Solving finite element problems involving elasto-plasticity coupled with damage softening faces two major difficulties: mesh dependence and volumetric locking. In this study, a specific finite element is proposed which allows to solve simultaneously both problems within the small strain framework. It combines a mixed treatment based on a three-field formulation (displacements, assumed pressure, and assumed dilatation) to solve the volumetric locking and a nonlocal implicit gradient-enhanced formulation to avoid localization of damage. Simulations on a double-notched specimen are presented which allow to compare the nonlocal formulation and the mixed nonlocal formulation. Triangular elements with quadratic shape functions for the displacements and linear shape functions for the assumed dilatation, assumed pressure, and nonlocal variable are used. First results show that the mixed nonlocal method regularizes the problem and allows to obtain smoother stress fields than the nonlocal method at the same time.
Aero, E.L. and Kuvshinskii, E.V. ( 1960). Fundamental Equations of the Theory of Elastic Materials with Rotationally Interacting Particles, Fizika Tverdogo Tela , 2: 1399-1409.
2.
Babuska, I. ( 1971). Error Bounds for Finite Element Methods, Numerischa Mathematik, 16: 322-333.
3.
Babuska, I. ( 1973). The Finite Element Method with Lagrange Multipliers, Numerischa Mathematik, 20: 179-192.
4.
Bargellini, R., Besson, J., Lorentz, E. and Michel-Ponnelle, S. ( 2009). A Non-local Finite Element Based on Volumetric Strain Gradient: Application to Ductile Fracture, Computational Materials Science, 45: 762-767.
5.
Bazant, Z.P. ( 1984). Imbricate Continuum and its Variational Derivation, Journal of Engineering Mechanics, ASCE, 110: 1693-1712.
6.
Bellet, M. ( 1999). Finite Element Analysis of Compressible Viscoplasticity Using a Three-field Formulation: Application to Metal Powder Hot Compaction , Computer Methods in Applied Mechanics Engineering, 175: 19-40.
7.
Besson, J. and Foerch, R. ( 1997). Large Scale Object-oriented Finite Element Code Design , Computer Methods in Applied Mechanics Engineering, 142: 165-187.
8.
Brezzi, F. ( 1974). On the Existence, Uniqueness and Approximation of Saddle Point Problems Arising from Lagrangian Multipliers, Revue Française D’Automatique, Informatique, Recherche Opérationnelle Analyse Numerique, 8-R2: 129-151.
9.
Cosserat, E. and Cosserat, F. ( 1909). Théorie des Corps Déformables, Paris, A. Herrman et Fils.
10.
de Borst, R., Sluys, L.J., Mühlhaus, H.-B. and Pamin, J. ( 1993). Fundamental Issues in Finite Element Analyses of Localization of Deformation, Engineering Computation, 10: 99-121.
11.
Engelen, R.A.B., Geers, M.G.D. and Baaijens, F.P.T. ( 2003). Nonlocal Implicit Gradient-enhanced Elasto-plasticity for the Modelling of Softening Behaviour, International Journal of Plasticity, 19: 403-433.
12.
Eringen, A.C. ( 1966). A Unified Theory of Thermomechanical Materials, International Journal of Engineering Science, 4: 179-202.
13.
Forest, S. and Sievert, R. ( 2006). Nonlinear Microstrain Theories, International Journal of Solids Structures, 43: 7224-7245.
14.
Geers, M.G.D., Engelen, R.A.B. and Ubachs, R.J.M. ( 2001). On the Numerical Modelling of Ductile Damage with an Implicit Gradient-enhanced Formulation, Revue européenne des éléments finis, 10: 173-191.
15.
Grimaldi, R., Addessi, D. and Ciampi, V. ( 2007). Localization and Regularization Behavior of Mixed Finite Element for 2D Structural Problems with Damaging Material, Computer Methods in Applied Mechanics and Engineering, 197: 255-264.
16.
Grioli, G. ( 1960). Elasticità Asimmetrica, Annali di matematica pura ed applicata, 50: 389-417.
17.
Hughes, T.J.R. ( 1980). Generalization of Selective Integration Procedures to Anisotropic and Non Linear Media, International Journal for Numerical Methods in Engineering, 15: 1413-1418.
18.
Lorentz, E., Besson, J. and Cano, V. ( 2008). Numerical Simulation of Ductile Fracture with The Rousselier Constitutive Law, Computer Methods in Applied Mechanics and Engineering, 197: 1965-1982.
19.
Malkus, D.S. and Hughes, T.J.R. ( 1978). Mixed Finite Element Methods in Reduced and Selective Integration Techniques: A Unification of Concepts, Computer Methods in Applied Mechanics and Engineering, 15: 63-81.
20.
Mediavilla, J., Peerlings, R.H.J. and Geers, M.G.D. ( 2006). A Robust and Consistent Remeshing-transfer Operator for Ductile Fracture Simulations, Computers and Structures, 84: 604-623.
21.
Michel-Ponnelle, S. ( 2009). Elements Finis Traitant la Quasi-incompressibilité, Documentation Code_Aster R3.06.08. Available at: www.code-aster.org
22.
Peerlings, R.H.J., de Borst, R., Brekelmans, W.A.M. and de Vree, J.H.P. ( 1996). Gradient-enhanced Damage for Quasi-brittle Materials , International Journal for Numerical Methods in Engineering , 39: 3391-3403.
23.
Pijaudier-Cabot, G. and Bazant, Z.P. ( 1987). Nonlocal Damage Theory, Journal of Engineering Mechanics , ASCE, 113: 1512-1533.
24.
Pijaudier-Cabot, G., Bazant, Z.P. and Tabbara, M. ( 1988). Comparison of Various Models for Strain Softening, Engineering Computation, 5: 141-150.
25.
Rajagopal, E.S. ( 1960). The Existence of Interfacial Couples in Infinitesimal Elasticity, Annalen des Physik, 6: 192-201.
26.
Simo, J.C. and Taylor, R.L. ( 1985). Consistent Tangent Operators for Rate-independent Elastoplasticity , Computer Methods in Applied Mechanics and Engineering, 48: 101-118.
27.
Taylor, R.L. ( 2000). A Mixed-enhanced Formulation for Tetrahedral Finite Elements , International Journal for Numerical Methods in Engineering , 47: 205-227.
28.
Truesdell, C.A., III and Toupin, R. (1960) The classical field theories . In: Flügge’s Handbuch der Physik, vol. 3, part 1, 226-793. Berlin : Springer-Verlag.
29.
Zienkiewicz, O.C. and Nakazawa, S. ( 1984). On Variational Formulations and its Modification for Numerical Solution, Computer Structuring, 19: 303-313.
30.
Zienkiewicz, O.C. and Taylor, R.L. ( 2000a). The Finite Element Method, Vol. 1: Its Basis and Fundamentals, 5th edn, chapters 2 and 3, Oxford, Butterworth-Heinemann.
31.
Zienkiewicz, O.C. and Taylor, R.L. ( 2000b). The Finite Element Method, Vol. 2: Solid Mechanics, 5th edn, chapter 2, Oxford, Butterworth-Heinemann .
32.
Zienkiewicz, O.C., Taylor, R.L. and Zhu, J.Z. ( 2005). The Finite Element Method, Vol. 1: Its Basis and Fundamentals, 6th edn, chapters 10-12, Oxford, Butterworth-Heinemann.
