Gradient‐enhanced ductile damage — a finite deformation framework with application to DP800

Abstract

Gradient-enhanced ductile damage is implemented in the finite element method by means of an additional field variable, representing non-local damage which is linked to the local damage variable. At material point level the isotropic damage formulation uses exponential damage functions to circumvent further constraints on the value set of the local damage variable. The ductility of the model is achieved by a coupling of damage to finite strain plasticity. In a multisurface approach the onset of damage and plasticity is each governed by damage and yield criteria respectively. Linear and exponential isotropic hardening is implemented. With the help of the F-bar method the results of the finite element simulation exhibit mesh independent behaviour. In order to model the behaviour of DP800 with the material model discussed here, elastic and plastic parameters are identified. Model descriptionThere are various possibilities to enhance a material model with gradients. The approach chosen here follows the small strain formulation [1, 2] and, in particular, the finite deformation gradient-enhanced framework established in [3,4]. Modelling details of a related small strain model are discussed in [5]. In the framework an additional scalar field variable φ, representing non-local damage, is introduced and linked to the local damage variable d in penalty form. Furthermore, only the gradient of the field variable, ∇ X φ, and not the gradient of the local damage, ∇ X d, enters the energy functional. Thereby, in the context of finite elements, a local formulation of the material model at quadrature point level is possible despite the gradient-enhancement. The local part of the free Helmholtz energy is formulated in terms of logarithmic elastic strains, ε e = ln(b e ) with b e = F e · F e t , assuming a multiplicative split of the deformation gradient into elastic and plastic parts, i.e. F = F e · F p . The energy functional then takes the formwhere the volumetric and isochoric contributions are each weighted by a so-called damage function f vol and f iso . The regularisation parameter c d has to be chosen sufficiently large and controls the width of the damage zone. The damage function f maps the positive but otherwise unbounded local damage variable d to the interval ]0, 1], i.e.Additional constraints on the link between φ and d can thus be avoided. Damage driving force q, isotropic hardening stress β and Mandel stresses m are derived within the framework of generalised standard dissipative materials. Following the postulate of strain equivalence, the effective Mandel stress tensor m eff is defined asIn this multisurface formulation the evolution of plasticity and damage is each governed by its own potential. For the plastic evolution a von Mises yield criterion Φ p based on the effective Mandel stresses including non-linear isotropic hardening is used. The damage potential Φ d simply compares the damage driving force q with a variable threshold in the form ofIn terms of the algorithmic implementation, d and α are integrated by...

A Finite Plasticity Gradient-Damage Model for Sheet Metals during Forming and Clinching

“…Resulting from the plasticity-damage coupling most of the deformation localises inside the (softer) damage zone, which is governed by the internal length scale and typically of small size. Therefore, the localised deformations inside this small zone result in large deformations that combined with the plastic incompressibility cause locking phenomena [15], even when applying small strains. To avoid locking, we used, as already mentioned, standard fully-integrated quadratic hexahedral elements being accurate but expensive.…”