The present paper discussed the critical depth, i.e., the depth at which the thermal tensile stress reaches a maximum, caused by the frictional excitation of a fast moving asperity. In the study, the critical depth was computed directly by maximizing the thermal tensile stress with respect to positions under the asperity inside the material. The relationship between critical depth and Peclet number for all materials in the two-dimensional formulation may be simplified to satisfy the exponential form

R(η_{cr})^{2.275} = 20.4368.

Stellite III was chosen as the indicator material. Other parametric effects including mechanical properties and thermal properties were tested with materials having diverse property values. These tests confirmed that for the two-dimensional formulation, the Peclet number is the only one which dominates the critical depth.