The evolution of the fracture process zone and its influence on dynamic crack propagation are investigated using a simple constitutive assumption to model the material in the fracture process zone. The mode III problem of anti-plane shear is examined using a finite difference scheme to obtain the full field solution. Unlike the classical approach of determining the steady state solutions for imposed constant speed of crack propagation, here the fracture criterion is imposed along with the formulation of the boundary-initial value problem and the coupled equations are solved numerically to determine the generation and growth of the fracture process zone as well as the crack tip. The results of the simulation indicate that the constitutive behavior of the process zone material (or equivalently the evolution of the process zone itself) plays a key role in determining the dynamics of fast fracture.