Indiana University-Purdue University Indianapolis (IUPUI) / Zirconia (ZrO2) is an important ceramic material with a broad range of applications. Due to its high melting temperature, low thermal conductivity, and high-temperature stability, zirconia based ceramics have been widely used for thermal barrier coatings (TBCs). When TBC is exposed to thermal cycling during real applications, the TBC may fail due to several mechanisms: (1) phase transformation into yttrium-rich and yttrium-depleted regions, When the yttrium-rich region produces pure zirconia domains that transform between monoclinic and tetragonal phases upon thermal cycling; and (2) cracking of the coating due to stress induced by erosion. The mechanism of erosion involves gross plastic damage within the TBC, often leading to ceramic loss and/or cracks down to the bond coat. The damage mechanisms are related to service parameters, including TBC material properties, temperature, velocity, particle size, and impact angle. The goal of this thesis is to understand the structural and mechanical properties of the thermal barrier coating material, thus increasing the service lifetime of gas turbine engines. To this end, it is critical to study the fundamental properties and potential failure mechanisms of zirconia. This thesis is focused on investigating the structural and mechanical properties of zirconia. There are mainly two parts studied in this paper, (1) ab initio calculations of thermodynamic properties of both monoclinic and tetragonal phase zirconia, and monoclinic-to-tetragonal phase transformation, and (2) image-based finite element simulation of the indentation process of yttria-stabilized zirconia. In the first part of this study, the structural properties, including lattice parameter, band structure, density of state, as well as elastic constants for both monoclinic and tetragonal zirconia have been computed. The pressure-dependent phase transition between tetragonal (t-ZrO2) and cubic zirconia (c-ZrO2) has been calculated using the density function theory (DFT) method. Phase transformation is defined by the band structure and tetragonal distortion changes. The results predict a transition from a monoclinic structure to a fluorite-type cubic structure at the pressure of 37 GPa. Thermodynamic property calculations of monoclinic zirconia (m-ZrO2) were also carried out. Temperature-dependent heat capacity, entropy, free energy, Debye temperature of monoclinic zirconia, from 0 to 1000 K, were computed, and they compared well with those reported in the literature. Moreover, the atomistic simulations correctly predicted the phase transitions of m-ZrO2 under compressive pressures ranging from 0 to 70 GPa. The phase transition pressures of monoclinic to orthorhombic I (3 GPa), orthorhombic I to orthorhombic II (8 GPa), orthorhombic II to tetragonal (37 GPa), and stable tetragonal phases (37-60 GPa) are in excellent agreement with experimental data. In the second part of this study, the mechanical response of yttria-stabilized zirconia under Rockwell superficial indentation was studied. The microstructure image based finite element method was used to validate the model using a composite cermet material. Then, the finite element model of Rockwell indentation of yttria-stabilized zirconia was developed, and the result was compared with experimental hardness data.