AbstractIn this paper an implicit peridynamic (PD) framework is devised for the solution of quasi-static problems involving brittle and quasi-brittle fracture. A common way to deal with such problems, in explicit frameworks, is to incorporate artificial damping into the PD equation of motion. This may contribute to non-physical behavior of the system. Also, implicit frameworks, based on the Newton-Raphson method, fail to trace the snap-back behavior. To this end, we develop an implicit peridynamic framework to deal with quasi-static problems involving snap-back/snap-through instabilities. This is achieved through two different approaches. The first approach is based on the global arc-length method applicable to problems with zero displacement constraints. The second one is based on a displacement-controlled arc-length method, suitable for problems with non-zero prescribed displacements. The framework is developed for linear elastic as well as softening (degrading) material damage models, respectively appropriate for brittle and quasi-brittle fracture modeling. The robust performance of the proposed framework, within various loading scenarios and damage models, is demonstrated. To boost the numerical performance of the proposed PD framework, a hybrid integration scheme is employed. We demonstrate the advantages of this scheme over the conventional (standard) one in terms of efficiency for quasi-static problems.