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{
"Title": "Torsional rigidity of classical planar domains",
"Definition": "For a bounded simply connected planar domain $D$, $T(D)=\\int_Du_D\\,dA$, where $-\\Delta u_D=1$ and $u_D\\in H^1_0(D)$ CITE{VBP}. The entries give this same quantity in two invertible scale conventions, $T(D)$ and $T(D)/|D|^2$.",
"Parameters": {
"domain": {
"type": "Symbolic",
"title": "planar domain",
"display": "$D$",
"constraints": "$D$ is a bounded simply connected planar domain with a specified reference geometry.",
"values": {
"disk": "unit disk",
"equilateral-triangle": "equilateral triangle, side one",
"right-isosceles-triangle": "right isosceles triangle, legs one",
"half-disk": "half-disk, radius one",
"slit-disk": "slit disk, radius one",
"rectangle-1-1": "unit square",
"rectangle-4-3": "rectangle, sides $4/3$ and $1$",
"rectangle-3-2": "rectangle, sides $3/2$ and $1$",
"rectangle-5-3": "rectangle, sides $5/3$ and $1$",
"rectangle-2-1": "rectangle, sides $2/1$ and $1$",
"rectangle-5-2": "rectangle, sides $5/2$ and $1$",
"rectangle-3-1": "rectangle, sides $3/1$ and $1$",
"rectangle-4-1": "rectangle, sides $4/1$ and $1$",
"rectangle-5-1": "rectangle, sides $5/1$ and $1$",
"rectangle-6-1": "rectangle, sides $6/1$ and $1$",
"ellipse-3-2": "ellipse, semiaxes $3/2$ and $1$",
"ellipse-2-1": "ellipse, semiaxes $2/1$ and $1$",
"ellipse-3-1": "ellipse, semiaxes $3/1$ and $1$",
"ellipse-4-1": "ellipse, semiaxes $4/1$ and $1$"
}
},
"normalization": {
"type": "Symbolic",
"title": "scale convention",
"constraints": "The reference geometry or its rescaling to area one.",
"values": {
"unit-length": "reference geometry",
"unit-area": "area one"
}
}
},
"Comments": {
"boundary": "The function $u_D$ is the unique weak solution for the Euclidean Laplacian with zero Dirichlet values on the entire boundary.",
"scales": "The reference-geometry convention stores $T(D)$ for the dimensions specified in CITE{geometries}. 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.",
"geometries": "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.\n\nThe 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.\n\nA 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 $T$.",
"elasticity": "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 CITE{Love}, §224. The half-disk, slit-disk and right-triangle constants in CITE{Half}, CITE{Slit} and CITE{Triangle} use $J$. The polar second moment of area and the maximum of $u_D$ are different quantities."
},
"Formulas": {
"scale": "For $c>0$, $T(cD)=c^4T(D)$, and hence $T(A^{-1/2}D)=T(D)/A^2$ when $A=|D|$ CITE{VBP}, equation (1.5).",
"ellipse": "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 $T=1/(8\\pi)$ CITE{Love}, §225(a,b).",
"equilateral": "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 $\\sqrt3/60$.",
"sectors": "The half-disk has $T=\\pi/8-1/\\pi$ CITE{Half}. The slit disk has $T=\\pi/4-16/(9\\pi)$ CITE{Slit}.",
"rectangle": "For the rectangle $(0,r)\\times(0,1)$, $T=r/12-(16/\\pi^5)\\sum_{k\\geq1,\\ k\\ {\\rm odd}}\\tanh(\\pi r k/2)/k^5$ CITE{CM}, Proposition 2.1.",
"triangle": "For the right isosceles triangle with perpendicular legs one, $T=1/48-(4/\\pi^5)\\sum_{k\\geq1,\\ k\\ {\\rm odd}}\\coth(\\pi k/2)/k^5$ CITE{Triangle}.",
"acceleration": "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 $r>0$,\n$S_+(r)=\\frac{31}{32}\\zeta(5)-2\\sum_{k\\geq1,\\ k\\ {\\rm odd}}[k^5(e^{\\pi r k}+1)]^{-1}$ and\n$S_-(r)=\\frac{31}{32}\\zeta(5)+2\\sum_{k\\geq1,\\ k\\ {\\rm odd}}[k^5(e^{\\pi r k}-1)]^{-1}$.\nThe value HREF{Values_of_the_Riemann_zeta_function_at_rational_numbers#5}[$\\zeta(5)$] supplies the common algebraic tail.",
"bound": "The Saint-Venant inequality gives $T(D)/|D|^2\\leq1/(8\\pi)$, with equality for a disk CITE{VBP}, equations (1.4)–(1.6)."
},
"Tags": [
"analysis"
],
"Keywords": [
"torsion constant",
"Saint-Venant",
"Poisson equation",
"torsion function",
"disk",
"equilateral triangle",
"right isosceles triangle",
"half-disk",
"slit disk",
"rectangle",
"ellipse"
],
"Similar tables": [
{
"table": "HREF{Dirichlet_eigenvalues_of_the_classical_planar_domains}[Dirichlet eigenvalues of the classical planar domains]",
"relation": "Laplacian eigenvalues on classical domains; the torsion integral is a weighted reciprocal spectral sum."
},
{
"table": "HREF{Steklov_eigenvalues_of_the_classical_planar_domains}[Steklov eigenvalues of the classical planar domains]",
"relation": "Boundary spectra for several of the same reference geometries."
}
],
"References": {
"VBP": {
"bib": "Michiel van den Berg, Giuseppe Buttazzo and Aldo Pratelli, On the relations between principal eigenvalue and torsional rigidity (2019).",
"arxiv": "1910.14593"
},
"CM": {
"bib": "Joseph Comer and Patrick McDonald, Torsional rigidity and isospectral planar sets (2021), Proposition 2.1.",
"arxiv": "2105.07477"
},
"Love": {
"bib": "A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, second edition, Cambridge University Press (1906), §§224–225, pp. 310–311.",
"url": "https://brittlebooks.library.illinois.edu/brittlebooks_open/Books2011-02/loveau0001tremat/loveau0001tremat.pdf"
}
},
"Links": {
"Half": {
"title": "OEIS A180310: engineering torsion constant for a half-disk",
"url": "https://oeis.org/A180310"
},
"Slit": {
"title": "OEIS A180311: engineering torsion constant for a slit disk",
"url": "https://oeis.org/A180311"
},
"Triangle": {
"title": "OEIS A180314: engineering torsion constant for a right isosceles triangle",
"url": "https://oeis.org/A180314"
}
},
"Programs": {
"rectangle-program": {
"language": "Python (mpmath)",
"code": "# One rectangle of side lengths r and 1, with -Delta u=1.\n# This numerical snippet is not the certified generator.\nimport mpmath as mp\nmp.mp.dps = 150\nr = mp.mpf(3) / 2\ns = 31 * mp.zeta(5) / 32 - 2 * mp.fsum(\n mp.exp(-mp.pi*r*k)/(k**5*(1+mp.exp(-mp.pi*r*k)))\n for k in range(1, 161, 2))\nt = r/12 - 16*s/mp.pi**5\nprint(mp.nstr(t, 100)) # reference geometry\nprint(mp.nstr(t/r**2, 100)) # area one\n# The attached generate.py encloses the omitted tail and covers all domains."
}
},
"Data properties": {
"type": "R",
"digits": "100",
"rigour": "proven",
"complete": "no",
"complete-note": "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",
"rigour details": "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.\n\nThe 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.\n\nComer–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 CITE{Triangle}, checked against the independently integrated triangle eigenfunctions."
},
"Display properties": {
"number-header": "$T(D)$ or $T(D)/|D|^2$",
"group parameters": [
[
"domain"
],
[
"normalization"
]
]
},
"Numbers": {
"disk": {
"unit-length": "0.3926990816987241548078304229099378605246461749218882276218680740384770507857761248285043531677646335",
"unit-area": "0.03978873577297383394222094084312859050861491143511411218691683601472419940855663377252845069156327149"
},
"equilateral-triangle": {
"unit-length": "0.005412658773652741542273269817205851146696266418157439462674396810787290677840000115878581833616401799",
"unit-area": "0.02886751345948128822545743902509787278238008756350634380093011632419888361514666728468576977928747626"
},
"right-isosceles-triangle": {
"unit-length": "0.006522412927878237769549483982233887849768386959143599648177129438447688447536557365237229384234212781",
"unit-area": "0.02608965171151295107819793592893555139907354783657439859270851775379075379014622946094891753693685112"
},
"half-disk": {
"unit-length": "0.07438919551493348327006289616490913645572688344097533012653338592068345551732305464827674763525846159",
"unit-area": "0.03014880535909737903119555261825960766220630115054414180934045519656055045624457693721214352799100581"
},
"slit-disk": {
"unit-length": "0.2195139212929315602151852427176024338156580538777090819186967025342099322054134582254929631666294058",
"unit-area": "0.02224141033137079822324712135103284574067278083026719906791061031285451118245461947710060838834339625"
},
"rectangle-1-1": {
"unit-length": "0.03514425373878842889711718268443277881689827970752306701848953722821897821913010930142943728243245389",
"unit-area": "0.03514425373878842889711718268443277881689827970752306701848953722821897821913010930142943728243245389"
},
"rectangle-4-3": {
"unit-length": "0.06015242557729066455729721838475935157781642221442221008448696685014799122986419275430988587895112203",
"unit-area": "0.03383573938722599881347968534142713526252173749561249317252391885320824506679860842429931080691000614"
},
"rectangle-3-2": {
"unit-length": "0.07341026582832900937179089835065613159370491413834416715216232339707316312607345522854936307815067964",
"unit-area": "0.03262678481259067083190706593362494737497996183926407428984992150981029472269931343491082803473363540"
},
"rectangle-5-3": {
"unit-length": "0.08692167144116837407690913438063403947445054161330713629094946582643376048466065262934301981861783055",
"unit-area": "0.03129180171882061466768728837702825421080219498079056906474180769751615377447783494656348713470241900"
},
"rectangle-2-1": {
"unit-length": "0.1143408385597853825128004797625789736895041898896650940262085228993049658853478879730270357749487687",
"unit-area": "0.02858520963994634562820011994064474342237604747241627350655213072482624147133697199325675894373719217"
},
"rectangle-5-2": {
"unit-length": "0.1558531716926164872226462767339573658538511721131313144096922917633579408578968163918488235361447291",
"unit-area": "0.02493650747081863795562340427743317853661618753810101030555076668213727053726349062269581176578315666"
},
"rectangle-3-1": {
"unit-length": "0.1974876982511250798152918368445814328393239640232965557971735253745416346889820708769352443662309356",
"unit-area": "0.02194307758345834220169909298273127031548044044703295064413039170828240385433134120854836048513677062"
},
"rectangle-4-1": {
"unit-length": "0.2808129583076773784824632411598978785739065260602522565068463692187538634353764600309945603867151386",
"unit-area": "0.01755080989422983615515395257249361741086915787876576603167789807617211646471102875193716002416969616"
},
"rectangle-5-1": {
"unit-length": "0.3641459427349860857969735336572619150436447717510009827382248635912373122246232036309989766410197316",
"unit-area": "0.01456583770939944343187894134629047660174579087004003930952899454364949248898492814523995906564078926"
},
"rectangle-6-1": {
"unit-length": "0.4474792609906705066558127072414405679171593222649354655718340353344281854473215790466193721724027055",
"unit-area": "0.01242997947196306962932813075670668244214331450735931848810650098151189404020337719573942700478896404"
},
"ellipse-3-2": {
"unit-length": "0.8156057850665809369085708783514094026281112863762293958300336922337600285550734900284321181176650080",
"unit-area": "0.03672806379043738517743471462442639123872145670933610355715400247513003022328304655925703140759686599"
},
"ellipse-2-1": {
"unit-length": "1.256637061435917295385057353311801153678867759750042328389977836923126562514483599451213930136846827",
"unit-area": "0.03183098861837906715377675267450287240689192914809128974953346881177935952684530701802276055325061719"
},
"ellipse-3-1": {
"unit-length": "2.120575041173110435962284283713664446833089344578196429158087599807776074243191074073923507105929021",
"unit-area": "0.02387324146378430036533256450587715430516894686106846731215010160883451964513398026351707041493796289"
},
"ellipse-4-1": {
"unit-length": "2.956793085731570106788370243086590949832630022941276066799947851583827205916431998708738659145521946",
"unit-area": "0.01872411095198768656104514863206051318052466420475958220560792283045844678049723942236632973720624541"
}
}
}