A quasi-linear viscoelastic relation that stems from an implicit viscoelastic constitutive body containing a crack is considered. The abstract form of the response function is given first in [Formula: see text], [Formula: see text], due to power-law hardening; second in [Formula: see text] due to limiting small strain. In both the cases, sufficient conditions on admissible response functions are formulated, and corresponding existence theorems are proved rigorously based on the variational theory and using monotonicity methods. Due to the presence of a Volterra convolution operator, an auxiliary-independent variable of velocity type is employed. In the case of limiting small strain, the generalized solution of the problem is provided within the context of bounded measures and expressed by a variational inequality.