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Material failure plays an important role in human life. By investigating the failure mechanisms, people can more precisely predict the failure conditions to develop new products, to enhance product performances, and most importantly, to save lives. This work consists of three parts corresponding to three different failure mechanisms in metals, i.e., the localized necking in sheet metals, the bifurcation in bulk and sheet metals, and the ductile fracture induced by the void nucleation, growth, and coalescence. The objective of the first part is to model the localized necking in anisotropic sheet metals to demonstrate that localized geometric softening at a certain stage of deformation rather than the initial defects is the main cause of localized necking. The sheet is assumed to have no initial geometric defects. The deformation process is divided into two stages. The critical strains for a neck to form are obtained from a Considère-type criterion. The defect ratio at the neck formation is obtained using an energy-based approach. The neck evolution is considered. A novel failure criterion is proposed. Two types of necks are fond to be most competitive to cause material failure during continued deformation. The forming limit curves are hereby found to exhibit different characteristics in different region. The predicted forming limit curve for 2036-T4 aluminum is found to fit with the experimental results well. The sheet thickness, the strain hardening behavior, and plastic anisotropy are found to affect the sheet metal formability. More realistic yield criterions and strain hardening behaviors can be implemented into the proposed model. This part provides an alternative approach to modeling the localized necking in anisotropic sheet metals. The objective of the second part is to model the bifurcation in anisotropic bulk and sheet metals to couple plastic anisotropy and the strain hardening/softening behavior and also to identify different bifurcation modes in sheet metals. The material is assumed to exhibit a non-linear strain hardening/softening behavior and to obey the Hill-type Drucker-Prager yield criterion along with a non associated flow rule. The constitutive relations and the conditions for bifurcation in bulk and sheet metals are derived. The internal friction coefficient, plastic anisotropy, the terms introduced by the co-rotational stress rates, and the terms introduced by the stress resultant equilibrium are found to affect the onset of bifurcation. Two bifurcation modes are found to exist in sheet metals. More realistic material properties can be implemented into the proposed model. This part provides an applicable approach to modeling the bifurcation in anisotropic bulk and sheet metals. The objective of the third part is to derive the constitutive relations for porous metals using generalized Green’s functions to better understand the micromechanism of the ductile fracture in metals. The porous metals are assumed to consist of an isotropic, rigid-perfectly plastic matrix and numerous periodically distributed voids and to be subject to non-equal biaxial or triaxial extension. Two types of hollow cuboid RVEs are employed represent the typical properties of porous metals with cylindrical and spherical voids. The microscopic velocity fields are obtained using generalized Green’s functions. The constitutive relations are derived using the kinematic approach of the Hill-Mandel homogenization theory and the limit analysis theory. The macroscopic mean stress, the porosity, the unperturbed velocity field, and the void distribution anisotropy are found to affect the macroscopic effective stress and the microscopic effective rate of deformation field. The proposed model is found to provide a rigorous upper bound. More complicated matrix properties (e.g., plastic anisotropy) and void shapes can be implemented into the proposed model. This part provides an alternative approach to deriving the constitutive relations for porous metals.

Glacial isostatic adjustment (GIA) is the process by which the solid Earth responds to past and present-day changes in glaciers, ice caps, and ice sheets. This thesis focuses on vertical crustal motion of the Earth caused by GIA, which is influenced by several factors including lithosphere thickness, mantle viscosity profile, and changes to the thickness and extent of surface ice. The viscosity of the mantle beneath Antarctica is a poorly constrained quantity due to the rarity of relative sea-level and heat flow observations. Other methods for obtaining a better-constrained mantle viscosity model must be investigated to obtain more accurate GIA model predictions. The first section of this study uses seismic wave tomography to determine mantle viscosity. By calculating the deviation of the P- and S-wave velocities relative to a reference Earth model (PREM), the viscosity can be determined. For Antarctica mantle viscosities obtained from S20A (Ekstrom and Dziewonski, 1998) seismic tomography in the asthenosphere range from 1016 Pa∙s to 1023 Pa∙s, with smaller viscosities beneath West Antarctica and higher viscosities beneath East Antarctica. This agrees with viscosity expectations based on findings from the Basin and Range area of North America, which is an analogue to the West Antarctic Rift System. Section two compares bedrock elevations in Antarctica to crustal thicknesses, to infer mantle temperatures and draw conclusions about mantle viscosity. Data from CRUST 2.0 (Bassin et al., 2000), BEDMAP (Lythe and Vaughan, 2001) and specific studies of crustal thickness in Antarctica were examined. It was found that the regions of Antarctica that are expected to have low viscosities agree with the hot mantle trend found by Hyndman (2010) while the regions expected to have high viscosity are in better agreement with the trend for cold mantle. Bevis et al. (2009) described new GPS observations of crustal uplift in Antarctica and compared the results to GIA model predictions, including IJ05 (Ivins and James, 2005). Here, we have generated IJ05 predictions for a three layered mantle (viscosities ranging over more than four orders of magnitude) and compared them to the GPS observations using a χ2 measure of goodness-of-fit. The IJ05 predictions that agree best with the Bevis et al. observations have a χ2 of 16, less than the null hypothesis value of 42. These large values for the best-fit model indicate the need for model revisions and/or that uncertainties are too optimistic. Equally important, the mantle viscosities of the best-fit models are much higher than expected for West Antarctica. The smallest χ2 values are found for an asthenosphere viscosity of 1021 Pa•s, transition zone viscosity of 1023 Pa∙s and lower mantle viscosity of 2 x 1023 Pa∙s, whereas the expected viscosity of the asthenosphere beneath West Antarctica is probably less than 1020 Pa∙s. This suggests that revisions to the IJ05 ice sheet history are required. Simulated annealing was performed on the ice sheet history and it was found that changes to the recent ice load history have the strongest effect on GIA predictions. / Graduate