Dear fellow contact mechanicians,
I just stumbled over a problem in analytic mechanics of plane Cattaneo problems in the presence of bulk stress.
It is said that the Ciavarella-Jäger principle for "small enough bulk stress" applies to this problem in the following form:
q(x) = \mu*(p(x; P, beta = 0) - p(x; P - Q/mu, beta)),
where q(x) is the tangential contact traction distribution, p(x) the pressure distribution, P the normal line load, Q the tangential line load, mu the friction coefficient and beta a "rotation angle" proportional to the bulk stress, which I will discuss in a minute.
The second term on the right side in above equation corresponds to a "fictious" normal contact problem of the same contacting bodies under the load (P - Q/mu) and with a relative rotation by beta.
The condition of "small enough bulk stress" is basically that the contact area for this "fictious" problem (which corresponds to the stick region in the actual Cattaneo problem), completely lies within the actual contact area. Moreover, a non-zero value of beta will increase the contact length on one side and decrease it on the other side. So, e.g., for Q = 0 the condition of "moderate bulk stress" is actually that beta = 0, i.e., there is no bulk stress.
Now, we know that tangential contact problems have a loading history. Even the Cattaneo problem has a history: first the normal load is applied, and then an increasing tangential load. However, when beginning to apply the tangential load, Q equals zero, so any (constant) bulk stress will violate the "moderate bulk stress condition".
Or to put it more generally: For any non-zero constant bulk stress, the "moderate bulk stress condition" is violated at the beginning of tangential loading.
Does that change anything about the final contact configuration at the end of the Cattaneo loading?
Or am I missing something?
Thank you very much for your help!
Kind regards,
Emanuel
I have two questions:
1. Do I understand correctly that the "moderate bulk stress" condition will always be violated for non-zero bulk stress at the beginning of tangential loading?
2. If yes, does that change the contact solution for stages with higher tangential loads, when the condition is nominally fulfilled?
Because, as I see it, any tangential problem with friction is incremental between all instances, that a point changes its "state" from slip (or non-contact) to stick. E.g., the classical Cattaneo problem is not incremental, because points only change from stick to slip. However, if the answer on the first question is "yes", how can we be sure that during tangential loading no point changes from slip to stick (during the phase, for which the "moderate bulk stress" condition is violated)? And that is what we assume using the Ciavarella-Jäger solution with moderate bulk stress (for ranges where it is nominally applicable), because it is a non-incremental solution.
no, i think you are missing something, see
https://www.sciencedirect.com/science/article/pii/S0020768319303476
Well thank you, but that paper doesn't answer my question, as the authors just put moderate bulk stress ("We restrict attention to cases in which the direction of slip is the same at either end of the contact") and calculate from there. And for now, I am only interested in the Cattaneo loading, not the cyclic case.
My question is, what happens, if the contact switches from non-moderate to moderate bulk stress? Consider Q = 0. Then, the "moderate bulk stress condition" is that there is actually no bulk stress; you can look it up in Barber's book, but it is logical, as any bulk stress will lead to opposite slip directions at the two contact ends.
Now, does that change anything about the contact solution if Q gets large enough during the Cattaneo loading for the bulk stress to be actually moderate?
I guess, the point is, that slip will always just propagate inside from the contact edges, unless loading is reversed, even if the bulk stress is not moderate, and so, no point will ever switch from slip to stick (although at one edge the slip direction is continously reversed), but I am not able to prove that rigorously.
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