added to the classical equilibrium equations and associated weak forms derived. This leads to discretized IBVP governed by two algebric systems. The first one, associated with equilibrium equations, is highly non linear and can be solved by an iterative Newton Raphson method. The second one, related to the non local damage, is a linear algebric system and can be solved directly to compute the non local damage variable at each load increment. Two fields, linear interpolation triangular element with additional degree of freedom is terms of the non local damage variable is constructed. The non local damage variable is then transferred from mesh nodes to the quadrature (or Gauss) points to affect strongly the elastoplastic behavior. Two simple 2D examples are worked out in order to investigate the ability of proposed approach to deliver a mesh independent solution in the softening stage.

Keywords

Elastoplastic, Damage Behaviour Coupling, Isotropic Hardening, Damage Gradient, Finis

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	G. Pijaudier-Cabot and Z. Bažant, “Nonlocal Continuum Damage, Localization Instability and Convergence,” International Journal of Applied Mechanics, Vol. 55, No. 4, 1988, pp. 287-293.
[2]	K. Saanouni, “Sur l’Analyse de la Fissuration des Mili- eux Elasto-Viscoplastique par la Théorie de l’Endom- magement Continue,” Thèse de Doctorat, Université de Technologie de Compiègne, 1988.
[3]	T. Svedberg and K. Runesson, “An Adaptive Finite Element Algorithm for Gradient Theory of Plasticity with Coupling to Damage,” International Journal of Solids Structures, Vol. 37, No. 48-50, 2000, pp. 7481-7499.
[4]	R. H. J. Peerlings, “Enhanced Damage Modelling for Fracture and Fatigue,” Ph. D. Dissertation, Technische Universiteit Eindhoven, 1999.
[5]	C. Comi, “A Nonlocal Model with Tension and Com- pression Damage Mechanics,” European Journal of Mechanics A/Solids, Vol. 20, No. 1, 2001, pp. 1-22.
[6]	M. Geers, R. Ubachs and R. Engelen, “Strongly Non- Local Gradient-Enhanced Finit Strain Elastoplasticity,” International Journal for Numerical Methods in Engin- eering, Vol. 56, No. 14, 2003, pp. 2039-2068.
[7]	R. Abu-Al-Rub and G. Voyiadjis, “A Physically Based Gradient Plasticity Theory,” International Journal of Plasticity, Vol. 22, No. 4, 2006, pp. 654-684.
[8]	K. Saanouni and J.-L. Chaboche, “Comprehencive Struc- tural Integrity,” ISBN: 0-08-043749-4, Vol. 10, 2003.
[9]	R. Mindlin and N. Eshel, “On First Strain Gradient Theories in Linear Elasticity,” International Journal of Solids and Structures, Vol. 4, No. 1, 1968, pp. 109-124.
[10]	P. Germain, “The Method of Virtual Power in Continuum Mechanics, Part 2: Microstructure,” SIAM Journal on Applied Mathematics, Vol. 25, No. 3, 1973, pp. 556-575.
[11]	G. Maugin, “Nonlocal Theories or Gradient-Type Theories: A Matter of Convenience,” Archives of Mechanics, Vol. 31, No. 1, 1979, pp. 15-26.
[12]	S. Forest and R. Sievert, “Elastoviscoplastic Constitutive Frameworks for Generalized Continua,” Acta Mechanica, Vol. 160, No. 1, 2003, pp. 71-111.
[13]	A. Eringen, “Microcontinuum Field Theories: Vol. I Foundations and Solids; Vol. II Fluent Media,” Springer, New York, 1999.
[14]	R. Chambon, D. Caillerie and T. Matsuchima, “Plastic Continuum with Microstructure, Local Second Gradient Theories for Geomaterials,” International Journal of Solids and Structures, Vol. 38, 2001, pp. 8503-8527.
[15]	S. Forest, “Micromorphic Approach for Gradient Elasticity, Viscoplasticity and Damage,” Journal of Engineering Mechanics, Vol. 135, No. 3, 2009, pp. 117-131.
[16]	R. Peerlings, T. Massart and M. Geers, “A Thermodynamical Motivated Implicit Gradient Damage Framework and its Application to Brick Masonry Cracking,” Computer Methods in Applied Mechanics and Engineering, Vol. 193, No. 30-32, 2004, pp. 3403-3417.
[17]	K. Saanouni, C. Forster and F. Ben-Hatira, “On the Anelastic Flow with Damage,” International Journal of Damage Mechanics, Vol. 3, No. 2, 1994, pp. 140-169.
[18]	K. Saanouni, H. Badreddine and M. Ajmal, “Advances in Virtual Metal Forming Including the Ductile Damage Occurrence: Application to 3D Sheet Metal Deep Drawing,” Journal of Engineering Materials and Technology, Vol. 130, No. 2, 2008.
[19]	D. Sornin, “Sur les Formulations Elastoplastiques Non Locales en Gradient d’Endommagement,” Thèse de Doc- torat, Université de Technologie de Troyes, 2007.
[20]	S. Boers, P. Schreurs and M.Geers, “Operator-Split Damage-Plasticity Applied to Groove Forming in Food Can Lids,” International Journal of Solids and Structures, Vol. 42, No. 14, 2005, pp. 4154-4178.
[21]	A. Simone, G. Wells and L. Sluys, “From Continuous to Discontinuous Failure in a Gradient-Enhanced Conti- nuum Damage Model,” Computer Methods in Applied Mechanics and Engineering, Vol. 192, No. 41, 2003, pp. 4581-4607.
[22]	Y. Hammi, “Simulation Numérique de l’Endommagement dans les Procédés de Mise en Forme,” Université de Technologie de Compiègne, 2000.
[23]	D. R. J. Owen and E. Hinton, “Finite Elements in Plasticity, Theory and Practice,” Pineridge Press Limited, Swansea, 1988.
[24]	M. Almansba, K. Saanouni and N. E. Hannachi, “Régu- larisation d'un Modèle Elastoplastique par Introduction d'un Gradient d'endommagement,” XIXème Congrès Français de Mécanique, CFM’09, France, 2009.
[25]	B. Nedjar, “Elastoplastic-Damage Modelling Including the Gradient of Damage: Formulation and Computational Aspects,” International Journal of Solids and Structures, Vol. 38, No. 30-31, 2001, pp. 5421-5451.