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The purpose of this work is to study physically possible crack propagation at constant velocity inside a discrete solid by means of theoretical analysis supported by numerical simulations. Analytical solutions are delivered for fracture problems in one dimensional chains, a double chain and square lattices. Evaluation of obtained solutions required implementation of numerical algorithms for computation of integral transforms. Consideration of one-dimensional cases, namely a simple chain of oscillators and a chain of masses with non-local interactions, allowed to examine the validity of derived formulae by a complementary computer simulation of a corresponding dynamic system. Starting from simple models, the analysis of physically admissible and forbidden fracture regimes has been performed. The analytical predictions of possible steady states found a good agreement with a purely numerical scheme. The work discusses the advantages of different approaches to study steady-state failure processes: either with energetic or load characteristics. These attributes of fracture mechanics are shown to be effient for quantifying global predictions, e.g. a choice a particular loading condition for achieving a certain value of a crack speed. However, it was demonstrated that derivation of these characteristics is not enough and consideration of the displacement or stress fields should be performed. The results on chains with non-local interactions between the oscillators illustrated the features of failure at micro-level. Namely, different combinations of microscopic parameters, that result in the same bulk quantities, reflect different patterns of crack propagation in discrete solids. A problem of a separation a double chain compounded by two chains with different properties shows the peculiarities of parameters mismatch. Particularly, it was established that, contrary to quasi-static problems, a steady-state separation is necessarily caused by forces, applied to each chain, of different values. Furthermore, distinct material parameters of chains give a chance for the observation of the supersonic fracture of the structure. Increasing the problem dimension from chains to lattices, several new features emerged. For instance, the behaviour of displacements along a crack path changes. Moreover, the admissibility analysis is expanded to the consideration of possible fracture behind a crack tip. The outcomes predict crack propagation regimes with high energy release rates be accompanied by snapping of the springs on the faces of the original moving crack. The evaluation of displacement eld in the direction orthogonal to a crack path is also presented. The contrast in material properties in anisotropic lattices and mismatch of material properties in dissimilar lattices unveiled different scenarios of admissible regimes. Furthermore, the question of the choice of a particular fracture criterion is addressed. Two history-dependent criteria are compared to the classical one of threshold elongation for linear bonds. The results show that steadystate regimes can be reached in the low subsonic crack speed range which can not be according to the classical criterion. Repercussions in terms of load and crack opening versus velocity are explained in details. Once known the steady-state regimes of fracture propagation, a procedure for applying history-dependent criteria emerges as not restricted to the two examined ones and opens the way to dierent and more complex problems.