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This paper presents a methodology for the development of discontinuity-capturing operators for general elastodynamics. These operators are indicated for problems with sharp gradients in the space and in the time. The development here presented is based on the methodology for obtaining discontinuity-capturing operators developed by Dutra do Carmo and Galeão (1986) for diffusion–convection problems and is inspired in the works presented by Hughes and Hulbert (1988 and 1990). It is shown that their operator belongs to the families of operators developed here. The formulation is applied to one-dimensional and two-dimensional problems. The results show that the method produces better results than classic methods for the one dimensional case and presents robust performance for the two-dimensional case.