{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T05:07:52Z","timestamp":1774933672531,"version":"3.50.1"},"reference-count":65,"publisher":"MDPI AG","issue":"22","license":[{"start":{"date-parts":[[2022,11,9]],"date-time":"2022-11-09T00:00:00Z","timestamp":1667952000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>This article deals with the problem of predicting the brittle fracture of rocks and similar materials, which can also include frozen sandy soils. Such materials, due to the diversity of their conditions of origin, are characterized by natural heterogeneity at the micro-, meso-, and macro-levels, which makes it difficult to develop sufficiently universal criteria for their strength. Despite a number of known models and criteria of strength and fracture, the search for such criteria remains an urgent problem. In this paper, using the energy approach to the mathematical modeling of mechanical systems, the fracture criterion is justified, which differs from the known criteria that do not require integration to calculate the strain energy We and dissipation energy Wd. The well-known relation for the input energy W=We+Wd is used. The object of the study was the ratio of dW=dWe+dWd. The main research question concerned what the ratio of dWe and dWd would be at the point of brittle failure. The search for an answer to the question led to the justification of a differential energy criterion for the failure of brittle materials on the descending branch of the full stress\u2013strain curve. It was found that the point of predicted fracture is determined by the equality \u03c3=0.5\u00a0\u03b5Etangential (if there is an inflection point on the ascending branch) or \u03c3=0.5\u00a0\u03b5Esecant_secant. The main result of the work was ascertaining the differential strength and fracture criteria of brittle materials in the form of inequalities and equations, which were oriented for application in engineering calculations. Examples of application of the developed criteria are given; their consistency with the experimental data known from the literature confirmed.<\/jats:p>","DOI":"10.3390\/ma15227907","type":"journal-article","created":{"date-parts":[[2022,11,10]],"date-time":"2022-11-10T02:11:15Z","timestamp":1668046275000},"page":"7907","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Energy Criterion for Fracture of Rocks and Rock-like Materials on the Descending Branch of the Load\u2013Displacement Curve"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9694-0264","authenticated-orcid":false,"given":"Gennady","family":"Kolesnikov","sequence":"first","affiliation":[{"name":"Institute of Forestry, Mining and Construction Sciences, Petrozavodsk State University, Lenin Pr., 33, 185910 Petrozavodsk, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2781-2706","authenticated-orcid":false,"given":"Vitali","family":"Shekov","sequence":"additional","affiliation":[{"name":"Institute of Geology, Karelian Research Centre, Russian Academy of Sciences, Pushkinskaya St., 11, IG KarRC RAS, 185610 Petrozavodsk, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1016\/0148-9062(70)90007-0","article-title":"A Study of brittle Rock Fracture in Laboratory Compression Experiments","volume":"7","author":"Wawersik","year":"1970","journal-title":"Int. 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