Abstract In the mid 1960s, powerful pulse electron beam accelerators having a voltage of some millions of volts were invented and later used to fracture various materials. Experimental data analysis allowed discovery of a new mode of fracture in several ductile crystals caused by a specific energy supply to the crack tip. The mode differs from well known thermomechanical modes of fracture caused by the “ heat-thermostress-crack ” mechanism. This new mode is called the electron fracture mode (EFM). It is characterized by the following three special features, (i) Initial macrocracks in a specimen do not affect the threshold of fracture; that is, the value of the beam intensity at which the specimen breaks, (ii) The fracture of different materials, which can be very ductile at usual mechanical loads, occurs in a brittle manner; that is, the specimen usually splits by a crack without any residual deformation, (iii) The splitting cracks propagate with supersonic velocities. These data are controversial from the point of view of common fracture mechanics and, hence, they cannot be understood or explained from the traditional position. The purpose of the present study is to create a simple practical model of the EFM. The basic viewpoint can be briefly summarized as follows: during irradiation of a solid by a high intensity electron beam, some solid plasma clots are formed and act as “ blades ” or “ wedges,” cutting the crystalline specimen. In the Introduction, experimental data on the EFM are analyzed and discussed, while the peculiarities of the EFM are specified. As a result, it is concluded that the processes caused by the EFM are unusual for the common concepts of fracture mechanics. In Section 2 the invariant Γ-integrals of an electromagnetic deformable medium are modified for supersonic singularities. The basic model and some problems serving to explain and describe the EFM are formulated. In Section 3, the relativistic electron interactions in beams are considered. Using Γ-integrals, we derive the law of the interaction of two moving relativistic charges; that is, the generalized Coulomb's law for relativistic charges. In particular, when two relativistic electrons, e, move with the same velocity, v, one behind the other along a rectilinear trajectory, the force, F, acting upon the rear electron is equal to: where R is the distance between the electrons, c is the speed of light in the vacuum, and a is the phase-speed of light in a medium having electromagnetic constants, μ, ϵ, and ϵ'. It appears that two electrons moving faster than the phase-speed of light attract one to the other, as distinct from the common Coulomb law. Hence, the beams of such relativistic electrons tend to self-pack and self-compress. The latter problem is studied using a periodic chain model of the electron beam. In Section 4, the dynamic elastic problem of supersonic cutting by a thin wedge is formulated and solved, and the drag force is calculated. In Section 5, the problem of deceleration of the moving wedge is solved in quasi-steady approximation. The length of a resulting cut, that is, the final crack, is determined. Some applications of the analytical solutions are given. In Section 6, the theoretical results are analyzed and compared with experimental results. The role of relativistic electrons is estimated and some parameters of solid-state electron plasma clots are defined. In the Conclusion, the necessity of further study of this mysterious phenomenon is emphasized.