  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.
