Reply to Comment on `What Determines the Static Force Chains in Stressed Granular Media?'
Oleg Gendelman, Yoav G. Pollack, Itamar Procaccia, Shiladitya Sengupta, Jacques Zylberg
RReply to Comment on ‘What Determines the Static Force Chains in StressedGranular Media?’
Oleg Gendelman , Yoav G. Pollack , Itamar Procaccia , Shiladitya Sengupta and Jacques Zylberg Faculty of Mechanical Engineering, Technion, Haifa 32000, Israel Dept. of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel
The determination of the normal and tangential forcesbetween frictional disks from visual data was consideredinsoluble for three main reasons: (i) the tangential forcesthat accumulate at contacts are history-dependent andwere believed not to be obtainable from a visual [1], (ii)the number of mechanical constraints, i.e the vanishingof the net force and the torque on each disk, is muchsmaller than the number of inter-disk normal and tan-gential forces, and the problem is thus under-determined.(iii) In many realistic granular systems (sand, metallicdisks etc.) the compression is so small that the changein the distances between centers of mass cannot be mea-sured accurately. In the context of an array of disks of di-ameters σ i , one can determine the positions of the centerof mass r i relatively easily. But if the disks are highly in-compressible, it is not possible to determined accuratelythe difference between the nominal distance σ i + σ j andthe actual distance | r i − r j | . In Ref. [2] it was shown thatgiven the directions of the vectors connecting the centersof masses of the disks (but not the actual distances be-tween the center of mass) and the external forces on thedisks, all the normal and tangential forces can be de-termined exactly provided the normal forces are linear.There is no need to know the tangential force law. Thesolution of all the aforementioned difficulties is achievedby adding geometric constraints in the form of the mini-mal polygons that connect the centers of mass of adjacentdisks.In a comment on that paper, DeGiuli and McElwaineshowed that if the radii of the disks are not known withproper accuracy, this results in errors in the determinedforces [3]. This is obvious; given highly incompressibledisks in contact, introducing errors in the their radiichanges their positions and the vector distances betweenthe centers of mass. Of course, experimental errors are unavoidable, and care should be taken to diminish themas much as possible. The theoretical solution of the con-ceptual difficulties (i)-(iii) still requires experimental ef-forts to achieve the highest possible precision. For ex-ample using larger and stiffer disks will automaticallyreduce the relative error in the radii. The theory inRef. [2] aimed at finding the forces when provided withgood measurement of the disk radii; the comment [3]addresses another problem: the statistics of forces in un-certain configuration. The obtained forces may dependon inaccuracies, but this fact does not make the solutionof Ref. [2] “false” in any conceivable way as they claim.Even with experimental errors one can improve the de-termination of the inter-particle forces by noticing thatthe predicted forces do not annul the net force on eachparticle. An iterative procedure to achieve such an im-provement was proposed in Ref. [4]. The idea is to movethe particles in the direction of the net force, and re-compute the inter-particle forces. For systems with onlynormal forces this procedure converges extremely well.An equivalent procedure for frictional assemblies of diskswill be provided elsewhere. [1] Photoelastic disks are the only exception where infomra-tion about tangential forces may be extracted from exper-iments.[2] O. Gendelman, Y. G. Pollack, I. Procaccia, S. Sengupta,and J. Zylberg, Physical Review Letters , 078001(2016).[3] E. DeGiuli and J. N. McElwaine, Phys. Rev. Lett., pre-ceding comment.[4] O.Gendelman, Y. G. Pollack and I. Procaccia, Phys. Rev.E , 060601(R) (2016). a r X i v : . [ c ond - m a t . s o f t ] J unun