This thesis aims at applying new concepts from solid state physics and statistical mechanics - chaos theory and fractal geometry - to the study of nonlinear dynamic systems. More precisely, it deals with a two-dimensional continuum elastoplastic Mohr-Coulomb model, commonly used to simulate pressure-sensitive materials (e.g., soils, rocks and concrete) subjected to stress-strain fields, normally found in general soil or rock mechanics problems (e.g., slope stability and underground excavations). It is shown that such many-body system is spontaneously driven to a state at the edge of chaos, called self- organized criticality (SOC), capable of developing long- range interactions in space and long-range memory in time. A new entropic form proposed by C. Tsallis is presented and shown that it is the suitable theoretical framework to deal with these problems. Furthermore, the index q of the Tsallis entropy, which measures the degree of non- additivity of the system, is calculated, for the first time, for an elastoplastic model. In addition, as is usual in non-equilibrium systems with threshold dynamics, the model changes its symmetry, from translational to fractal (that is, self-similar), leading to what is called discrete scale invariance. It is shown that this special type of scale invariance, characterized by systematic oscillatory deviations from the fundamental power-law behavior, can be used to predict the failure of heterogeneous materials, while the process is still being build-up, i.e., from precursory signals, typical of progressive failure processes. Specifically, this framework was applied, for the first time, not only to the elastoplastic geomechanical model, but to laboratory tests in sedimentary rocks as well. Finally, it is interesting to realize that the above- mentioned behaviors are also displayed by the binomial multifractal function, known to adequately describe multiplicative cascading processes.