On the numerical modelling of ductile damage with an implicit gradient-enhanced formulation
Keywords:
ductile damage, fracture, gradient damage, hyperelasto-plasticity, higher-order regularization, nonlocalityAbstract
This paper presents a phenomenological framework to incorporate ductile damage in an elasto-plastic madel. Based on a graduai reduction of the yield stress upon damage evolution, the theory isfirst presented in a small deformation context, after which a generalization to a geometrically nonlinear hyperelasto-plastic formulation is made. ln bath cases, an isotropie damage variable is used that is computed from a non local field variable. This approach guarantees the well-posedness of the underlying mathematical problem, and thus prevents any form ofpathological mesh dependence. Special attention isfocused on the type ofnonlocality that is used to drive the ductile damage. Lagrangian and Eulerian nonlocal kernels are compared and the ir influence on the mate rial length scale is evaluated. Examples are given for a small and a large deformation problem, which highlights the applicability in view of its practical implementation to simulate forming processes.
Downloads
References
[ALF 98] ALFANO G., ROSATI L., VALOROSO N., "A displacement-like finite element mode!
for h elastoplasticity: variational formulation and finite-step solution", Computer Methods
in Applied Mechanics and Engineering, vol. 155, 1998, pp. 325-358.
[BAZ 88] BAZANT Z. P., PIJAUDIER-CABOT G., "Nonlocal continuum damage, localization
instability and convergence", Journal of Applied Mechanics, vol. 55, 1988, pp. 287-293.
[DB 92] DE BORST R., MÜHLHAUS H. B., "Gradient-dependent plasticity: Formulation and
algorithmic aspects", International Journal for Numerical Methods in Engineering, vol. 35,
, pp. 521-539.
[ENG 01] ENGELEN R. A. B., GEERS M. G. D., BAAIJENS F.P.T., "Nonlocal implicit
gradient-enhanced softening plasticity", International Journal of Plasticity. Accepted.
[GAN 99] GANGHOFFER J.F., SLUYS L.J ., DE BORST R., "A reappraisal of nonlocal mechanics",
European Journal of Mechanics- A/Solids, vol. 18 1-2, 1999, pp. 17-46.
(GB 99] G. BüRINO P. FUSCHI, POLIZZOTTO C., "Thermodynamic approach to nonlocal
plasticity related variational principles", Journal of Applied Mechanics, vol. 66 4, 1999, pp.
-963.
[GEE 98] GEERS M. G. D., DE BüRST R., BREKELMANS W. A. M., PEERLINGS R. H. J.,
"Strain-based transient-gradient damage model for failure analyses", Computer Methods in
Applied Mechanics and Engineering, vol. 160 1-2, 1998, pp. 133-154.
[GEE 00] GEERS M. G. D., PEERLINGS R. H. J., BREKELMANS W. A. M., DE BüRST
R., "A comparison of phenomenological nonlocal approaches based on implicit gradientenhanced
damage", Acta Mechanica. Accepted.
[GUR 77] GURSON A.L., "Continuum theory of ductile rupture by void nucleation and
growth: Part 1 - yield criterion and flow rules for porous ductile media", Transactions
of ASME Journal of Engineering Materials Technology, vol. 99 1, 1977, pp. 2-15.
[KRA 96] KRAUSZ A. S., KRAUSZ K., eds., Uni.fied Constitutive laws of plastic deformation,
San Diego, Academie Press, 1996.
[LEM 90] LEMAITRE 1 ., CHABOCHE 1 .L., Mechanics of Solid Materials, Cambridge University
Press, 1990.
[MIE 98a] MIEHE C., "A constitutive frame for elastoplasticity at large strains based on the
notion of a plastic metric", International Journal of Solids and Structures, vol. 30, 1998,
pp. 3859-3897.
[MIE 98b] MIEHE C., "A formulation of finite elastoplasticity based on dual co- and contravariant
eigenvector triads normalized with respect to a plastic metric", Computer Methods
in Applied Mechanics and Engineering, vol. 159, 1998, pp. 223-260.
[MIK 99] MIKKELSEN L. P., "Necking in rectangular tensile bars approximated by a 2-d
gradient dependent plasticity mode!", European Journal of Mechanics A/Solids, vol. 18,
, pp. 805-818.
[NAG 90] NAGHDI P. M., "A cri ti cal review of the state of fini te plasticity", Journal of Applied
Mathematics and Physics, vol. 41, 1990, pp. 315-387.
[NIL 98] NILS SON C., "On nonlocal rate-independent plasticity", International Journal of
Plasticity, vol. 14 6, 1998, pp. 551-575.
[PAM 94] PAMIN 1 ., Gradient-dependent plasticity in numerical simulation of localization
phenomena, Ph.D. thesis, Delft University of Technology, Delft, 1994.
[PEE99] PEERLINGS R. H. 1., GEERS M. G. D., DE BüRST R., BREKELMANS W. A. M.,
"A critical comparison of nonlocal and gradient-enhanced continua", International Journal
of Solids and Structures. Submitted.
[RAM 98] RAMASWAMY S., AVARAS N., "Finite element implementation of gradient plasticity
models. part ii: Gradient-dependent evolution equations", Computer Methods in Applied
Mechanics and Engineering, vol. 163, 1998, pp. 33-53.
[SIM 85] SIMO 1. C., ÜRTIZ M., "A unified approach to finite deformation plasticity based on
the use of hyperelastic constitutive equations", Computer Methods in Applied Mechanics
and Engineering, vol. 49, 1985, pp. 221-245.
[SIM 88a] SIMO J. C., "A framework for fini te strain elastoplasticity based on maximum
plastic dissipation and the multiplicative decomposition: Part i. continuum formulation",
Computer Methods in Applied Mechanics and Engineering, vol. 66, 1988, pp. 199-219.
[SIM 88b] SIMO J. C., "A framework for finite strain elastoplasticity based on maximum
plastic dissipation and the multiplicative decomposition: Part ii. computational aspects",
Computer Methods in Applied Mechanics and Engineering, vol. 68, 1988, pp. 1-31.
[SIM 98] SIMO J. C., HUGHES T. J. R., Computationalinelasticity, Interdisciplinary Applied
Mathematics, Springer-Verlag, 1998.
[STE 99] STEINMANN P., "Formulation and computation of geometrically non-linear gradient
damage", International Journal for Numerical Methods in Engineering, vol. 46, 1999, pp.
-779.
[SYE 97] SVEDBERG T., RUNESSON K., "A thermodynamically consistent theory of gradientregularized
plasticity coupled to damage", International Journal of Plasticity, vol. 13 6-7,
' pp. 669-696.
[TVE 95] TVERGAARD V., NEEDLEMAN A., "Effects of nonlocal damage in porous plastic
solids", International Journal of Solids and Structures, vol. 32 8-9, 1995, pp. 1063-1077.
