Soft materials undergo fracture under large deformations, experiencing both geometric and material nonlinearities. As the stress fields at their crack tips are highly nonlinear, the fracture process is not adequately described by traditional linear elastic fracture mechanics (LEFM). Several experimental studies with neo-Hookean hydrogels have demonstrated the existence of a nonlinear length scale $\ell_\mathrm{NL}$ that governs crack propagation [1]. This length scale is material-specific, as soft materials exhibit varying levels of strain-stiffening or softening. In contrast, geometric nonlinearity is universal, arising from the kinematic description of materials undergoing finite strains. Recent numerical simulations have shown its effects on a propagating crack within a linear elastic material under Mode I loading [2]. Driven by geometric nonlinearities, the crack tip accelerates from the sub-Rayleigh to the super-shear wave speed range, exceeding the theoretical limit predicted by LEFM. Consequently, both material and geometric nonlinearities have been identified as key factors in explaining diverse aspects of dynamic fracture phenomena. Therefore, given the limitations of LEFM, we aim to elucidate the fundamental role of these nonlinearities in the fracture process. In this work, we present a numerical and theoretical study on the Mode I fracture of the Generalized Mooney-Rivlin (GMR) material model. This represents a general class of incompressible materials suitable for describing strain-softening and strain-stiffening behavior in the large-deformation regime. By employing a CTOD-based numerical procedure, we determine the singular values for the elastic fields as well as the intrinsic nonlinear length scale, finding excellent agreement with theoretical predictions. Furthermore, we investigate the contributions of geometric nonlinearity to the material's nonlinear elastic response within the GMR framework. Ultimately, we establish the relationship between the evolution of the elastic modulus and fracture phenomena, such as super-shear propagation speeds and oscillatory propagation instabilities. [1] E. Bouchbinder, T. Goldman, and J. Fineberg. The Dynamics of Rapid Fracture: Instabilities, Nonlinearities and Length Scales. Reports on Progress in Physics 77, no. 4, 2014. [2] M. Pundir, M. Adda-Bedia, and D. S. Kammer. Transonic and Supershear Crack Propagation Driven by Geometric Nonlinearities. Physical Review Letters 132, no. 22, 2024.