What is the theoretical upper limit on the rigidity of a material? - Physics Stack Exchange most recent 30 from physics.stackexchange.com 2022-01-21T04:23:42Z https://physics.stackexchange.com/feeds/question/89562 https://creativecommons.org/licenses/by-sa/4.0/rdf https://physics.stackexchange.com/q/89562 5 What is the theoretical upper limit on the rigidity of a material? Mark Dominus https://physics.stackexchange.com/users/9062 2013-12-09T03:26:05Z 2014-12-12T18:52:20Z <p>Take a perfectly rigid metal rod of length $2\ell$ and some uniform linear density. Place one end (‘south’) at $(0,-\ell)$ and the other (‘north’) at $(0, \ell)$. Over some reasonably short time interval $t$, perhaps on the order of a fraction of a second, displace the center of the rod eastward from $(0,0)$ to $(1,0)$. In practice it's very easy to do this so that the entire rod moves one unit eastward; in particular the north end moves from $(0, \ell)$ to $(1, \ell)$.</p> <p>But this is actually a classical view of the situation. To see this, make $\ell$ very long, say on the order of ten light-seconds, and large enough to be bigger than $t\cdot c$. Then at time $t$ the center of the rod is at $(1,0)$ but the north end is still at $(0,\ell)$, because the north end can't have noticed yet that the middle has moved.</p> <p>But this has an implication for the material properties of the bar. I claimed in the first paragraph that it was perfectly rigid, but it now appears that it isn't as rigid as all that. Purely from speed-of-light considerations we can conclude that even a perfectly elastic bar must temporarily deform in the process of being translated from $x=0$ to $x=1$.</p> <p>It seems to me that if one assumed that the rod had length $2\ell$ and uniform linear density $\rho$, then one could calculate the amount of force required to translate it from $x=0$ to $x=1$ by pushing on the midpoint. Then supposing that the rest of the rod followed as quickly as speed-of-light propagation allows, one could calculate the stiffness of the rod, and this would be a theoretical upper bound on the maximum stiffness of any material whatever.</p> <p>But I don't have enough expertise or understanding of materials calculations to do actually perform this one. Also I suspect I must have left out something important, for the same reason.</p> <p>My questions are:</p> <blockquote> <ol> <li>Can this calculation be done, or is there some reason the whole idea is unsound?</li> <li>If it does make sense, what upper limit on material stiffness does this method produce?</li> </ol> </blockquote> <p>I suppose that if it does work, the upper bound is vastly greater than the stiffness of any real material, but I don't mind that.</p> <p>(I found the question <a href="https://physics.stackexchange.com/questions/48392/extended-rigid-bodies-in-special-relativity">Extended Rigid Bodies in Special Relativity</a>, which is clearly related to this, but doesn't get at what I want. My earlier question <a href="https://physics.stackexchange.com/questions/28137/behavior-of-shock-waves-at-relativistic-speeds">Behavior of shock waves at relativistic speeds</a> started out as an attempt to ask this one, and somehow went in a completely different direction by the time I posted it.)</p> https://physics.stackexchange.com/questions/89562/-/152958#152958 2 Answer by Floris for What is the theoretical upper limit on the rigidity of a material? Floris https://physics.stackexchange.com/users/26969 2014-12-12T18:52:20Z 2014-12-12T18:52:20Z <p>I believe that one could rephrase the question as "if the limit of the speed of sound in a medium must be the speed of light in vacuum, what does that mean for the limit on rigidity of an object?"</p> <p>Speed of sound is given by $$c=\sqrt{\frac{E}{\rho}}$$ - it depends on both density and Young's modulus. I would consider "rigidity" to be just the modulus, and if there is no theoretical limit on density then there is no theoretical limit on rigidity (following your logic).</p> <p>Of course from a materials science and quantum mechanics perspective there is always going to be a finite force-distance relationship for atoms - this sets realistic limits on elastic modulus that are well below the theoretical one calculated above. At 12,000 m/s, diamond (a very rigid material) is still far away from the limit.</p>