In former papers, a concept of crack energy density ε was proposed as the most fundamental parameter in fracture mechanics and it was shown that εεεεε can be evaluated by a path independent integral ε_J without any restriction on constitutive law. In this paper, ε_J -integral is applied to elasto-plastic crack problems under monotonic or cyclic loading; and the availabilities of ε_J -integral are demonstrated. First, it is shown that the physical meanings and applicable ranges of J and ΔJ can be clarified in the light of ε_J. Next, the method of the estimation of ε based on ε_J is established through the finite element analysis of elasto-plastic crack under monotonic loading. Moreover, an analysis under cyclic loading is carried out, and the basic characteristic of ε and the relations between ε and fatigue parameters ΔK and ΔJ are discussed.