1. Field of application#

1.1. Purpose#

The ductile damage of metallic materials results in broad strokes from three mechanisms: the germination of cavities, their growth within an elastoplastic matrix and their coalescence. [Gurson, 1977] proposed a micromechanically-inspired behavior model able to account for cavity growth in a simple way. It was completed phenomenologically in [Tvergaard &Needleman, 1984] to also represent the stages of germination and coalescence: we then speak of model GTN.

The evolution of porosity (volume of cavities in relation to the total volume) results in a softening of the material, i.e. a decrease in the ability to withstand stresses. This often results in the localization of deformations and porosity in thin strips. The resulting strong gradients then require an extension of the law of behavior to take into account interactions at a distance: we speak of non-local formulations of behavior. Such an extension has been proposed for the model GTN [Zhang et al., 2018] which is based on the gradient of the internal work-hardening variable (GRAD_VARI, GRAD_INCO) according to [Lorentz & Andrieux, 1999]. This is the wording introduced in code_aster.

These phenomena generally take place at deformation levels that require non-linear kinematics. Here, it relies on logarithmic deformation (GDEF_LOG). Moreover, at these levels of deformation, plasticity initially induces an almost incompressible behavior. To get rid of the numerical locking problems that may result, it is recommended to use a mixed formulation (GRAD_INCO), possibly supplemented by a penalization term. However, the model is also available in small deformations (PETIT) as well as in the absence of a mixed formulation (GRAD_VARI), or even in local formulation of the behavior (standard finite elements).

The effects of loading speed can be taken into account through a viscoplastic formulation based on a Norton law (VISC_GTN) or ignored (GTN). Let’s add that viscous regularization [R5.03.34] is also recommended to stabilize the behavior of completely broken points and therefore without resistance to stress.

Finally, it should be noted that the elastic part of the model is expressed explicitly, that is to say that the stress is expressed as a function of deformation and plastic deformation in the current state. There is therefore no incremental evolution of the constraint; it does not contribute to defining the mechanical state of the system; it is a consequence of this.

Note: for a broken point in a viscoplastic regime, the stress is not cancelled out instantly but it no longer depends on the deformation, as will be seen later. In this very particular case, the evolution of the constraint is purely incremental, which justifies the constraint appearing (computationally) among the internal variables. On the other hand, it is not useful to enter it in the initial state given in STAT_NON_LINE/ETAT_INITpar SIGM; this value is ignored.

1.2. Material parameters#

The modular expression of the behavioral relationship makes it possible to group the various parameters of the model by categories:

the elastic part is isotropic and depends on the Young’s modulus \(E\) and the Poisson’s ratio \(\nu\) defined under the keyword factor ELAS (E, NU) of the command DEFI_MATERIAU;
the parameters of isotropic work hardening are defined under the keyword factor ECRO_NL (R0, RH, R1, GAMMA_1, R2, GAMMA_2, RK, RK, P0, GAMMA_K). Note that it is not possible to define a Lüders plateau, contrary to model VMIS_ISOT_NL;
model damage parameters are grouped under the keyword factor GTN
- Cavity growth: PORO_INIT, Q1 and Q2
- Germination with Gauss distribution: NUCL_GAUSS_PORO \({f}_{N}\),, NUCL_GAUSS_PLAS \({\kappa }_{N}\), NUCL_GAUSS_DEV \({s}_{N}\)
- Germination with uniform distribution over an interval: NUCL_CRAN_PORO \({c}_{0}\),, NUCL_CRAN_INIT \({\kappa }_{i}\), NUCL_CRAN_FIN \({\kappa }_{f}\)
- Germination proportional to the cumulative plastic deformation starting from a threshold: NUCL_EPSI_INIT \({E}_{c}^{p}\), NUCL_EPSI_PENTE \({b}_{0}\)
- Coalescence: COAL_PORO \({f}_{c}\), COAL_ACCE \({\delta }_{c}\) and broken porosity PORO_RUPT \({f}_{F}\). The user defines either \({\delta }_{c}\) or \({f}_{F}\), the two quantities being linked by:

(1.1)#\[ {f} _ {F} =\ left (1-\ frac {1} {\ frac {1}} {\ mathrm {\ delta}}\ right) {f} _ {c} +\ frac {1} {\ mathrm {\ delta}} {\ mathrm {\ delta}}}\ frac {\ delta}}}\]

- Numerical regularization of the model: ENDO_CRIT_VISCdésigne the damage threshold \({d}_{v}\) beyond which a coupling between viscoplasticity and damage is activated; ENDO_CRIT_RUPTdésigne the damage \({d}_{f}\) from which it is considered that a point is broken.
the parameters of Norton’s viscosity law, when present, are defined under the keyword factor NORTON (coefficient K and exponent N)
in non-local formulation [R5.04.01], the corresponding parameters must also be defined under the keyword factor NON_LOCAL (C_ GRAD_VARI, PENA_LAGRet PENA_LAGR_INCO).

The parameters may depend on the temperature (keywords factor *_FO). The law of behavior is expressed in mechanical deformations, i.e. it is compatible with the usual shrinkage terms, in particular thermal deformation.

1.3. Internal variables#

The law of behavior is based on 25 internal variables, some of which are only there as post-processing. They are grouped together in the following table:

EPSEQ	V1	Work hardening variable \(\kappa\)
POROSITE	V2	Porosity
INDIPLAS	V3 (post)	Status of the current time step (0, 1, 2, or 3)
EPSPXX - EPSPYZ	V4- V9	Components of plastic deformation
EPCUM	V10	Cumulative plastic deformation \({E}_{\mathit{eq}}^{p}\)
PORO_NUC	V11	Porosity linked to germination \({f}^{\mathit{nu}}\)
ENDO	V12	Damage \(d\)
VIT_ENDO	V13	Damage rate \(\dot{d}\)
SIXX - SIYZ	V14-V19	Constraint (used for broken points)
ENDO_EX	V20 (post)	Extrapolated damage \({d}_{\mathit{ex}}\)
SIEQ_ERX	V21 (post)	Constrained extrapolation error
SIEQ_ECR	V22 (post)	Share of the stresses associated with work hardening
SIEQ_VSC	V23 (post)	Share of constraints related to viscosity
SIEQ_NLC	V24 (post)	Share of constraints linked to the non local
ARRET	V25 (post)	Variable available to the user


EPSEQ	V1	Work hardening variable \(\kappa\)
POROSITE	V2	Porosity
INDIPLAS	V3	State of the current time step (0, 1, 2, or 3)
EPSPXX - EPSPYZ	V4- V9	Components of plastic deformation
EPCUM	V10	Cumulative plastic deformation \({E}_{\mathit{eq}}^{p}\)
PORO_NUC	V11	Porosity linked to germination \({f}^{\mathit{nu}}\)
PORO_LOG	V12	Logarithmic porosity \(\mathrm{ln}f-\mathrm{ln}{f}_{0}\)

Note that we also talk about « damage ». We will see later how damage is linked to porosity through three contributions: germination, growth and the coalescence of cavities.