In this study, the concept of the fracture mechanics is used to solve the: (i) frictionless purely normal contact and (ii) the similar material contact under the mutual actions of the normal and tangential load. Considering the contact region is simply connected, the out-of-contact regions can be treated as periodic collinear cracks. Through evaluating the stress intensity factor (SIF), we are able to obtain the size and location of the contact/out-of-contact region. Then, the normal traction, shear traction and interfacial gap can be directly determined by the Green's function of the periodic collinear crack. In the case of frictionless purely normal contact, the new approach is applied to two classic problems, namely, the Westergaard problem (sinusoidal waviness punch) and the periodic flat-end punch problem. Then, the sinusoidal waviness contact pair in the full stick and the partial slip conditions under the mutual actions of the normal and tangential loads are solved by the newly developed approach.

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