This part addresses some fundamental aspects about the use of standard constitutive equations to model strong discontinuities (cracks, shear bands, slip lines, etc.) in solid mechanics analyzes. The so called strong discontinuity analysis is introduced as a basic tool to derive a general framework, in which different families of constitutive equations can be inscribed, then allowing to extract relevant aspects for the intended analysis. In particular, a ling between continuum ad discrete approaches to the strain localization phenomena is obtained. Applications to standard continuum damage and elasto-plastic constitutive equations are presented. Relevant aspects to be considered in the numerical simulation of the problem are alo presented.