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Prestress stability

Lecture VISession on Granular Matter

Institut Henri Poincaré R. Connelly

Cornell UniversityDepartment of Mathematics

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Potential functions

How is the stability of a structure determinedwhen it is statically indeterminant or evennot statically rigid?

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Potential functions

How is the stability of a structure determinedwhen it is statically indeterminant or evennot statically rigid?

Think of each member as a spring at rest (abar), in tension (a cable), or in compression(a strut). In the spirit of Hooke’s Lawdefine an energy function as follows:

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Member potentials

For each member {i,j} define (or determine)Eij(x) its potential at length x1/2. So the total

energy for any configuration q=(q1, q2, …qn) in Ed is

E(q)=Si

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The individual energy functions

A bar: A local minimum,concave up

2ij ji

E (|p - p | )

ji|p - p | 2

ji|p - p | 2

2|p - p | i j

A strut: Monotone decreasing,concave up

2ij ji

E (|p - p | )

A cable: Monotone increasing,concave up

2ij ji

E (|p - p | )

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The critical calculation

Define wij = E¢ij(|pi-pj|2) the first derivative of Eij atx= |pi-pj|2. Then a calculation shows that p is acritical point for the energy functional E if andonly if w = (…, wij,…) is a proper self stress forthe tensegrity G(p).

Our goal is to determine when the configuration p isa local minimum for E, up to congruences. So wecalculate the second derivative.

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Stiffness

For each member {i,j}, definecij = E¢¢ij(|pi-pj|2) > 0,

the second derivative of Eij at |pi-pj|2 as thestiffness coefficient of member {i,j}. We assumethat these coefficients are all positive. Define C asthe e-by-e diagonal matrix whose ij, ij diagonalentry is cij, where e is the number of members inthe tensegrity.

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HessianLet p¢= (p1¢, p2¢, … pn¢) be a set of directions, where

each pi¢ is in Ed. So E(p + tp¢), for 0 ≤ t, hasderivative 0 when t = 0, for all p¢. A calculationshows that the second derivative in this direction p¢is given by the sum of the following two quadraticforms:

4 Si≤j cij[(pi- pj)(pi¢- pj¢)]2 + 2 Si≤j wij(pi¢- pj¢)2.The matrix S = 4 R(p)TCR(p) is what we call the

stiffness matrix, where R(p) is our old friend therigidity matrix. Note that S is always positive semi-definite, and 4 (p¢)T S p¢ is the term on the left.

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Stress matrixThe matrix of the form on the right is 2 W, where

each non-zero off-diagonal entry of W is definedto be -wij corresponding to each coordinate, andsuch that the row and column sums are 0. Thematrix W is called the stress matrix and is veryinteresting. It can provide a lot of globalinformation about the tensegrity, although it canhave negative eigenvalues.

Define the matrix H = 4 S + 2 W. So 2 (p¢)TWp¢ isthe term on the right for the second derivativeabove.

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Prestress stability

Corresponding to given stress coefficients w,and stiffness coefficients, we say that G(p)is prestress stable if H is positive semi-definite with only the trivial infinitesimalflexes in its kernel. The following is in Connelly-Whiteley 1995.

Theorem: If a tensegrity G(p) is prestressstable in Ed, then it is rigid in Ed.

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Examples of prestress stabletensegrities

The pattern of tensions and compressions is onthe right. Neither of these structures isinfinitesimally rigid.

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Displacements and forces

Given a prestress stable tensegrity structureG(p) and an equilibrium force F applied tothe configuration p, how does p deform toaccommodate F, and how is the forceresolved?

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Displacements and forces

Given a prestress stable tensegrity framework G(p)and an equilibrium force F applied to theconfiguration p, how does p deform toaccommodate F, and how is the force resolved?

As a first approximation to the displacement Dp,solve F + 2HDp = 0, which is possible since Hhas maximal rank, and 2H is the gradient of theapproximation to the energy form. Then theresolving stress can itself be determined from theedge lengths and the energy formulas.

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Example of a displacement

it deforms into a configurationthat does resolve the force.

When the equilibrium force above isapplied to this tensegrity it cannotresolve the force as is, but. . .

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An application to packings

What sort of model of rigidity is appropriatefor packings of ellipses in the plane orellipsoids in space?

Even for a single ellipse in a triangle, whatdoes a maximal ellipse with a fixed axisratio look like?

(Joint work with A. Donev, S. Torquato, F. Stillinger, Chaiken, et. al.)

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An ellipse in a triangle

If the lines through the points ofcontact do not intersect in a point,then a rotation (counterclockwisehere) about the triangle enclosedmoves the ellipse into the triangle.

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The critical case

When the 3 lines meet at a point there is an"infinitesimal flex" of the ellipse inside thetriangle. So there is no occasion when theellipse is "infinitesimally rigid" inside thetriangle.

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Modeling ellipses

At each point of contact between ellipses place a circlewhose curvature is the same at the curvature of theellipse at that point. Join these centers in each ellipsewith a statically rigid bar framework. Join the centersof touching circles with a strut.

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Consequences of the model

• The tensegrity paradigm applies to irregularlyshaped particles.

• The notion of being infinitesimally rigid applies,BUT, unlike the case when all the particles arecircles (or spheres in space), being rigid does notalways imply infinitesimal rigidity. (Thecanonical push does not work in general.)

• If the structure is infinitesimally rigid, then thereis a minimum number of contacts that arenecessary. We calculate that next.

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Contacts for ellipses

Suppose a packing of ellipses in the plane is such thatall are pinned except for n which are allowed tomove, and the system is infinitesimally rigid. Eachellipse has 3 degrees of freedom and each contactcorresponds to 2 ellipses, except for the boundary.So if n is large enough so that the boundary effectsare negligible, and if Z is the average degree of ajammed ellipse packing, then

Z ≥ 6 - O(n-1/2),which seems unlikely for congruent ellipses.

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Contact numbers for ellipsoids

Doing the same calculation in 3-space, whereeach ellipsoid has 3 distinct axis lengths, thedegrees of freedom of each ellipsoid is 6.Then the average degree is Z ≥ 12 - O(n-2/3),which seems almost impossible forcongruent ellipsoids unless they are a smallperturbation of one of the standard mostdense packings of congruent spheres.

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Almost spherical ellipsoids

There was a famous argument between Newton andGregory about how many spheres can touch asingle sphere, all with the same radius in apacking. Newton said 12, Gregory 13. Newtonwas right as shown in the 1950’s. This must alsobe true for ellipsoids if the axis ratios are closeenough to 1. So the coordination number for suchellipsoids cannot be greater than 12 either. So ifany sort of random packing of such packings isjammed, its coordination number should be closerto 6 than 12 and it cannot be statically rigid.

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Contact numbers for spheroids

When the ellipsoid has one degree ofrotational symmetry, i.e. when two of thethree axes are the same length, that subtractsfrom the degrees of freedom of theellipsoids. In that case, for infinitesimalrigidity, one needs Z ≥ 10 -O(n-2/3) for thecoordination number.

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Experimental results

In the 8 author joint paper in Science, it isshown that in dimension 3 with congruent(monodispersed) ellipsoids with an axisratio chosen in the range close to 1, but notequal to, have coordination number lessthan 12 in the case when all axes aredifferent and less than 10 (about 9.5 forsome) when two of the axes are the same.

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The moral of the story

• Random, even mildly random, packings ofellipsoids must not be infinitesimally rigid.They must be prestress stable.

• “Random” packings of monodispersedellipsoids with the appropriately chosen axisratios tend to have surprisingly highpacking densities.

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Packing densities andcoordination numbers

9.80.671 : 1 : 0.6(m&m’s)

11.40.73 1 : 1.3 : 0.77

9.50.70 1 : 1 : 1.5

60.631 : 1 : 1

CoordinationnumberPacking densityAxis ratios

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The m&m container andsimulation

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Remarks

Theorem (L. Fejes Toth 1940’s): The most densepacking of any centrally symmetric convex bodyin the plane is achieved with a lattice packing.

In particular, for any ellipse, it has maximumpacking density p/√12=0.90699…, the same as themost dense packing of congruent circles, and for aperiodic packing it is achieved by a one parameterfamily of ellipses obtained by some rotation of thestandard circle packing dilated in one directiononly.

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An ellipse packing with maxdensity

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An ellipse packing with maxdensity

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An ellipse packing with maxdensity

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Packing congruent ellipsoids

A. Bezdek and W. Kuperberg showed thatthere are packings of congruent ellipsoidswhose packing density exceedsp/√18=0.7408…, the most dense packing ofcongruent spheres in 3-space. This wasimproved by Donev, Chaiken, Stillinger,Torquato to 0.7704…

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The first layer

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The ellipsoid packing

Each layer is as in the previous slide. But each layeris turned 90º and laid down onto the next layer. Itthen fits better than laying them down in a parallelmanner. The axis ratio of the ellipsoids in eachlayer is √3. The length in the third direction isarbitrary. So this 77% density can be achievedwhen any of the 3 axis ratios is √3. Each ellipsoidis in contact with 14 others.

Note that the most dense random arrangement ofellipsoids was one where one of the axis ratiosway 1.68, where as √3=1.732….

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Question

Is there an upper bound a < 1 for the bestpacking density of congruent ellipsoids,independent of the axis ratios?