The combined effects of large deformation and material inertia on the displacement gradients are analyzed for rapid mode III crack propagation. The constitutive model is linear at small strains, but the stress level cannot increase beyond a critical level. Steady-state dynamic strain fields are presented, which are valid on the crack line, from the moving crack tip to the boundary with the zone of linear behavior. The large-deformation crack-line fields have been matched to appropriate small-strain elastic fields at a transition boundary between the zones of linear and nonlinear behavior, and thus to the elastodynamic stress-intensity factor of linear elastic fracture mechanics. For the constitutive model considered here, the effect of inertia removes the strain singularity at the crack tip. In addition, a first-order asymptotic field around the crack tip has been constructed. This field relates the crack-opening angle to the crack-line field, and hence to the elastodynamic stress-intensity factor. A criterion of critical total strain has been applied to obtain curves relating the small-strain elastodynamic stress-intensity factor to the crack-tip speed. The balance of rates of energies is briefly discussed in the final section.