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The present article establishes a general theory of frictional moving contact of orthotropic materials indented by a moving rigid punch with various punch profiles. The punch moves to the right or left at a constant speed with the shear stress arising inside the contact region. The motion should be subsonic. By using Galilean transformation and Fourier transform, a singular integral equation of the second kind is obtained, solution of which has a non-square-root or unconventional singularity. Numerical results are presented to show the influences of relative moving velocity and the friction coefficient on surface in-plane stress for each case of the four types of punches, which demonstrates that the surface crack initiation and propagation in load transfer components more likely occur at the trailing edge. The present theory provides a basis for explaining the surface damage mechanism of orthotropic materials under an indentation loading and for exploiting the physics behind the different punch profiles.