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We present a novel constitutive model using the framework of strain-limiting theories of elasticity for an evolution of quasi-static anti-plane fracture. The classical linear elastic fracture mechanics (LEFM), with conventional linear relationship between stress and strain, has a well documented inconsistency through which it predicts a singular cracktip strain. This clearly violates the basic tenant of the theory which is a first order approximation to finite elasticity. To overcome the issue, we investigate a new class of material models which predicts uniform and bounded strain throughout the body. The nonlinear model allows the strain value to remain small even if the stress value tends to infinity, which is achieved by an implicit relationship between stress and strain. A major objective of this paper is to couple a nonlinear bulk energy with diffusive crack employing the phase-field approach. Towards that end, an iterative L-scheme is employed and the numerical model is augmented with a penalization technique to accommodate irreversibility of crack. Several numerical experiments are presented to illustrate the capability and the performance of the proposed framework We observe the naturally bounded strain in the neighborhood of the crack-tip, leading to different bulk and crack energies for fracture propagation.