An experimental and numerical study of the isothermal and non-isothermal warm formability of an AA3003 aluminum alloy brazing sheet is presented. Forming limit diagrams were determined using warm limiting dome height (LDH) experiments with in situ strain measurement based on digital image correlation (DIC) techniques. Forming limit curves (FLCs) were developed at several temperature levels (room temperature, 100ºC, 200ºC, 250ºC, and 300ºC) and strain-rates (0.003, 0.018, and 0.1s-1). The formability experiments demonstrated that temperature has a significant effect on formability, whereas forming speed has a mild effect within the studied range. Elevating the temperature to 250C improved the formability more than 200% compared to room temperature forming, while forming at lower speeds increased the limiting strains by 10% and 17% at room temperature and 250ºC, respectively. Non-isothermal deep draw experiments were developed considering an automotive heat exchanger plate. A parametric study of the effects of die temperature, punch speed, and blank holder force on the formability of the part was conducted. The introduction of non-isothermal conditions in which the punch is cooled and the flange region is heated to 250C resulted in a 61% increase in draw depth relative to room temperature forming. In order to develop effective numerical models of warm forming processes, a constitutive model is proposed for aluminum alloy sheet to account for temperature and strain rate dependency, as well as plastic anisotropy. The model combines the Barlat YLD2000 yield criterion (Barlat et al., 2003) to capture sheet anisotropy and the Bergstrom (1982) hardening rule to account for temperature and strain rate dependency. Stress-strain curves for AA3003 aluminum alloy brazing sheet tested at elevated temperatures and a range of strain rates were used to fit the Bergstrom parameters, while measured R-values were used to fit the yield function parameters. The combined constitutive model was implemented within a user defined material subroutine that was linked to the LS-DYNA finite element code. Finite element models were developed based on the proposed material model and the results were compared with experimental data. Isothermal uniaxial tensile tests were simulated and the predicted responses were compared with measured data. The tensile test simulations accurately predicted material behaviour. The user material subroutine and forming limit criteria were then applied to simulate the isothermal warm LDH tests, as well as isothermal and non-isothermal warm deep drawing experiments. Two deep draw geometries were considered, the heat exchanger plate experiments developed as part of this research and the 100 mm cylindrical cup draw experiments performed by McKinley et al. (2010). The strain distributions, punch forces and failure location predicted for all three forming operations were in good agreement with the experimental results. Using the warm forming limit curves, the models were able to accurately predict the punch depths to failure as well as the location of failure initiation for both the isothermal and non-isothermal deep draw operations.

An experimental and numerical study of the isothermal and non-isothermal warm formability of an AA3003 aluminum alloy brazing sheet is presented. Forming limit diagrams were determined using warm limiting dome height (LDH) experiments with in situ strain measurement based on digital image correlation (DIC) techniques. Forming limit curves (FLCs) were developed at several temperature levels (room temperature, 100ºC, 200ºC, 250ºC, and 300ºC) and strain-rates (0.003, 0.018, and 0.1s-1). The formability experiments demonstrated that temperature has a significant effect on formability, whereas forming speed has a mild effect within the studied range. Elevating the temperature to 250C improved the formability more than 200% compared to room temperature forming, while forming at lower speeds increased the limiting strains by 10% and 17% at room temperature and 250ºC, respectively. Non-isothermal deep draw experiments were developed considering an automotive heat exchanger plate. A parametric study of the effects of die temperature, punch speed, and blank holder force on the formability of the part was conducted. The introduction of non-isothermal conditions in which the punch is cooled and the flange region is heated to 250C resulted in a 61% increase in draw depth relative to room temperature forming. In order to develop effective numerical models of warm forming processes, a constitutive model is proposed for aluminum alloy sheet to account for temperature and strain rate dependency, as well as plastic anisotropy. The model combines the Barlat YLD2000 yield criterion (Barlat et al., 2003) to capture sheet anisotropy and the Bergstrom (1982) hardening rule to account for temperature and strain rate dependency. Stress-strain curves for AA3003 aluminum alloy brazing sheet tested at elevated temperatures and a range of strain rates were used to fit the Bergstrom parameters, while measured R-values were used to fit the yield function parameters. The combined constitutive model was implemented within a user defined material subroutine that was linked to the LS-DYNA finite element code. Finite element models were developed based on the proposed material model and the results were compared with experimental data. Isothermal uniaxial tensile tests were simulated and the predicted responses were compared with measured data. The tensile test simulations accurately predicted material behaviour. The user material subroutine and forming limit criteria were then applied to simulate the isothermal warm LDH tests, as well as isothermal and non-isothermal warm deep drawing experiments. Two deep draw geometries were considered, the heat exchanger plate experiments developed as part of this research and the 100 mm cylindrical cup draw experiments performed by McKinley et al. (2010). The strain distributions, punch forces and failure location predicted for all three forming operations were in good agreement with the experimental results. Using the warm forming limit curves, the models were able to accurately predict the punch depths to failure as well as the location of failure initiation for both the isothermal and non-isothermal deep draw operations.

In the forming process of steel, large plastic deformation evolves as a complex mechanism. Simulations with polycrystal material modeling provide detailed insight into the material characteristics. However, sheet-bulk metal forming is a complicated process that needs comprehensive information on strain history and complex loading states before and during service life. In general, the transient hardening, Bauschinger effects, and induced anisotropic plastic character make this process even more challenging under non-monotonic loadings. In order to quickly simulate the elastoplastic process under these circumstances without compromising accuracy, one needs to consider sophisticated elastoplastic material models at coarser length scales motivated by microscopic length scale material modeling. Plastic deformation is a microscopic length scale phenomenon that involves the dislocation activities within the grains of a polycrystal. Therefore, a physically motivated crystal plasticity model is developed to consider the plastic transformation based on the mobility of mean dislocation densities on multiple active slip planes. Following the resistance due to non-parallel (forest) and parallel (piled-up) dislocations, the evolution of persistent plastic state is also dealt with in implicit and explicit manners, respectively. To validate the influence of back stresses resulting from the incompatibility of plastic deformation within polycrystals, three statistically informed representative volume elements of DC04 material with different strain histories are deformed under cyclic loadings and compared with experimental data. Due to the higher geometrical resolution, it becomes difficult to solve the prescribed complex plastic transformation process for the entire domain in larger geometries. This leads to the development of an effective material model based on the insights of the microscopic approach. Plastic transformation is driven by a total dislocation density equivalent state variable in an effective material modeling approach. Its evolution describes the non-linear isotropic hardening mechanism. Additionally, the Bauschinger effect can also be calculated with the Armstrong-Fredric kinematic hardening law. However, the absence of the microstructural feature, i.e., texture at this length scale, makes it challenging to include structural anisotropy in the effective material modeling. Advanced anisotropic yield models such as Barlat Yld2004-18p can tackle this problem. However, a further challenging experimental setup is required to predict the 18 parameters of these yield functions. A simulation strategy is proposed in the current work, which utilizes the homogenized stress tensors calculated by microscopic polycrystal material simulations to predict the anisotropic state of DC04 material. Two transformation tensors are optimized to reproduce an accurate representation of the distorted material symmetry. In the end, the effective material simulations are validated, which use the anisotropic yield functions along with the transformed tensors.:1 Introduction 2 Continuum mechanics 2.1 Kinematics 2.2 Strainrates 2.3 Balancelaws 2.3.1 Conservationofmass 2.3.2 Conservationoflinearmomentum 2.3.3 Conservationofangularmomentum 2.3.4 Balanceofenergy 2.3.5 Balanceofentropy 2.4 ConstitutiveTheory 2.4.1 Principleofmaterialcausality 2.4.2 Principleofdeterminism 2.4.3 Principleoflocalaction 2.4.4 Principle of material frame-indifference (material objectivity) 3 Microscopic material model 3.1 Crystallographicnatureofmetallicmaterials 3.1.1 Crystallatticeformetallicmaterials 3.1.2 Latticeplanesanddirections 3.2 Dislocationbasedplasticdeformation 3.2.1 Geometricdescriptionofdislocations 3.2.2 Continuum framework for a crystal plasticity model 3.3 Elasticconstitutivemodel 3.4 Crystalplasticitymaterialmodeling 3.4.1 Criteriafordislocationmobility 3.4.2 Resistancetodislocationmotion 3.4.3 Evolutionofplasticstate 3.5 Integrationoflocalplasticstate 3.5.1 Trustregionmethods 3.6 Homogenization 3.7 NumericalmethodswithinFEMscheme 3.7.1 Weakform 3.7.2 Temporaldiscretization 3.7.3 Domaindiscretization 3.8 Development of statistically informed polycrystal microstructure model 3.8.1 Statisticaldescription 3.8.2 Geometricaldescription 3.8.3 Crystallographicdescription 3.8.4 Representativevolumeelement 3.9 Validationofmicroscopicmaterialmodeling 4 Physically motivated material modeling at macroscopic length scale 4.1 Effectivematerialmodel 4.1.1 Phenomenologicalplasticitymodels 4.1.2 Extension to anisotropic effective material model (Barlat Yld2004-18p) 4.1.3 Evolution of macroscopic plastic transformation 4.2 PredictionandvalidationofBarlateffectivemodel 4.2.1 Methodologyofinvestigation 4.2.2 Virtual polycrystal simulation approach to fit effective material model 4.2.3 Optimization framework to calibrate Barlat anisotropic coefficients 4.3 Resultsanddiscussion 4.3.1 Validationofanisotropicyieldstates 4.3.2 Validation of work hardening with the effective material model Summary and Outlook Bibliography Curriculum Vitae / Bei der Umformung von Stahl kommt es zu großen plastischen Verformungen, die einen komplexen Mechanismus darstellen. Simulationen mit polykristallinen Materialmodellen geben einen detaillierten Einblick in die Materialeigenschaften. Die Blechmassivumformung ist jedoch ein komplizierter Prozess, der umfassende Informationen über die Verformungshistorie und komplexe Belastungszustände vor und während der Lebensdauer erfordert. Im Allgemeinen machen die transiente Verfestigung, Bauschinger-Effekte und der induzierte anisotrope plastische Charakter diesen Prozess unter nicht-monotonen Belastungen zu einer noch größeren Herausforderung. Um den elastoplastischen Prozess unter diesen Umständen schnell zu simulieren, ohne die Genauigkeit zu beeinträchtigen, muss man hochentwickelte elastoplastische Materialmodelle auf gröberen Längenskalen berücksichtigen, die durch die mikroskopische Längenskalen-Materialmodellierung motiviert sind. Plastische Verformung ist ein Phänomen auf mikroskopischer Längenskala, das die Versetzungsaktivitäten innerhalb der Körner eines Polykristalls betrifft. Daher wird ein physikalisch motiviertes Kristallplastizitätsmodell entwickelt, um die plastische Umwandlung auf der Grundlage der Mobilität der mittleren Versetzungsdichten auf mehreren aktiven Gleitebenen zu berücksichtigen. In Anlehnung an den Widerstand durch nicht-parallele (Wald) und parallele (aufgetürmte) Versetzungen wird auch die Entwicklung des dauerhaften plastischen Zustands auf implizite bzw. explizite Weise behandelt. Um den Einfluss von Rückspannungen, die aus der Unverträglichkeit plastischer Verformung innerhalb von Polykristallen resultieren, zu validieren, werden drei statistisch informierte repräsentative Volumenelemente des DC04-Materials mit unterschiedlichen Dehnungsverläufen unter zyklischer Belastung verformt. Aufgrund der höheren geometrischen Auflösung wird es bei größeren Geometrien schwierig, den vorgeschriebenen komplexen plastischen Umwandlungsprozess für den gesamten Bereich zu lösen. Dies führt zu der Entwicklung eines effektiven Materialmodells, das auf den Erkenntnissen des mikroskopischen Ansatzes basiert. Die plastische Umwandlung wird durch eine äquivalente Zustandsvariable der Gesamtversetzungsdichte in einem effektiven Materialmodellierungsansatz gesteuert. Ihre Entwicklung beschreibt den nichtlinearen isotropen Verfestigungsmechanismus. Zusätzlich kann der Bauschinger-Effekt auch mit dem kinematischen Verfestigungsgesetz von Armstrong-Fredric berechnet werden. Das Fehlen der mikrostrukturellen Merkmale, d. h. der Textur auf dieser Längenskala, macht es jedoch schwierig, strukturelle Anisotropie in die effektive Materialmodellierung einzubeziehen. Fortgeschrittene anisotrope Fließmodelle wie das Barlat Yld2004-18p können dieses Problem angehen. Allerdings ist ein weiterer anspruchsvoller Versuchsaufbau erforderlich, um die 18 Parameter dieser Fließfunktionen vorherzusagen. In der vorliegenden Arbeit wird eine Simulationsstrategie vorgeschlagen, die die durch mikroskopische Polykristall-Materialsimulationen berechneten homogenisierten Spannungstensoren nutzt, um den anisotropen Zustand des DC04-Materials vorherzusagen. Zwei Transformationstensoren werden optimiert, um eine genaue Darstellung der verzerrten Materialsymmetrie zu reproduzieren. Am Ende werden die effektiven Materialsimulationen validiert, die die anisotropen Fließfunktionen zusammen mit den transformierten Tensoren verwenden.:1 Introduction 2 Continuum mechanics 2.1 Kinematics 2.2 Strainrates 2.3 Balancelaws 2.3.1 Conservationofmass 2.3.2 Conservationoflinearmomentum 2.3.3 Conservationofangularmomentum 2.3.4 Balanceofenergy 2.3.5 Balanceofentropy 2.4 ConstitutiveTheory 2.4.1 Principleofmaterialcausality 2.4.2 Principleofdeterminism 2.4.3 Principleoflocalaction 2.4.4 Principle of material frame-indifference (material objectivity) 3 Microscopic material model 3.1 Crystallographicnatureofmetallicmaterials 3.1.1 Crystallatticeformetallicmaterials 3.1.2 Latticeplanesanddirections 3.2 Dislocationbasedplasticdeformation 3.2.1 Geometricdescriptionofdislocations 3.2.2 Continuum framework for a crystal plasticity model 3.3 Elasticconstitutivemodel 3.4 Crystalplasticitymaterialmodeling 3.4.1 Criteriafordislocationmobility 3.4.2 Resistancetodislocationmotion 3.4.3 Evolutionofplasticstate 3.5 Integrationoflocalplasticstate 3.5.1 Trustregionmethods 3.6 Homogenization 3.7 NumericalmethodswithinFEMscheme 3.7.1 Weakform 3.7.2 Temporaldiscretization 3.7.3 Domaindiscretization 3.8 Development of statistically informed polycrystal microstructure model 3.8.1 Statisticaldescription 3.8.2 Geometricaldescription 3.8.3 Crystallographicdescription 3.8.4 Representativevolumeelement 3.9 Validationofmicroscopicmaterialmodeling 4 Physically motivated material modeling at macroscopic length scale 4.1 Effectivematerialmodel 4.1.1 Phenomenologicalplasticitymodels 4.1.2 Extension to anisotropic effective material model (Barlat Yld2004-18p) 4.1.3 Evolution of macroscopic plastic transformation 4.2 PredictionandvalidationofBarlateffectivemodel 4.2.1 Methodologyofinvestigation 4.2.2 Virtual polycrystal simulation approach to fit effective material model 4.2.3 Optimization framework to calibrate Barlat anisotropic coefficients 4.3 Resultsanddiscussion 4.3.1 Validationofanisotropicyieldstates 4.3.2 Validation of work hardening with the effective material model Summary and Outlook Bibliography Curriculum Vitae

In an attempt to improve the quality in finite element analysis of thermoformed components, a method for predicting the thickness distribution is presented. The strain induced anisotropic material behaviour in the amorphous polymers of concern is also taken into account in the method. The method comprises of obtaining raw material data from experiments, followed by a simulation of the vacuum thermoforming process where hyperelastic material behaviour is assumed. The theory of hyperelasticity that was applied was based on the Ogden model and implemented in the FE-software LS-DYNA. Material behaviour from thermoformed prototypes is examined by experiments and implemented together with the mapped results from the thermoforming simulation in a succeeding FE-model. For the latter, the three-parameter Barlat model was suggested, giving the possibility to account for anisotropic material behaviour based on an initial plastic strain.