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Dislocation Density Based Modelling Extensions
An alternative choice on internal state variable, the dislocation density ratio, is chosen to replace the evolving internal stress like variable of the original mechanical threshold stress model in Chapter 5. The model implementation using the alternative formulation of the temperature and rate dependent model of Chapter 3 is now considered using the islocation density ratio. Within the new modelling environment, various extensions are discussed and implemented to include further physical phenomena present during metal forming.
The inclusion of geometrically necessary dislocations and stage IV hardening is achieved by including an additional internal state variable to represent the average slip plane lattice incompatibility. Extending the microstructural evolution equation to include thermal or static recovery of statistical dislocations is also included. By viewing the microstructure as a system of channels or regions of low dislocation density, separated by parallel narrow walls with a high density of segmented edge dislocations, cyclic eects are also included.
To model cyclic eects, a back stress and additional internal state variable is considered. The additional internal state variable represents the density of dislocations trapped in the walls but partially recoverable upon stress reversal. Both back stress and partially recoverable dislocation density ratio internal state variables have their own evolution equations and are eected by load reversal. The dislocation density ratio based model with cyclic behaviour is implemented numerically into the combined hardening model framework validated in Chapter 2. The characterisation of the dislocation density based combined hardening model is nally illustrated on digitised experimental data for two dierent metal alloys.
An extension on the isotropic dislocation density ratio model is also covered to include recrystallisation. Some of the foundation theory on recrystallisation modelling is rst discussed as well as the assumptions made in the current modelling approach.
The model describes multiple waves of recrystallisation and each recrystallised or unrecrystallised volume fraction has its own set of internal state variables. The numerical implementation as well as choices made to keep track of, initialise or shift internal state variables as needed at the onset or completion of a specic wave of recrystallisation are discussed. The recrystallisation model is nally calibrated to dynamic recrystallisation data on Cobalt and Copper.
1 Introduction
1.1 Structure
1.1.1 Chapter 2: Verication of the User Material Framework
1.1.2 Chapter 3: Temperature and Rate Eects .
1.1.3 Chapter 4: Characterising Imperfect Compression Data for Cemented Carbides .
1.1.4 Chapter 5: Dislocation Density Based Modelling Extensions .
1.1.5 Chapter 6: Roughing of a Steel Alloy
1.1.6 Chapter 7: Conclusions .
1.1.7 Appendices
2 Verication of the User Material Framework
2.1 Comparison exampl
2.1.2 Compression of a cylindrical specimen .
2.2 Software package comparison .
2.3 Incremental hypo-elastoplasticity: an Abaqus user material framework .
2.3.1 Deformation theory .
2.3.2 Displacement and velocity .
2.3.3 Incremental solution update in Abaqus
2.3.4 Hypo-elastoplasticity
2.3.5 Isotropic Hardening
2.3.6 Consistent Tangent Formulation .
2.3.7 Combined Hardening .
2.3.8 Numerical Implementation .
2.4 Material Framework Verication .
3 Temperature and Strain Rate Eects
3.1 The Kinetic Equation
3.2 Kocks-Mecking work hardening
3.3 The Mechanical Threshold Stress Model .
3.4 Numerical Implementation .
3.4.1 Determining the evolving thermal stress .
3.4.2 Analytical Gradients
3.5 Characterisation to Experimental Data
3.6 Temperature and rate dependent compression of a cylindrical test specimen
4 Characterising Imperfect Compression Data for Cemented Carbides
4.1 Experimental Data
4.1.1 Simple Room Temperature Characterisation
4.1.2 Characterisation Accuracy
4.1.3 Finite element based inverse analysis
4.2 Simultaneous Estimation of Experimental and Material Parameters
4.3 Compression of a high pressure high temperature capsule
4.4 Conclusions
5 Dislocation Density Based Modelling Extensions
5.1 Dislocation density based model variation
5.2 Geometrically necessary dislocations and stage IV hardening
5.3 Thermal recovery .
5.4 Cyclic eects .
5.5 Numerical Implementation
5.6 Model calibration on cyclic data:
5.7 Recrystallisation .
5.8 Numerical Implementation .
5.9 Model calibration on recrystallisation data
6 Roughing of a Steel Alloy.
7 Conclusions
Bibliography
