Gurson-Tvergaard-Needleman-Ben Bettaïeb model (GTNB)

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Formability characteristics

The Dual-Phase steels present an excellent candidate for the car body structural components. These are often produced for safety-critical parts (see Figure II.4) which maintain passenger surviving space in crash events. The DP steels present a good balance of strength, formability, energy absorption and durability. Also the employability of this kind of steel provides the possibility of reducing the weight of the vehicle.
DP is sometimes selected for visible body parts and closures, such as doors, hoods, front and rear rails. Other well known applications include: beams and cross members; rocker, sill, and pillar reinforcements; cowl inner and outer; crush cans; shock towers, fasteners, and wheels [TAM 2011].
Figure II.4: DP steels used for safety-critical body parts (General Motors, [BUI 2011]).
However, the excellent structural properties of this AHSS group are limited by the fracture phenomena. Fractures are different from ordinary steel and depend strongly on the DP steel grades used. The Figure II.5 below shows a shear fracture, an edge cracking and a crack during a hole expansion test. Numerous factors can affect the outstanding DP properties such as number of forming stages, tool geometry for each stage, boundary constraints, lubrication conditions, material variability and eventually the product changes [KEE 2009].
Figure II.5: Fracture type in automotive part for DP steel in different solicitations [MAT 2012]. Another crucial cause (not studied in this document) for the fabrication of inconsistent sheet metal part is springback i.e. the elastic strain recovery in the DP steel after the tooling is removed. Springback of sheet metal parts after forming causes deviation from the designed target shape and produces downstream quality problems and assembly difficulties as seen in Figure II.6.

X-ray tomography principle

The technique of X-Ray tomography started in the end of the 80’s in the medical field by improving the detection of brain and breast cancer tumor. This was available by using 2D radiographies. In the middle of the 90’s the X-ray tomography was adapted and found its respective place in materials science to analyze debris from plane and cars crashes for the insurance companies. Since 2000 the method has become an efficient investigation tool for various materials.
The principle of X-ray tomography used by Landron and co-workers shown in Figure II.8 is composed of X-ray beam, the sample to analyze, the camera and the projections acquisition. The object to characterize rotates about a single axis while a series of 2D X-ray absorption images is recorded. Using mathematical principles of tomography, this series of images is reconstructed to produce a 3D digital image where each voxel (volume element or 3D pixel) represents the X-ray absorption at that point.

In-situ tensile tests

Due to the fact that the X-ray tomography is a non destructive test as mentioned previously, this technique has been coupled with an in-situ tensile test as seen in Figure II.9(a). One specimen is pulled for a given deformation. The specimen presented in Figure II.9(b) is unloaded. The tensile device is placed on the beam line. During imaging, the deformation is stopped but maintained constant.
Landron also performed a so called in-situ continuous test. It is the same procedure as the previous one but with no interruption of the deformation during the tensile test. This characterization requires a small scan time to have clear images of the reconstructed volume (as shown in Figure II.9(c)). The displacement speed of the tensile device is between 1µm/s and 5µm/s and the reconstructed volume is equal to 0.3×0.3×0.3mm³.

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Yield function and the associative normality rule

The yield surface or plastic yield criterion defined in the stress space as seen Figure III.1, models the elastic limit and the beginning of the plastic flow. It is written as: Fp = σ eqvσ − σ y , Fp ≤ 0 (III.1).
Where σ eqvσis the equivalent stress and σ y is the material flow stress. On one hand, when
σ eqvσ is smaller than σ y Fp 0, the deformation is purely elastic. On the other hand when σ eqvσ is equal to σ y Fp 0, the border is reached and the DP steel starts to plastically deform. At initial state and before hardening function takes place, the yield surface is written as following:0+000.

Table of contents :

Chapter I. :Introduction
I.1 Context
I.1.1 Automotive history
I.1.2 Production and development of new materials
I.I.3 Dual-Phase in automotive industry
I.2 Objectives of the thesis
I.3 Contents
Chapter II. :Dual-Phase steels
II.1 Description of DP steels
II.1.1 Microstructures
II.1.2 Mechanical properties
II.I.3 Formability characteristics
II.2 Experimental Damage Investigation of DP steels
II.2.1 Experimental techniques to study fracture
II.2.2 X-ray tomography principle
II.2.3 In-situ tensile tests
II.3 Selected DP steels grades
Chapter III. :State of the art
III.1 Introduction
III.2 Plasticity modeling
III.2.1 Yield functions
III.2.2 Hardening functions
III.3 Damage modeling
III.3.1 Ductile damage mechanism
III.3.2 The Gurson model and its extensions
III.4 Fracture modeling
III.4.1 Micromechanical fracture criteria
III.4.2 Empirical fracture criteria
III.5 Conclusions
Chapter IV. : Model description
IV.1 Introduction
IV.2 Gurson-Tvergaard-Needleman-Ben Bettaïeb model (GTNB)
IV.2.1 Constitutive equations
IV.2.2 Physically-based void nucleation and growth
IV.2.3 Experiments and GTNB model comparisons
IV.3 GTNB model extension (GTNBF)
IV.3.1 Landron’s physically-based void nucleation and growth
IV.3.2 Void coalescence law
IV.3.3 Fracture initiation modelling
IV.3.4 Flowcharts of the extended GTN models
IV.4 Conclusions
Chapter V. :Material parameters identification
V.1 Elastic-Plastic parameters
V.1.1 Elastic parameters
V.1.2 Anisotropic plastic parameters
V.1.3 Isotropic hardening parameters
V.1.4 Kinematic hardening parameters
V.2 Damage and fracture parameters
V.2.1 GTN model
V.2.2 GTNB model
V.2.3 GTNBF model
Conclusions
Chapter VI. :Implementation of the GTNBF constitutive model
VI.1 Introduction
VI.2 The implementation of the GTNBF model in Abaqus/Explicit
VI.2.1 The GTNBF model implementation in Lagamine framework
VI.2.2 The Lagamine/Abaqus-Explicit interfacing via VUMAT
VI.3 VUMAT programming validations
VI.3.1 Introduction
VI.3.2 Validation on single element
VI.4 Conclusions
Chapter VII. : Potentialities and limitations of the GTNBF model 
VII.1 Introduction
VII.2 Sensitivity study
VII.2.1 Material parameters influence on the GTNBF predictions
VII.2.2 Element size influence on GTNBF model
VII.3 GTNBF model validation on a notched tensile test
VII.3.1 Stress-strain curve and triaxiality evolution
VII.3.2 Nucleation laws comparison
VII.3.3 Growth laws comparison
VII.3.4 GTN Abaqus-explicit and GTNBF models responses
VII.3.5 Back stress definition
VII.4 Conclusions
Chapter VIII. : Applications
VIII.1 Introduction
VIII.2 Tensile tests
VIII.2.1 Experiments and Finite element model
VIII.2.2 Results and comparisons
VIII.2.3 Conclusions
VIII.3 Cross-Die Drawing test
VIII.3.1 Experimental and finite element model
VIII.3.2 Results and comparisons
VIII.4 Conclusions
Chapter IX. :Conclusions and Further works
IX.1 Conclusions
IX.2 Further works
IX.2.1 Improvement of the hardening modeling
IX.2.2 Improvement of the porosity evolution for vanishing triaxiality .
Appendix
Nomenclature
Abbreviations
References

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