Some Characteristic behavior of the T*ε is identified in this paper through an extensive numerical study. T*ε is a near tip contour integral and has been known to measure the magnitude of singular deformation field at crack tip for arbitrary material models. In this paper, T*ε is found to behave quite differently for different choices of near tip integral contours. If the integral contour moves with advancing crack tip (moving contour), then T*ε measures primarily the energy release rate at the crack tip. It is very small for metallic materials, and tends to zero in the limit as Δa→0 for low hardening materials. Thus, T*ε evaluated on a moving contour tends to zero as ε→0 and Δa→0, for low hardening materials. If the integral contour elongates as crack extends (elongating contour), then T*ε measures total energy inside the volume enclosed by Γε [i.e., the energy dissipated in the extending wake] plus the energy release at the crack tip. Furthermore, the difference in the behavior of CTOA and T*ε, when the applied load is slightly perturbed, is identified. The CTOA is found to be quite insensitive to applied load change. T*ε is found to be roughly proportional to the square of the applied load. The functional shape of T*ε in terms of the size ε of integral contour (for the elongating contour case), is identified, using the crack tip asymptotic formula of Rice (1982). Also, the behaviors of CTOA and T*ε are discussed from the view point of Rice's asymptotic solution. It is recommended that as a crack tip parameter for ductile materials, T*ε with elongating path be used. CTOA is sometimes not very sensitive to the applied load change, therefore it may create some numerical problems in application phase crack propagation analysis.