BUILDING A SANDCASTLE ===================== An independent implementation, written from published equations. WHAT THIS IS ------------ Building a sandcastle is a folk activity. It has no author, no year of publication, no publisher and no trademark, and this app is not a replica of anyone's software. Nothing here is copied from any other simulation: the sphere packing, the capillary-bridge solver, the Young-Laplace oracle, the wedge minimiser, the buckling eigensolve, the game and the renderer are all written from the physics. What IS derived from other people's work is the physics itself, and those people are named below. Where a formula is used, the source that was actually read is given. Where the primary source could not be obtained, that is said plainly and the restatement that was read is named instead. TAGS ---- DOCUMENTED read in the source named, in full text MEASURED produced by this app's own code, by this app's own harness DERIVED algebra from a documented item RECONSTRUCTED a modelling choice with no source behind it qualified documented, but of something adjacent: a restatement rather than the original, or a different material SOURCES USED ------------ IAPWS Release R1-76(2014), "Surface Tension of Ordinary Water Substance". Surface tension of water: 0.072736 N/m at 20 C. DOCUMENTED https://iapws.org/public/documents/CH-L9/Surf-H2O-2014.pdf H. Rumpf (1962), tensile strength of an agglomerate: sigma_t = phi k F / (pi d^2), and the closure k = 3.1/(1-phi). DOCUMENTED, qualified The 1962 chapter is offline; four independent modern full texts restate it: Mitarai & Nori arXiv:cond-mat/0601660 Eq. 11; Bianchi et al. arXiv:1709.00205 Eq. 5; Badetti et al. arXiv:1802.08172 Eq. 2; Kimura et al. arXiv:2006.05107. A. G. Greenhill (1881), self-weight buckling of a column. DOCUMENTED, qualified The constant 7.8373 is standard; this app does not quote it. It solves the buckling eigenproblem by shooting and obtains 7.8373474. MEASURED Culmann's planar-wedge method for slope stability, as set out in M. Murthy, "Geotechnical Engineering: Principles and Practices of Soil Mechanics and Foundation Engineering", Chapter 10. DOCUMENTED H_c = (4c/gamma) sin(beta) cos(phi) / (1 - cos(beta - phi)), critical plane at (beta + phi)/2, and the infinite-slope result tan(beta_max) = tan(phi'). V. Richefeu, M. S. El Youssoufi and F. Radjai, "Shear strength properties of wet granular materials", Physical Review E 73, 051304 (2006). DOCUMENTED https://arxiv.org/pdf/cond-mat/0604368 Friction angle of sand ~33 degrees and independent of water content; Coulomb cohesion saturating at ~600 Pa above ~3% water by mass. C. Semprebon, M. Scheel, S. Herminghaus, R. Seemann and M. Brinkmann, "Liquid morphologies and capillary forces between three spherical beads", arXiv:1512.07668. DOCUMENTED Read in place of Scheel et al., Nature Materials 7, 189 (2008), which is paywalled and was NOT read. Scheel is a co-author of the open paper, which restates the plateau result with numbers. F. Radjai, "Force and fabric states in granular media", arXiv:0801.4722, and arXiv:0711.2732. DOCUMENTED Why the bulk friction angle is not the grain friction coefficient. M. Pakpour, M. Habibi, P. Moller and D. Bonn, "How to construct the perfect sandcastle", Scientific Reports 2, 549 (2012). DOCUMENTED Open institutional copy: https://pure.uva.nl/ws/files/1675811/120167_380812.pdf Elastic modulus G = alpha a^(-1/3) E^(2/3) gamma^(1/3) and the buckling height. N. Lu, T.-H. Kim, S. Sture and W. J. Likos, "Tensile strength of unsaturated sand", Journal of Engineering Mechanics 135, 1410 (2009). DOCUMENTED Measured tensile strengths, 744 Pa to 1655 Pa. B. Sharma and A. Sauret (2025), review of wet granular media, arXiv:2501.10830; Y. Feng (2025), arXiv:2501.08344; Charlaix & Ciccotti, arXiv:0910.4626; Gladkyy & Schwarze, arXiv:1403.7926. DOCUMENTED The capillary-bridge force law, its contact limit and its volume dependence. S. Torquato and F. Stillinger, arXiv:1008.2982. DOCUMENTED Random close packing, and why the number is protocol-dependent. Contact angle of water on untreated natural sand, 61 degrees: arXiv:2009.09528. DOCUMENTED NOT OBTAINED, AND NOT QUOTED ---------------------------- Scheel et al., Nature Materials 7, 189 (2008) -- paywalled (303 to SSO). Herminghaus, Advances in Physics 54, 221 (2005) -- paywalled, not on arXiv. Willett, Adams, Johnson & Seville, Langmuir 16, 9396 (2000) -- ACS paywall. Rumpf (1962), the original chapter -- offline. Pierrat & Caram (1997) -- ScienceDirect 403. Nothing in this app is quoted from any of them. Where their results are used, a full-text restatement is cited above instead, and facts.md records which. WHAT DIFFERS FROM THE REAL THING -------------------------------- * Real sand grains are angular and graded. Here they are smooth equal spheres, which is known to over-state the strength by roughly a factor of two. * Surface roughness is one number, an effective minimum separation of 1% of the grain radius. RECONSTRUCTED. * The model is pendular: it describes isolated liquid necks. Past the point where bridges merge it says so and stops, and the wet tail of the curve is an extrapolation, drawn dashed. * Rumpf's mean field and the plane-cut sum both assume every bridge breaks at once. Real failure runs as a crack, so both are upper bounds. * The game's clock runs six times faster than real time, and the tide is a great deal faster than a real one. * Grains are not simulated individually in the game; the packing model produces a constitutive curve and the castle is a continuum on top of it. LICENCE ------- See LICENSE.txt. The app is offered under the MIT licence. The physics is not anybody's to licence.