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Lemaitre's damage nonlocal model performance under crack initiation and propagation

Seabra M., Cesar de Sa J.

Computer Methods in Materials Science

2013

Vol. 13, No. 1

30--35

Modelling of ductile fracture is a rather challenging task due to the complexity of the physical phenomena involved. In ductile metals, material progressive degradation, which is strongly associated with large plastic straining, may be captured using the Lemaitre damage model. Nevertheless, to model the propagation of macro-cracks, the use of a discontinuous approach is in general imperative. Therefore, in this work, the Lemaitre Damage model is combined with the XFEM for a complete description of ductile failure. In addition, as in general continuous softening models suffer from pathological mesh dependence, a non-local formulation is implemented and its effects on crack initiation and propagation are evaluated.

Modelowanie pęknięć plastycznych jest zagadnieniem trudnym ze względu na złożoność występujących zjawisk fizycznych. W metalach plastycznych postępujące zniszczenie materiału, związane ściśle z dużymi odkształceniami plastycznymi, może być opisane modelem opracowanym przez Lemaitre’a. Niemniej jednak model propagacji makro uszkodzeń wymaga uwzględnienia nieciągłości tych zjawisk. Dlatego w niniejszej pracy model uszkodzeń Lemaitre’a został połączony z modelem XMES, co pozwoliło na kompletny opis zjawiska pęknięcia plastycznego. Ponadto, ponieważ modele ciągłe stosowane do opisu mięknięcia materiału są uzależnione od siatki elementów skończonych, zastosowano sformułowanie nielokalne, co umożliwiło modelowanie zarówno inicjacji jak i rozprzestrzeniania się pęknięć.

A comparison of meshless and finite element approaches to ductile damage in forming processes

Cesar de Sa J., Zheng C.

Computer Methods in Materials Science

2007

Vol. 7, No. 2

262-268

In metal forming processes the damage associated with large deformations is a phenomenon that should be minimized or simply avoided as it usually leads to flawed parts. The initiation of plasticity and damage is caused by movement and accumulation of dislocations in metals but their nature and evolution is different. Ductile damage evolution in metals is usually associated with the initiation and growth of micro cracks and cavities, resulting in a progressive material softening. Damage growing influences indirectly the plastic behaviour by locally reducing the elementary area of resistance and therefore plasticity and damage should be coupled at the constitutive level. In the theory of Continuum Damage Mechanics the damage is represented by internal variables (of scalar, vectorial or tensor type) which give a measure of the deteriorated state at each representative volume of the material. This variable may then be used to define the effective stress state. Another important aspect is related with the fact that in ductile damage localization is similar to that associated with plastic strain. These physical phenomena are characterised by the accumulation of damage and large deformations within narrow bands. In experiments these localization zones display a finite width which may be related to the micro structure of the material. Classical theories of plasticity and damage mechanics, based on internal variable approaches, are local theories and do not include size effects associated to a characteristic dimension of the material. Their implementation in a finite element setting shows a pathologic effect of spatial mesh dependence because the constitutive models are unable to capture the limitation of the localization upon mesh refinement. In fact, the original hypothesis of homogeneous continuous models does not take into account large changes in the internal variables, like plastic strain and damage, in the localization zone. The aforementioned effect can be adequately explained by micro mechanical theories but their numerical implementation is still rather expensive. Non-local models have been proposed to bridge the gap between classical continuum theories and the micromechanical ones. In these models the evolution of some internal variables describing strain and damage in a specific point is also determined by the history of the surrounding material by including in the formulation averages or gradients of part or all of them. Some of theses models have proved to be effective when implemented in a finite element framework. Some claims have been made that the new class of computational methods, i.e. meshless methods, could be more effective when dealing with localization problems. Typically these new methods use a set of points and local support functions to represent the problem domain with no need of an additional mesh. This local support functions could then be broadened for the evaluation of the evolution of the internal variables, giving a non-local character to the solution. Therefore, in this work, an incursion was made into the application of these methods to this particular type of problems in order to investigate how meshless methods deal with ductile damage phenomena, if the unacceptable discretization dependence is also present and to assess how effectively the non-local and gradient models work in these settings. The chosen meshless method was the Reproducing Kernel Particle Method (RKPM). The material model was extended in order to include ductile damage effects by coupling the elastoplastic constitutive law with the damage evolution equations. Non-local and related gradient (explicit and implicit) models were also implemented using the RKPM. A set of numerical examples showed that the meshless solution scheme on ductile damage, exhibits the same type of dependence of solutions upon refinement of the geometrical discretization. Both implicit and explicit gradient and non-local models can alleviate this pathological behaviour. Nevertheless the explicit gradient model still presents a local behaviour by concentrating the damage on a narrower zone.

Możliwość wykorzystania metody bez siatkowej Reproducing Kernel Particle Method (RKPM) do symulacji plastycznego pękania w procesach przeróbki plastycznej jest tematem niniejszej pracy. Zalety metody RKPM są porównane z konwencjonalnymi modelami MES, szczególnie pod względem problemów z dyskretyzacją badanego obszaru. Zastosowany model pękania bazuje na podejściu Lemaitre z uwzględnieniem rozgraniczenia pękania dla lokalnych obszarów rozciąganych i spęczanych. Zaimplementowane lokalne i globalne modele w formie jawnej i niejawnej są porównane i omówione w niniejszej pracy.