On shakedown and ratchetting of conforming frictional systems

Conforming contact is found on many important areas: materials with cracks, inclusions, layered structures, pinned geometries, etc. Coulomb friction is non-associative i.e. does not satisfy Drucker’s postulates for stability in plasticity theory, and this does not permit to use the classical Melan and Koiter theorems for cyclic loading which however remain obvious necessary conditions. On the other hand, the Coulomb friction cone has a peculiarity, its linear self-similarity, and this permits few general results, quite different but also powerful. For example, we show that for conforming contact, the Bree interaction diagram is open and sectors-shaped, and therefore there is no lower and upper bounds as in plasticity, since there is linear dependence on a load parameter \(\lambda\), and not a limit on \(\lambda\). As there is dependence on initial conditions in general, the Bree diagram doesn’t have the same profound meaning as in plasticity, since multiple steady states are possible. However, there is a complete scaling of the possible states (elastic shakedown, cyclic slip, ratchetting) with load factor \(\lambda\), and hence a worst-scenario can be defined. However, in order to develop residual stresses (which generally will take many cycles of steady cyclic loading), the load cannot go to zero, at least for friction coefficient below the critical value for "wedging". More in general, there needs to be a "permanent stick region" in the steady state.Almost conforming contacts, and in particular, in the case of initial interference, loose the independence on \(\lambda\) at low \(\lambda\) only, but "gain" greater possibilities to develop residual stresses. However, their study can be greatly simplified by developing a master curve approach to study the dependence on \(\lambda\). The theorems can also be considered the extension to cyclic loadings of the Dundurs results for conforming contacts under monotonic loading, in which case the contact area doesn’t vary in time and hence the problem is linear (from which the denomination "receding contacts"). However, here the problems remain fully non-linear, and the contact are not in general receding, nor there is a unique steady state, not even in the case of proportional loading. In the latter case it is nevertheless possible to derive some special results.