Earthquakes are dynamic shear ruptures that nucleate, propagate, and arrest. The arrest of earthquakes is often attributed to the interaction of these dynamic rupture fronts with barriers of either negative stress drop or higher fracture energy. However, it is not clear which one of these is the most effective way of stopping a rupture. In this thesis, we primarily aim to investigate the arrest of dynamic ruptures in heterogeneous fault interfaces by comparing observations from laboratory stick-slip experiments with Linear Elastic Fracture Mechanics (LEFM). To simulate rupture arrest in the laboratory, we use a tougher inclusion at parts of a 760 mm Poly Methyl Methacrylate (PMMA) fault. An array of eight slip sensors positioned across the fault is used to measure fault slip. We then use these slip measurements to constrain the dynamic rupture fronts both in time and space. Dynamic rupture fronts after the inclusion consistently show an apparent time delay ranging from $\mathrm{50\ \mu s}$ to $\mathrm{250\ \mu s}$ for different experiments. For the experiment and LEFM comparison, we adopt the fracture energy of the bare PMMA interface given previously [Svetlizky and Fineberg, 2014] and apply the classical equation of motion of shear cracks described in LEFM. We further use the spatiotemporal behavior of the rupture front to constrain the fracture energy and stress drop on the tougher inclusion, which are also the free parameters in our LEFM model. For cases where the rupture arrests, it is only possible to constrain a lower bound on fracture energy. On the other hand, for cases where the rupture nucleates close to the inclusion, and arrests on the inclusion, it again renucleates beyond the inclusion. This renucleation behavior cannot be captured with the simple LEFM models considered here. LEFM describes earthquakes from a local energy balance perspective. In which, a rupture nucleates as the nonlocal energy release rate balances the local fracture energy in a fault. The latter serves as the dissipative energy from the system that enables rupture propagation. While the concept of local fracture energy is well understood in the fault mechanics community, an ideal range of fracture energies responsible for natural earthquakes remains unknown. Seismological evidence suggests fracture energy spanning from $\mathrm{1\ J/m^{2}}$ to $\mathrm{10\ MJ/m^{2}}$. Recent discussions on the distinction between fracture energy and breakdown work indicate that such high fracture energy estimates are not entirely localized at the rupture tip. Instead, in addition to local dissipation at the rupture tip, there exists a nonlocal dissipation mechanism further from the rupture tip, arising from a long-tailed secondary weakening phase that scales with the final slip. However, the extent to which this nonlocal energy dissipation influences rupture dynamics and arrest remains an open question. To investigate this, we developed dynamic rupture models using the Spectral Boundary Integral Method, incorporating a secondary weakening phase following a primary weakening phase within a linear slip-weakening constitutive law. Our findings indicate that, the presence of a secondary weakening phase leads to higher rupture speeds compared to scenarios without it. This suggests that the additional stress drop associated with the secondary weakening do more to "fuel" the rupture than additional "total fracture energy" or "breakdown work" to arrest the rupture.