Fracture is one of the most common mechanisms of failure in engineering brittle materials. Numerical methods provide efficient tools to both predict the crack propagation path and also understand the fracture process. The phase-field method has proved to be an efficient numerical method due to its ability to simulate crack initiation and propagation without any ad-hoc criteria. It uses a scalar damage variable to model the discrete crack as a diffuse crack phase field. In this work, we have adopted the hybrid phase field model for simulating the benchmark problems in quasi-static elastic brittle fracture. Later, we have extended the hybrid model to dynamic brittle fracture and quasi-static thermo-elastic brittle fracture problems. Numerical examples involving dynamic crack propagation, crack branching and complex crack propagations in thermo-elastic cases have been studied. Functionally graded materials with linear material gradation have been adopted to improve the behavior of the specimen in thermo-elastic quenching problem and the results have been studied. Throughout this study, a staggered solution scheme is used to solve the system of nonlinear partial differential equations using an open-source finite element software FEniCS. The solution to this system of equations drives the crack propagation path, displacement, and temperature fields in our problems.