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## Fracture surfaces and strain ratios

For both uniaxial tensile and compression tests in which shear failure occurred, the fracture surfaces were oriented at approximately 45° with respect to the loading direction, but not in a random manner. If one considers that the cross section of the specimen has an elliptic shape, Figure 3.16 shows two possible orientations (among infinitely many) of the 45° fracture surface with respect to the loading axis. In Figure 3.16 (a), the fracture surface is oriented at 45° to the loading direction and contains the semi-major axis of the ellipse, whereas in Figure 3.16 (b) it is also oriented at 45° to the loading direction but contains the semi-minor axis. Observations of the various fracture surfaces revealed that the latter case occurred consistently in uniaxial tension and preferentially in compression. A reasonable conclusion is that fracture surfaces are aligned with the orthotropic directions of the plate.

### Identification of the Yld2004‐18p material constants

The three parameters of the isotropic hardening law 0 , Q and C are calibrated based on the tensile test in the rolling direction (RD) and the two elastic parameters E and are chosen equal to nominal values for aluminium (see Table 4.1). The plastic anisotropy of the material is taken into account with the 18 parameters in C and C calibrated using the experimental results presented in Chapter 3.

#### The shape parameter m

The m parameter is not calibrated as the other parameters. Hosford (1972) and Hill (1979) showed that m8 is reasonable for f.c.c. crystal structures. Since aluminium is studied here, m8 was used in a first approach. Then, since the parameter m controles the shape of the yieldsurface, it was of interest to investigate its influence on the predicted behaviour of the different tests. Indeed, and as suggested by Barlat et al. (1991), the crystallographic texture and grain morphology of rolled aluminium plates could increase the exponent of the yield function compared to an isotropic microstructure, by changing the active slip systems during the plastic process. Then, a new set of anisotropic parameters was determined with m 12 . The two identifications led to about the same residual in the least squares approach used in the calibration.

**The anisotropy parameters**

From a tension test on a smooth specimen in a given in-plane direction , the yield stress and the strain in the transverse and normal directions are obtained. The flow stress ratio is defined as Thus, is the flow stress in the direction and 0 is the corresponding flow stress in the rolling direction (RD). The flow stress ratios were determined by use of the 0.2% proof stresses in the different directions before being adjusted by inverse identification using trial and error. By definition, the flow stress ratio represents a point on the yield surface and is equal to unity in the rolling direction (RD). Then, the strain ratio R defined in Eq.(13) is the ratio between p TS and p ND , the logarithmic plastic strain rates in the transverse direction of the specimen (TS) and the normal direction (ND) of the plate. As plastic flow normal to the yield surface is assumed, the strain ratio determines the normal to the yield surface at a given stress state.

**Numerical procedures and finite element models**

The constitutive relations described above were implemented as a user-defined material subroutine (see Grytten et al., 2008) in the non-linear finite element code LS-DYNA (LSTC, 2007). All tests presented in Chapter 3 were simulated using the explicit solver of LS-DYNA with 8-node fully integrated solid elements. To reduce the computational time, affected both by the element size and the number of elements, the mesh was only refined in the area exposed to large deformations and mass-scaling was applied. As boundary conditions, a function was applied to smoothly reach a constant velocity. The finite element geometries of the different specimens are shown in Figure 4.4, while the numbers and minimum size of the elements e h are given in Table 4.2. Compression tests were performed with 0 0 h D {1,1.5} but only the results for 0 0 h D 1 will be presented for the reasons explained in Chapter 3. A friction coefficient 0.02 f was used between the lubricated platens and the cylindrical specimen loaded in compression. This value was found by trial and error: the friction coefficient was not found to influence yielding but only the behaviour at very large strains.

**Table of contents :**

Preface

. Abstract

. Résumé

. Publications related to the thesis

. Acknowledgements

. Content

. Notations

**Chapter 1. Introduction **

1.1 Background

1.2 Plastic anisotropy towards failure

1.3 Failure mechanisms

1.4 Structural simulations

1.5 Objectives and scope

1.6 Invariants of the stress tensor

**Chapter 2. The AA7075-T651 aluminium alloy **

2.1 Introduction

2.2 Optical microscope analysis

2.3 Scanning electron microscope (SEM) analysis

2.4 Transmission electron microscope (TEM) analysis

**Chapter 3. Experimental study **

3.1 Introduction

3.2 Tensile tests on smooth axisymmetric specimens (uniaxial tension)

3.3 Tensile tests on notched axisymmetric specimens (multiaxial tension)

3.4 Compression tests on cylindrical specimens (uniaxial compression)

3.5 Shear tests on butterfly specimens

3.6 Fracture surfaces and strain ratios

**Chapter 4. Anisotropic plasticity model **

4.1 Introduction

4.2 Constitutive model

4.3 Identification of the Yld2004-18p material constants

4.3.1 The shape parameter m

4.3.2 The anisotropy parameters

4.4 Numerical procedures and finite element models

**Chapter 5. Numerical analysis **

5.1 Introduction

5.2 Macroscopic stress-strain curves

5.2.1 Isotropic version of Yld2004-18p

5.2.2 Anisotropic version of Yld2004-18p

5.2.3 Influence of plastic anisotropy

5.3 Local stress and strain along fracture surface

5.4 Conclusions

**Chapter 6. Analytical considerations **

6.1 Introduction

6.2 Void growth approach for anisotropic materials

6.2.1 Rice and Tracey analysis

6.2.2 Extension of the Rice and Tracey analysis for anisotropic matrix behaviour

6.2.3 Conclusions

6.3 Localization criteria

6.3.1 General constitutive framework

6.3.2 Localization analysis for a homogeneous material

6.3.3 Some applications

6.3.4 Conclusions

**Chapter 7. Anisotropic failure criterion **

7.1 Introduction

7.2 Extended Cockcroft–Latham (ECL) criterion

7.3 Anisotropic extended Cockcroft–Latham (AECL) criterion

7.4 Quasi-static tests at different stress triaxialities

7.4.1 Numerical aspects

7.4.2 Analysis of results

**Chapter 8. Structural impact of AA7075-T651 plates **

8.1 Introduction

8.2 Experimental and numerical results from Børvik et al. (2010)

8.3 Anisotropic thermoelastic-thermoviscoplastic constitutive relations

8.4 Numerical results and discussion

8.4.1 Ballistic limit curves and velocities

8.4.2 Anisotropy versus isotropy

8.4.3 Sensitivity study

8.5 Conclusions

**Chapter 9. Conclusions and further work **

9.1 Conclusions

9.2 Further work

**. References **