Torsional rigidity of classical planar domains

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Numbers

$D$

$normalization$

$T(D)$ or $T(D)/|D|^2$

unit disk

reference geometry:

0.3926990816987241548078304229099378605246461749218882276218680740384770507857761248285043531677646335

unit disk

area one:

0.03978873577297383394222094084312859050861491143511411218691683601472419940855663377252845069156327149

equilateral triangle, side one

reference geometry:

0.005412658773652741542273269817205851146696266418157439462674396810787290677840000115878581833616401799

equilateral triangle, side one

area one:

0.02886751345948128822545743902509787278238008756350634380093011632419888361514666728468576977928747626

right isosceles triangle, legs one

reference geometry:

0.006522412927878237769549483982233887849768386959143599648177129438447688447536557365237229384234212781

right isosceles triangle, legs one

area one:

0.02608965171151295107819793592893555139907354783657439859270851775379075379014622946094891753693685112

half-disk, radius one

reference geometry:

0.07438919551493348327006289616490913645572688344097533012653338592068345551732305464827674763525846159

half-disk, radius one

area one:

0.03014880535909737903119555261825960766220630115054414180934045519656055045624457693721214352799100581

slit disk, radius one

reference geometry:

0.2195139212929315602151852427176024338156580538777090819186967025342099322054134582254929631666294058

slit disk, radius one

area one:

0.02224141033137079822324712135103284574067278083026719906791061031285451118245461947710060838834339625

unit square

reference geometry:

0.03514425373878842889711718268443277881689827970752306701848953722821897821913010930142943728243245389

unit square

area one:

0.03514425373878842889711718268443277881689827970752306701848953722821897821913010930142943728243245389

rectangle, sides $4/3$ and $1$

reference geometry:

0.06015242557729066455729721838475935157781642221442221008448696685014799122986419275430988587895112203

rectangle, sides $4/3$ and $1$

area one:

0.03383573938722599881347968534142713526252173749561249317252391885320824506679860842429931080691000614

rectangle, sides $3/2$ and $1$

reference geometry:

0.07341026582832900937179089835065613159370491413834416715216232339707316312607345522854936307815067964

rectangle, sides $3/2$ and $1$

area one:

0.03262678481259067083190706593362494737497996183926407428984992150981029472269931343491082803473363540

rectangle, sides $5/3$ and $1$

reference geometry:

0.08692167144116837407690913438063403947445054161330713629094946582643376048466065262934301981861783055

rectangle, sides $5/3$ and $1$

area one:

0.03129180171882061466768728837702825421080219498079056906474180769751615377447783494656348713470241900

rectangle, sides $2/1$ and $1$

reference geometry:

0.1143408385597853825128004797625789736895041898896650940262085228993049658853478879730270357749487687

rectangle, sides $2/1$ and $1$

area one:

0.02858520963994634562820011994064474342237604747241627350655213072482624147133697199325675894373719217

rectangle, sides $5/2$ and $1$

reference geometry:

0.1558531716926164872226462767339573658538511721131313144096922917633579408578968163918488235361447291

rectangle, sides $5/2$ and $1$

area one:

0.02493650747081863795562340427743317853661618753810101030555076668213727053726349062269581176578315666

rectangle, sides $3/1$ and $1$

reference geometry:

0.1974876982511250798152918368445814328393239640232965557971735253745416346889820708769352443662309356

rectangle, sides $3/1$ and $1$

area one:

0.02194307758345834220169909298273127031548044044703295064413039170828240385433134120854836048513677062

rectangle, sides $4/1$ and $1$

reference geometry:

0.2808129583076773784824632411598978785739065260602522565068463692187538634353764600309945603867151386

rectangle, sides $4/1$ and $1$

area one:

0.01755080989422983615515395257249361741086915787876576603167789807617211646471102875193716002416969616

rectangle, sides $5/1$ and $1$

reference geometry:

0.3641459427349860857969735336572619150436447717510009827382248635912373122246232036309989766410197316

rectangle, sides $5/1$ and $1$

area one:

0.01456583770939944343187894134629047660174579087004003930952899454364949248898492814523995906564078926

rectangle, sides $6/1$ and $1$

reference geometry:

0.4474792609906705066558127072414405679171593222649354655718340353344281854473215790466193721724027055

rectangle, sides $6/1$ and $1$

area one:

0.01242997947196306962932813075670668244214331450735931848810650098151189404020337719573942700478896404

ellipse, semiaxes $3/2$ and $1$

reference geometry:

0.8156057850665809369085708783514094026281112863762293958300336922337600285550734900284321181176650080

ellipse, semiaxes $3/2$ and $1$

area one:

0.03672806379043738517743471462442639123872145670933610355715400247513003022328304655925703140759686599

ellipse, semiaxes $2/1$ and $1$

reference geometry:

1.256637061435917295385057353311801153678867759750042328389977836923126562514483599451213930136846827

ellipse, semiaxes $2/1$ and $1$

area one:

0.03183098861837906715377675267450287240689192914809128974953346881177935952684530701802276055325061719

ellipse, semiaxes $3/1$ and $1$

reference geometry:

2.120575041173110435962284283713664446833089344578196429158087599807776074243191074073923507105929021

ellipse, semiaxes $3/1$ and $1$

area one:

0.02387324146378430036533256450587715430516894686106846731215010160883451964513398026351707041493796289

ellipse, semiaxes $4/1$ and $1$

reference geometry:

2.956793085731570106788370243086590949832630022941276066799947851583827205916431998708738659145521946

ellipse, semiaxes $4/1$ and $1$

area one:

0.01872411095198768656104514863206051318052466420475958220560792283045844678049723942236632973720624541

Definition

For a bounded simply connected planar domain $D$, $T(D)=\int_Du_D\,dA$, where $-\Delta u_D=1$ and $u_D\in H^1_0(D)$ [1]. The entries give this same quantity in two invertible scale conventions, $T(D)$ and $T(D)/|D|^2$.

Parameters

$D$

— planar domain ($D$ is a bounded simply connected planar domain with a specified reference geometry.)

$normalization$

— scale convention (The reference geometry or its rescaling to area one.)

Formulas

(1)

For $c>0$, $T(cD)=c^4T(D)$, and hence $T(A^{-1/2}D)=T(D)/A^2$ when $A=|D|$ [1], equation (1.5).

(2)

For the ellipse $x^2/a^2+y^2/b^2<1$, $T=\pi a^3b^3/[4(a^2+b^2)]$. In particular, the unit disk has $T=\pi/8$ and its area-one form has $T=1/(8\pi)$ [3], §225(a,b).

(3)

For an equilateral triangle of side one, $u_D(x,y)=y(\sqrt3x-y)(\sqrt3(1-x)-y)/(2\sqrt3)$ and $T(D)=\sqrt3/320$. Its area-one form is $\sqrt3/60$.

(4)

The half-disk has $T=\pi/8-1/\pi$ [4]. The slit disk has $T=\pi/4-16/(9\pi)$ [5].

(5)

For the rectangle $(0,r)\times(0,1)$, $T=r/12-(16/\pi^5)\sum_{k\geq1,\ k\ {\rm odd}}\tanh(\pi r k/2)/k^5$ [2], Proposition 2.1.

(6)

For the right isosceles triangle with perpendicular legs one, $T=1/48-(4/\pi^5)\sum_{k\geq1,\ k\ {\rm odd}}\coth(\pi k/2)/k^5$ [6].

(7)

Write $S_+(r)=\sum_{k\geq1,\ k\ {\rm odd}}\tanh(\pi r k/2)/k^5$ and $S_-(r)=\sum_{k\geq1,\ k\ {\rm odd}}\coth(\pi r k/2)/k^5$. For $r>0$,
$S_+(r)=\frac{31}{32}\zeta(5)-2\sum_{k\geq1,\ k\ {\rm odd}}[k^5(e^{\pi r k}+1)]^{-1}$ and
$S_-(r)=\frac{31}{32}\zeta(5)+2\sum_{k\geq1,\ k\ {\rm odd}}[k^5(e^{\pi r k}-1)]^{-1}$.
The value $\zeta(5)$ supplies the common algebraic tail.

(8)

The Saint-Venant inequality gives $T(D)/|D|^2\leq1/(8\pi)$, with equality for a disk [1], equations (1.4)–(1.6).

Comments

(9)

The function $u_D$ is the unique weak solution for the Euclidean Laplacian with zero Dirichlet values on the entire boundary.

(10)

The reference-geometry convention stores $T(D)$ for the dimensions specified in (11). The area-one convention stores $T(D)/A^2$, where $A=|D|$. These are two invertible scale conventions for one quantity: knowing $A$ recovers either from the other. Both conventions are retained for the square, whose area is already one.

(11)

The reference domains are open. The disk is $x^2+y^2<1$, with area $\pi$. The equilateral triangle has vertices $(0,0)$, $(1,0)$ and $(1/2,\sqrt3/2)$, with area $\sqrt3/4$. The right isosceles triangle is $x>0$, $y>0$, $x+y<1$, with area $1/2$; its two perpendicular legs have length one.

The half-disk is $x^2+y^2<1$, $y>0$, with area $\pi/2$; the diameter has zero Dirichlet values. The slit disk is $0<r<1$, $0<\theta<2\pi$, with area $\pi$. Both banks of the positive radial slit have zero Dirichlet traces, and the angular edges are not identified.

A rectangle with reduced ratio $p/q\geq1$ is $(0,p/q)\times(0,1)$ and has area $p/q$. An ellipse with reduced ratio $p/q>1$ is $x^2/(p/q)^2+y^2<1$ and has area $\pi p/q$. These dimensions are semiaxes. Translation and rotation do not change $T$.

(12)

The geometric Saint-Venant torsion constant of a homogeneous simply connected section is $J=4T$ in this normalization; its dimensional torsional rigidity is $GJ$, where $G$ is the shear modulus [3], §224. The half-disk, slit-disk and right-triangle constants in [4], [5] and [6] use $J$. The polar second moment of area and the maximum of $u_D$ are different quantities.

Programs

(P1)

Python (mpmath)

# One rectangle of side lengths r and 1, with -Delta u=1.
# This numerical snippet is not the certified generator.
import mpmath as mp
mp.mp.dps = 150
r = mp.mpf(3) / 2
s = 31 * mp.zeta(5) / 32 - 2 * mp.fsum(
    mp.exp(-mp.pi*r*k)/(k**5*(1+mp.exp(-mp.pi*r*k)))
    for k in range(1, 161, 2))
t = r/12 - 16*s/mp.pi**5
print(mp.nstr(t, 100))          # reference geometry
print(mp.nstr(t/r**2, 100))     # area one
# The attached generate.py encloses the omitted tail and covers all domains.

References

[1]

Michiel van den Berg, Giuseppe Buttazzo and Aldo Pratelli, On the relations between principal eigenvalue and torsional rigidity (2019). (arXiv)

[2]

Joseph Comer and Patrick McDonald, Torsional rigidity and isospectral planar sets (2021), Proposition 2.1. (arXiv)

[3]

A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, second edition, Cambridge University Press (1906), §§224–225, pp. 310–311. https://brittlebooks.library.illinois.edu/brittlebooks_open/Books2011-02/loveau0001tremat/loveau0001tremat.pdf

Links

[4]

OEIS A180310: engineering torsion constant for a half-disk

[5]

OEIS A180311: engineering torsion constant for a slit disk

[6]

OEIS A180314: engineering torsion constant for a right isosceles triangle

Similar tables

Dirichlet eigenvalues of the classical planar domains — Laplacian eigenvalues on classical domains; the torsion integral is a weighted reciprocal spectral sum.

Steklov eigenvalues of the classical planar domains — Boundary spectra for several of the same reference geometries.

Data properties

Entries are of type: real number

digits: 100 (Unknown key)

How well the digits are known: proven

Table is complete: no (it holds 19 reference domains in both reference-geometry and area-one conventions: the disk, equilateral triangle, right isosceles triangle, half-disk and slit disk; rectangles with side ratios $1,4/3,3/2,5/3,2,5/2,3,4,5,6$; and ellipses with semiaxis ratios $3/2,2,3,4$. The circle, square and equilateral triangle are selected symmetric controls. The other named domains are classical shaft sections. The rectangles cover every reduced $p/q\geq1$ with $p\leq6$, $q\leq3$, as a noncanonical reference for separated-variable Poisson integrals from symmetric to elongated sections. The shorter ellipse selection compares smooth sections across a similar range without copying the rectangular mesh. These exact rational choices are independent of decimal notation and do not assert an application for each ratio. Smooth, polygonal, mixed and slit boundaries are covered before extending an aspect-ratio family more deeply)

How they were obtained:

The values are written to 100 significant digits from Arb ball enclosures. Closed forms use ball arithmetic throughout. The rectangular tanh sums and triangular coth sum separate the exact odd zeta tail, then sum 80 positive exponential correction terms. If $M=161$ is the first omitted odd integer and $c=\pi r$, the correction remainder before its factor two is bounded by $e^{-cM}/[M^5(1-e^{-2c})]$ for tanh and by $e^{-cM}/[M^5(1-e^{-cM})(1-e^{-2c})]$ for coth. These follow by geometric majorization. The bound is added as a ball radius before the remaining arithmetic.

The attached checks compare 501-bit production arithmetic with a 665-bit computation using 120 correction terms and test containment in every written interval. They check exact polynomial Poisson solutions for the disk, ellipses and equilateral triangle; a sine-series derivation for the two sectors; rectangle axis exchange; both scale conventions; and the Saint-Venant bound. Independent Laplacian eigenfunction sums with Parseval remainder bounds give coarser enclosures for all rectangles and the right triangle. The square's first sine mode is a nonzero control. The three OEIS engineering constants are checked after the factor-four conversion. These checks are repeated on all values read back from the draft.

Comer–McDonald's triangle projection coefficients on page 5 disagree with direct integration of the stated eigenfunctions. The attached derivation documents the discrepancy. Only their rectangle formula is used here; the right-triangle formula is [6], checked against the independently integrated triangle eigenfunctions.