Wear phenomena are caused by interactions between contacting structures. The control of these phenomena is of great importance to predict the remaining lifetime of engineering or biological structures. In this paper, a precise mixed formulation for unilateral frictional contact problems involving wear, based upon the use of Archard's model, is developed. Classical Archard's model is shown to be nonlocal in the vicinity of the contact edges and this issue is addressed by carrying out delocalizing wear simulations. Moreover, a wear geometry update methodology is suggested to account for material removal due to wear. An extension, in the Arlequin framework, of the resulting wear simulation tool is suggested to model wear of thin structures submitted to the action of very localized contact loads. Finally, we suggest a strategy for multiscale and evolution-type contact problems (such as scratching of thin structures) by using a new concept of ”Moving” Arlequin patch. Some numerical tests are carried out, showing the potential of our developments.