Frictional systems subjected to oscillating loads

Abstract

If a discrete elastic system with frictional interfaces is subjected to periodic loading, the eventual steady-state response may depend on the initial condition or an initial transient phase of the loading history. In cases where shakedown is possible, it is known that it will occur for all initial conditions if there is no coupling between slip displacements and normal contact tractions, but that when coupling is present, counter-examples can be developed where the steady-state depends on the initial conditions. In this paper, we explore the conjecture that this is a special case of a more general theorem that the time-varying terms in the steady-state solution for an uncoupled system are always independent of initial conditions. In such problems, the 'memory' of previous events can only be stored in the slip displacements at nodes that are presently strictly within the friction cone. If all the nodes slip at some point in the cycle, this memory must be continually exchanged between nodes, with a consequent degradation, or loss of memory, resulting in an asymptotic approach to a unique steady state. This behaviour is illustrated using a simple two-node example. When there exists a set of 'permanently stuck nodes', these constitute a repository for the system memory, but in uncoupled problems the displacements at these nodes have no effect on the normal tractions at the slipping nodes and hence on the time-varying terms in the solution These arguments are illustrated in the context of two examples: a random distribution of frictional microcracks in a block loaded in plane strain and a generalized Hertzian contact problem with friction.

Frictional elastic contact with periodic loading

“…It is conjectured (Barber, 2011) that this is a special case of a more general but unproven result that even above the shakedown limit the steady-state response depends on the initial conditions only to the extent of a similar set of slip displacements that are now restricted to a unique 'permanent stick zone' -i.e. a part of the contact area that does not slip during the steady state.…”

“…In particular, if the system is subjected to loads that are periodic in time, the steady state may depend on the initial conditions or on an initial transient loading phase. As a result, the effective damping in the system and/or damage due to fretting may exhibit variability due to differences in assembly procedures (Barber, 2011).…”

Determination of the shakedown limit for large, discrete frictional systems

“…This work has practical relevance because frictional contacts in engineering structures are very frequently subjected to a combination of static and cyclic loads resulting, at least transiently, in partial slip conditions. Although it is well known that partial-slip plays a significant role in determining component performance (Farris et al, 2000;Barber, 2011), it is not always clear if a given configuration will result in these conditions after several load cycles. The Linear Programming approach presented by Bj€ orkman and Klarbring (1987) provides an efficient way to determine the load range above which partial-slip is guaranteed.…”

Recent progress in understanding the properties of elastic contacts

“…In a series of papers, Barber has sought to find more about the consequences of this and has conjectured that there may be a load factor below which there is certain shakedown, regardless of the initial conditions present in the problem -in other words, wholly independently of the presence of a pre-existing interfacial slip displacement distribution, either adverse or helpful in character. 34,35 At a load factor higher than conjectured, there can certainly be no shakedown. However, between the two conditions conjectured above, there may exist a load factor range within which shakedown is conditional on the starting conditions.…”