Performance of selected methods to describe strain localization in ductile materials 

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Modeling of the structural failure caused by contact explosion

The PhD thesis of Geffroy [57] (LIMATB, UBS), see also [58], contributes to the numerical analysis of the crack arrest capability of ship hull materials subjected to contact explosion as it occurs in a terroristic attack. For the behavior of the undam-aged material the effects of strain and strain rate hardening and thermal softening were taken into account. The author evidenced the interest of using an advanced damage-plasticity material model, namely a modified version of the well-known Gurson-Tvergaard-Needleman (GTN) model, to reproduce the evolution of more or less diffuse damage what is in accordance with microstructural observations. The constitutive model was implemented as user material (VUMAT) within the indus-trial finite element software Abaqus (explicit integration, 3D, dynamic loading). The developed material model showed satisfactory agreement in reproducing the material response before crack onset.
Crack propagation is treated by means of an empirical method: the crack is assumed to initiate as soon as the critical porosity is attained. Using the classical FEM, the element which meets the initiation criterion (critical porosity) is deleted (element erosion method). A comparison between experimental and numerical results of an air-blast experiment is shown in Fig. I.3.
However by using the critical porosity criterion, the specific phenomena which lead to the formation of a macro-crack are not represented. Furthermore, the element erosion method has the inconvenience that the structural response is strongly sensitive with respect to the FE mesh size and orientation (see also Song [160]). This is reflected e.g. in that the crack path can not be properly captured (see Fig. I.3) and in that the crack propagates too fast compared to the experimental results (as stated in [57]). That gives rise to the conclusion that the residual strength of the structure after failure can not be reliably predicted, thus demanding for a more precise modeling of the failure response.

Modeling of the crack propagation in ductile plate structures using X-FEM

To this end, the thesis of Crété [45] (ICA, ISAE and LIMATB, UBS) was launched to focus on the numerical treatment describing the formation and propagation of a macro-crack involving the damage-plasticity pre-crack model developed by Geffroy. Due to the limitations of the standard FEM to reproduce crack propagation (mesh dependence, see above), the eXtended Finite Element Method (X-FEM) was applied to describe the kinematic consequences of crack propagation across the mesh. Moreover, a direct coupling method between the more or less diffuse damage and the enriched FE technology was elaborated – without taking into account the transition phase of strain localization. The crack propagation relies on three elements:
• an initiation criterion,
• the determination of the crack direction and.
• the computation of the crack length within one time increment.
The crack is assumed to propagate as soon as a critical stored energy is at-tained. The direction of propagation is evaluated based on the bifurcation analysis. The criteria involve the application of an averaging technique to reduce the mesh sensitivity in the softening regime.
The numerical model was implemented as user element (UEL) in Abaqus (implicit integration, 2D plane strain, dynamic loading). Virtual tests of a plate with pre-existing cracks have shown that the physical crack path can be well reproduced, see Fig. I.4. However, irrespective of the choice of the critical stored energy, the structural response deviates from the experimental curve in that the force abruptly drops as soon as the crack starts to propagate (Fig. I.4, rightmost) – a prediction which is indeed too conservative (i.e. too pessimistic). The sudden drop in load can potentially be traced back to the neglect of the phase of strain localization as a precursor of crack formation.

Modeling of the crack propagation in ductile shell structures using X-FEM

The PhD thesis of Jan [80] (LaMCoS, INSA de Lyon) is involved in the same industrial context of military vulnerability with the aim to numerically analyze the residual strength of ships exposed to extreme loads. Therein, a promising numerical approach is proposed capable of coupling the shell element formulation with the X-FEM in order to represent a propagating crack within the ship hull. This coupling pursues the objectives to reduce the model size as well as to handle the pathologic mesh dependence. The adopted crack propagation criterion is based on the (stress-based) approach by Haboussa [68] for the case of an elasto-plastic material behavior. However, the author states that – although this criterion is efficient and simple to implement – it must still be improved to better reproduce ductile materials. In the course of the degradation of the material resistance (stress softening) the risk 1is to not attain the yield stress at all.
The numerical model was implemented in the calculation code Europlexus (EPX) of the Commissariat à l’Energie Atomique (CEA) (explicit integration, 3D, dynamic loading). A comparison between experiments and numerical1 tests shows promising results when using very fine meshes. Therefore, this work constitutes another significant building brick towards a numerical model of the entire ductile failure process under highly dynamic loading.

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Damage concentration and strain localization

The local material models fail to describe the damage concentration and strain localization in a narrow band and the potential crack propagation. A corrective approach consists in controlling the numerical strain localization in the FE mesh in order to allow for capturing the genuine physical strain localization. The mesh dependence resulting from strain localization can be coped with by introducing a characteristic length into the formulation which serves as localization limiter. Non-local techniques (e.g. Bažant [17], Pijaudier-Cabot [127]) can be applied to attenuate the pathology, but accurate results require a very fine mesh and thus a big computational effort for large structures. Moreover, the physical strain localization itself as possible precursor of crack formation can not be properly represented.
A promising approach consists in using comparably large FEs and embedding the thin band of highly localized strain into the FE, e.g. by enriching the kinematic FE formulation (Ortiz [119], Belytschko [20]) or the material model (Longère [95]).
The development of an appropriate (embedded-band) method capable of reproducing the physical localization band and reducing the mesh sensitivity to a minimum is a significant objective of this dissertation.

Crack formation and propagation

The standard FEM is not suitable to reproduce the discontinuous kinematics of a crack. Complementary techniques as the element deletion method (Song [160], Autenrieth [11]) and the inter-element crack method (Xu [182]) may be used to describe the crack propagation but they suffer from mesh dependence. Adaptive remeshing techniques have been shown to better reproduce the crack propagation, see e.g. Bouchard [31], yet requiring a huge computational effort. Similar to the embedded-band approach to treat the (numerical and physical) strain localization, the crack can be embedded within the finite element by corre-spondingly enriching the kinematic FE formulation, as e.g. done in the eXtended FEM (X-FEM) (Belytschko [18]). Works devoted to show the performance of the X-FEM to reproduce the failure of elastic-(quasi-)brittle structures are numerous in literature (Moës [106], Dumstorff [50]), whereas works dealing with ductile, strongly nonlinear structures are still remarkably scarce (Crété [46], Broumand [32]). This method of embedded finite elements moreover allows for using a coarser mesh and is accordingly more suitable for large structures.

Table of contents :

I Introduction 
I.1 General context of the study
I.2 Problem statement and scientific challenges
I.3 Main hypotheses for numerical treatment
I.4 Outline of the dissertation
II Overview of ductile fracture 
II.1 Introduction
II.2 Physical aspects of ductile failure
II.3 Modeling of ductile failure and numerical issues
II.4 Summary
IIIPerformance of selected methods to describe strain localization in ductile materials 
III.1 Introduction
III.2 Preliminary considerations
III.3 Method 1: Strong discontinuity
III.4 Method 2: Weak discontinuity
III.5 Method 3: Regularized discontinuity
III.6 Assessment, discussion and decision for a final method
III.7 Conclusion
IVModeling of strain localization using cohesive X-FEM 
IV.1 Introduction
IV.2 Preliminary considerations
IV.3 Combining cohesive models and X-FEM
IV.4 Application
IV.5 Discussion and conclusion
V Conclusions and perspectives 
V.1 Conclusions
V.2 Perspectives
Appendices
Bibliography

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