{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T15:56:15Z","timestamp":1785426975504,"version":"3.56.0"},"reference-count":19,"publisher":"National Academy of Sciences","issue":"24","license":[{"start":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T00:00:00Z","timestamp":1781049600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100019180","name":"EC | Horizon Europe | Excellent Science | HORIZON EUROPE European Research Council","doi-asserted-by":"publisher","award":["101043968"],"award-info":[{"award-number":["101043968"]}],"id":[{"id":"10.13039\/100019180","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000183","name":"DOD | USA | AFC | CCDC | Army Research Office","doi-asserted-by":"publisher","award":["W911NF-22-2-0256"],"award-info":[{"award-number":["W911NF-22-2-0256"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.pnas.org"],"crossmark-restriction":true},"short-container-title":["Proc. Natl. Acad. Sci. U.S.A."],"published-print":{"date-parts":[[2026,6,16]]},"abstract":"\n Ultrathin films of widely different materials exhibit a dramatic enhancement of projectile penetration resistance under high\u2013velocity impact. Despite extensive simulations and experiments, a unifying physical explanation has remained elusive. Here we show that the specific penetration energy follows a general inverse-cube scaling law,\n \n \n \n \n E<\/mml:mi>\n p<\/mml:mi>\n \u2217<\/mml:mo>\n <\/mml:msubsup>\n \n (<\/mml:mo>\n h<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \n E<\/mml:mi>\n \n p<\/mml:mi>\n ,<\/mml:mo>\n \u221e<\/mml:mi>\n <\/mml:mrow>\n \u2217<\/mml:mo>\n <\/mml:msubsup>\n +<\/mml:mo>\n B<\/mml:mi>\n \n h<\/mml:mi>\n \n \u2212<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:inline-formula>\n , across chemically and structurally distinct systems. The inverse\u2013cube scaling is traced to a finite\u2013size correction to the effective shear modulus caused by suppression of long\u2013wavelength nonaffine modes. The scaling describes impact data for multilayer graphene, graphene oxide, and polymer thin films, revealing a common elastic contribution to nanoscale impact resistance.\n <\/jats:p>","DOI":"10.1073\/pnas.2609202123","type":"journal-article","created":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T17:38:23Z","timestamp":1781113103000},"update-policy":"https:\/\/doi.org\/10.1073\/pnas.cm10313","source":"Crossref","is-referenced-by-count":0,"title":["General inverse-cube thickness scaling of projectile penetration energy in ultrathin films"],"prefix":"10.1073","volume":"123","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6673-7043","authenticated-orcid":false,"given":"Alessio","family":"Zaccone","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/00wjc7c48","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Physics \u201cA. Pontremoli\u201d, University of Milan","place":["Milan, Italy"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1105-4931","authenticated-orcid":false,"given":"Timothy W.","family":"Sirk","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/011hc8f90","id-type":"ROR","asserted-by":"publisher"}],"name":"Polymers Branch, DEVCOM Army Research Laboratory, Aberdeen Proving Ground","place":["Aberdeen"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"341","published-online":{"date-parts":[[2026,6,10]]},"reference":[{"key":"e_1_3_3_1_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.2010787117"},{"key":"e_1_3_3_2_2","first-page":"032602","article-title":"Local inversion symmetry breaking controls the boson peak in glasses and crystals","volume":"95","author":"Milkus R.","year":"2017","unstructured":"R. Milkus, A. Zaccone, Local inversion symmetry breaking controls the boson peak in glasses and crystals. Phys. Rev. E 95, 032602 (2017).","journal-title":"Phys. Rev. E"},{"key":"e_1_3_3_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-031-24706-4"},{"key":"e_1_3_3_4_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.83.184205"},{"key":"e_1_3_3_5_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-005-9015-5"},{"key":"e_1_3_3_6_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevMaterials.5.035602"},{"key":"e_1_3_3_7_2","volume-title":"Impact Dynamics","author":"Zukas J. A.","year":"1982","unstructured":"J. A. Zukas, Impact Dynamics (Wiley, 1982)."},{"key":"e_1_3_3_8_2","volume-title":"Impact: The Theory and Physical Behaviour of Colliding Solids","author":"Goldsmith W.","year":"2001","unstructured":"W. Goldsmith, Impact: The Theory and Physical Behaviour of Colliding Solids (Dover, 2001)."},{"key":"e_1_3_3_9_2","first-page":"478","article-title":"Penetration of targets by high-speed projectiles","volume":"38","author":"Florence A. L.","year":"1967","unstructured":"A. L. Florence, Penetration of targets by high-speed projectiles. J. Appl. Phys. 38, 478\u2013485 (1967).","journal-title":"J. Appl. Phys."},{"key":"e_1_3_3_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.actamat.2006.10.053"},{"key":"e_1_3_3_11_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41598-018-25050-2"},{"key":"e_1_3_3_12_2","doi-asserted-by":"publisher","DOI":"10.1126\/science.1258544"},{"key":"e_1_3_3_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.carbon.2015.11.052"},{"key":"e_1_3_3_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.carbon.2016.02.033"},{"key":"e_1_3_3_15_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.eml.2025.102419"},{"key":"e_1_3_3_16_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.mattod.2018.07.014"},{"key":"e_1_3_3_17_2","doi-asserted-by":"publisher","DOI":"10.1038\/nmat1175"},{"key":"e_1_3_3_18_2","doi-asserted-by":"publisher","DOI":"10.1021\/ma060790i"},{"key":"e_1_3_3_19_2","doi-asserted-by":"publisher","DOI":"10.1021\/acs.nanolett.0c00066"}],"container-title":["Proceedings of the National Academy of Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pnas.org\/doi\/pdf\/10.1073\/pnas.2609202123","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T17:39:54Z","timestamp":1781113194000},"score":1,"resource":{"primary":{"URL":"https:\/\/pnas.org\/doi\/10.1073\/pnas.2609202123"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,10]]},"references-count":19,"journal-issue":{"issue":"24","published-print":{"date-parts":[[2026,6,16]]}},"alternative-id":["10.1073\/pnas.2609202123"],"URL":"https:\/\/doi.org\/10.1073\/pnas.2609202123","relation":{},"ISSN":["0027-8424","1091-6490"],"issn-type":[{"value":"0027-8424","type":"print"},{"value":"1091-6490","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,10]]},"assertion":[{"value":"2026-03-16","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-05-17","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-06-10","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"e2609202123"}}