This thesis examines the qualitative and quantitative variation in local plastic deformation and surface roughening due to crystallographic texture in body-centered cubic materials, specifically interstitial-free steel sheet and molybdenum foil and sheet. Complex forming operations currently used in industrial manufacturing lead to high material failure rates, due in part to the severity of the applied strain path. A multi-scale model was developed to examine the contribution of mesoscopic and local microscopic behaviour to the macroscopic constitutive response of bcc metals during deformation. The model integrated a dislocation-based hardening scheme and a Taylor-based crystal-plasticity formulation into the subroutine of an explicit dynamic FEM code, LS-DYNA. Numerical analyses using this model were able to predict not only correct grain rotation during deformation, but variations in plastic anisotropy due to initial crystallographic orientation. Simulations of molybdenum foil under uniaxial tension supported the existence of bending due to local variations in plastic anisotropy, confirmed with good quantitative agreement by experimental measurements of surface roughening. A series of two-stage strain-path tests were performed, revealing a prestrain-dependent softening of both the steel and molybdenum samples when an orthogonal secondary strain path is applied. Numerical analyses of these tests overestimate macroscopic hardening during complex loading, due in part to the dynamic nature of the FEM code used. / Thesis (Ph.D, Mechanical and Materials Engineering) -- Queen's University, 2009-04-15 11:51:04.518

Further understanding of mesoscale slip mechanics is crucial to future development of polycrystalline metals with improved performance. The research contained within this thesis aims to characterize localized mesoscale slip on slip bands further through two studies. First, a comprehensive comparison of slip system identification techniques was carried out to further validate each method as well as compare advantages and disadvantages of each. Second, slip bands in the presence of grain boundaries were studied to better characterize the dislocation content and behavior. In the first study, the use of SEM-DIC, AFM, ECCI, and HR-EBSD to characterize slip-system activity was assessed on the same material volume of Ti-7Al. This study presents a robust comparison of the various methods for the first time, including an assessment of their advantages and disadvantages, and how they can be used effectively in a complementary manner. The analysis of the different approaches was carried out in a blind manner independently at three different universities. A Ti-7Al specimen was deformed in uniaxial tension to approximately 3% axial strain, and the active slip systems were independently identified using (i) trace analysis; (ii) in-SEM digital image correlation, (iii) observations of residual dislocations from ECCI, and (iv) long-range rotation gradients through HR-EBSD, with consistent trace identification in all cases. Displacement data from AFM was used to augment the SEM-DIC displacement data by providing complementary out of plane displacement information. Furthermore, short-range dislocation gradients (measured by DIC) provided insight into the residual geometrically necessary dislocation (GND) content, and was consistent with the GND content extracted from EBSD data and ECCI images, confirming the presence of residual GNDs on the dominant slip systems resulting in visible slip bands. These approaches can be used in tandem to provide multi-modal information on slip band identification, strain and orientation gradients, out-of-plane displacements, and the presence of GNDs and SSDs, all of which can be used to inform and validate the development of dislocation-based crystal plasticity and strain gradient models. In the second study, shear strain profiles along slip bands in a modified Rolls-Royce nickel superalloy (RR1000) were analyzed for a tensile sample deformed by 2%. The strain increased with distance away from a grain boundary (GB), with maximum shear strain towards the center of the grain, indicating that dislocation nucleation generally occurred in the grain interior. The strain gradients in the neighborhood of the GBs were quantified and generally correlated with rotation about the active slip system line direction. This leads to an ability to determine the active slip system in these regions. The dislocation spacing and pileup stresses were inferred. The dislocation spacing closely follows an Eshelby analytical solution for a single ended pileup of dislocations under an applied stress. The distribution of pileup stress values for GBs of a given misorientation angle follows a log-normal distribution, with no correlation between the pileup stress and the GB misorientation angle. Furthermore, there is no observed correlation between various transmissivity factors and slip band pileup stress. Hence it appears that the obstacle strength of any of the observed GBs is adequate to facilitate the dislocation pileups present in the slip bands. However, slip band transmission does correlate with transmissivity factors, with the current study focusing on the Luster and Morris m'-factor. Observation of strain profiles of transmitted bands indicate dislocation nucleation locations.

BCC metals are commonly used throughout the world and understanding their deformation behavior, especially at the sub-grain level, is essential for their continued use in technological advancements. Correctly and confidently characterizing the active slip systems in BCC materials has been a difficult task throughout past research. The research described in this thesis utilizes high resolution digital image correlation (HRDIC) and relative displacement ratio (RDR) analysis to accurately characterize active slip systems in large grained BCC tantalum and provides new insights into dislocation nucleation sites, relative CRSS values for {110} and {112} slip systems, the correlation between GB transmission factors and strain gradients, the relative length of NBGZs, and slip transmission. A 99.99% pure tantalum oligo sample was sputtered with gold and remodeled to provide high resolution data points to be used in HRDIC. The high resolution of the gold remodeled samples combined with a RDR analysis made it possible to confidently identify active slip systems during tensile deformation at room temperature. One of the observations from this analysis was the discrepancy between the observed active slip systems and those predicted from a simple single-CRSS Schmid's Law. By considering the active systems observed in grains with a range of orientation, it was concluded that the {112} slip systems have a higher CRSS than the {110} by 6.7%. Independent CPFE simulations and experiments on single crystal samples of the same material, agreed with our findings establishing a range of increased CRSS for {112} of 3.9%-7.1%. These conclusions are compared with the small number of available estimates of the CRSS ratio, and lie in between the value of equal CRSS used by most modelers, and experimental estimates of 15-25% higher for {112}. The identified active slip systems were also used in the Luster and Morris equation to calculate each GBs transmissivity factor - an estimate of strain incompatibility between neighboring grains. Results indicate that there is an inverse correlation between GB transmissivity and strain gradient slope, as well as a positive correlation between GB transmissivity and slip trace reorientation for some GBs. Only one instance of slip transmission was observed from the 24 GBs analyzed, suggesting it is an uncommon occurrence in BCC tantalum. An analysis of the length of the NBGZ in relation to slip and strain gradients was compared to previous studies and suggests the relative and absolute length of the NBGZ changes with grain size, at least for large length scales. Strain gradients for each side of the GB were measured and results indicated steep negative strain gradient slopes that suggest dislocation nucleation in the GBs and propagation towards the interior of the grain. When compared against the transmissivity factor, an inverse relationship was found to exist between strain gradients and high transmissivity factors.

The ability to predict fatigue crack growth rates is essential in safety critical systems. The discovery of fatigue crack closure in 1970 caused a flourish of research in attempts to simulate this behaviour, which crucially affects crack growth rates. Historically, crack tip plasticity models have been based on one-dimensional rays of plasticity emanating from the crack tip, either co-linear with the crack (for the case of plane stress), or at a chosen angle in the plane of analysis (for plane strain). In this thesis, one such model for plane stress, developed to predict fatigue crack closure, has been refined. It is applied to a study of the relationship between the apparent stress intensity range (easily calculated using linear elastic fracture mechanics), and the true stress intensity range, which includes the effects of plasticity induced fatigue crack closure. Results are presented for all load cases for a finite crack in an infinite plane, and a method is demonstrated which allows the calculation of the true stress intensity range for a growing crack, based only on the apparent stress intensity range for a static crack. Although the yield criterion is satisfied along the plastic ray, these one-dimensional plasticity models violate the yield criterion in the area immediately surrounding the plasticity ray. An area plasticity model is therefore required in order to model the plasticity more accurately. This thesis develops such a model by distributing dislocations over an area. Use of the model reveals that current methods for incremental plasticity algorithms using distributed dislocations produce an over-constrained system, due to misleading assumptions concerning the normality condition. A method is presented which allows the system an extra degree of freedom; this requires the introduction of a parameter, derived using the Prandtl-Reuss flow rule, which relates the magnitude of slip on complementary shear planes. The method is applied to two problems, confirming its validity.

Die vorliegende Arbeit beschäftigt sich mit Festkörperkontinuumsmechanik und Metall- bzw. Kristallplastizität auf verschiedenen Längenskalen. Diesbezüglich besteht die Arbeit aus drei größeren Teilen. Im ersten Teil werden Verformungsvorgänge mit expliziter FEM (Finite-Elemente-Methode) und einem makroskopischen phänomenologischen Modell der Viskoplastizität simuliert. Hierbei wird sich auf das Gleichkanalwinkelpressen (ECAP) eines Metallbarrens und die Stauchung einer sogenannten Crashbox konzentriert. In beiden Fällen gelingt es, die im Experiment bereits beobachtete Lokalisierung der Verformung korrekt wiederzugeben. Da bei den Simulationen die konkrete Mikrostruktur des Materials vernachlässigt wird, werden diese Lokalisierungsphänomene als verformungsinduziert angesehen. Der zweite Teil beschäftigt sich mit der Erweiterung des viskoplastischen Modells, sodass mikroskopische Vorgänge der Gitterdefektstruktur des Materials berücksichtigt werden können. Dazu wird ein Modell des dynamischen Verhaltens von Versetzungspopulationen entwickelt und an das makroskopische viskoplastische Modell gekoppelt. Auf diese Weise können Aspekte der sogenannten Kornfeinung – einem komplexen Strukturbildungsprozess von Versetzungen und anderen Gitterdefekten – erfasst werden. Allerdings kann die für die makroskopischen Eigenschaften entscheidende Bildung von Subkorngrenzen auf diese Weise nicht abgebildet werden. Um dies zu erreichen, wird im dritten Teil der Arbeit eine mesoskopische Theorie der Kristallplastizität mit kontinuierlich verteilten Versetzungen verwendet und weiterentwickelt. Hierbei werden die für eine Subkornbildung wesentlichen Freiheitsgrade hinzugenommen, die Anzahl phänomenologischer Ansätze und zugehöriger Materialparameter aber so klein wie möglich gehalten. Mit dieser Kontinuumsversetzungstheorie (KVT) gelingt es, die Bildung von Subkorngrenzen bei großen plastischen Verformungen eines Kristallits zu verfolgen. Bei den impliziten FEM-Simulationen wird ebenfalls eine Lokalisierung beobachtet, allerdings in Bezug auf die Aktivität der Versetzungen in verschiedenen Gleitebenen. Dementsprechend wird dieses Lokalisierungsphänomen als versetzungsinduziert angesehen. Der Beitrag der vorliegenden Arbeit liegt zum einen in der Aufarbeitung und Gegenüberstellung unterschiedlicher methodischer Herangehensweisen zur Modellierung verformungs- und versetzungsinduzierter Lokalisierungsphänomene. Zum anderen wird eine Analyse und Vereinheitlichung der geometrisch linearen KVT nach Berdichevsky & Le vorgenommen. Wie sich dabei zeigt, verhindern inhärente kinematische Einschränkungen der Theorie die Simulation einer Subkornbildung. Aus diesem Grund wird die konsistente geometrisch nichtlineare KVT von Gurtin aufgegriffen und erweitert. Mit einem daraus abgeleiteten elastisch und plastisch anisotropen Modell der Einkristallviskoplastizität wird der Nachweis erbracht, dass die Subkornbildung damit simuliert werden kann. Darüber hinaus wird eine Aufbereitung und Synthese von Algorithmen zur numerischen Lösung der zugehörigen Feldgleichungen mittels der Methode der finiten Differenzen und der finiten Elemente geliefert. Zudem werden beide Näherungsverfahren in Bezug auf Vor- und Nachteile sowie thermodynamische Konsistenz bei der Anwendung auf Mehrfeldprobleme miteinander verglichen. / The present thesis deals with solid continuum mechanics applied to metal and crystal plasticity on different length scales. In this respect, the work consists of three larger parts. In the first part, deformation processes are simulated with explicit FEM (Finite Element Method) and a macroscopic phenomenological model of viscoplasticity. Here the focus is on the Equal-Channel Angular Pressing (ECAP) of a metal billet and the compression of a so-called crash box. In both cases it is possible to correctly reproduce the localization of the deformation as already observed in the experiment. Since the concrete microstructure of the material is neglected in the simulations, these localization phenomena are regarded as deformation-induced. The second part deals with the extension of the viscoplastic model so that microscopic processes of the lattice defect structure of the material can be considered. A model of the dynamic behavior of dislocation populations is developed and coupled to the macroscopic viscoplastic model. In this way, aspects of the so-called grain refinement – a complex structure formation process of dislocations and other lattice defects – can be captured. However, the formation of subgrain boundaries, which is decisive for the macroscopic properties, cannot be predicted in this way. To achieve this, a mesoscopic theory of crystal plasticity with continuously distributed dislocations is used and further developed in the third part of the thesis. Here, the degrees of freedom essential for subgrain formation are added, while the number of phenomenological approaches and associated material parameters are kept as small as possible. With this continuum dislocation theory it is possible to follow the formation of subgrain boundaries during large plastic deformations of a crystallite. In the implicit FEM simulations, localization is also observed, but with respect to the dislocation activity in different slip planes. Accordingly, this localization phenomenon is considered dislocation-induced. The contribution of the present work lies on the one hand in the review and comparison of different methodical approaches to the modeling of deformation- and dislocation-induced localization phenomena. On the other hand, an analysis and unification of the geometrically linear continuum dislocation theory according to Berdichevsky & Le is carried out. As it turns out, inherent kinematic limitations of the theory prevent the simulation of subgrain formation. For this reason the consistent geometrically non-linear continuum dislocation theory from Gurtin is adopted and extended. With the derived model of elastically and plastically anisotropic single crystal viscoplasticity it is proven that subgrain formation can be simulated. Moreover, a preparation and synthesis of algorithms for the numerical solution of the associated field equations using the method of finite differences and finite elements is provided. In addition, both approximation methods are compared in terms of advantages and disadvantages as well as thermodynamic consistency when applied to multi-field problems.