{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,23]],"date-time":"2026-08-23T04:31:49Z","timestamp":1787459509738,"version":"build-2736575974"},"reference-count":41,"publisher":"MDPI AG","issue":"15","license":[{"start":{"date-parts":[[2021,7,22]],"date-time":"2021-07-22T00:00:00Z","timestamp":1626912000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Polymers"],"abstract":"<jats:p>A theoretical solution of the problem of thick-walled shell optimization by varying the mechanical characteristics of the material over the thickness of the structure is proposed, taking into account its rheological properties. The optimization technique is considered by the example of a cylindrical shell made of high-density polyethylene with hydroxyapatite subjected to internal pressure. Radial heterogeneity can be created by centrifugation during the curing of the polymer mixed with the additive. The nonlinear Maxwell\u2013Gurevich equation is used as the law describing polymer creep. The relationship of the change in the additive content along with the radius r, at which the structure is equally stressed following the four classical criteria of fracture, is determined in an elastic formulation. Moreover, it is shown that a cylinder with equal stress at the beginning of the creep process ceases to be equally stressed during creep. Finally, an algorithm for defining the relationship of the additive mass content on coordinate r, at which the structure is equally stressed at the end of the creep process, is proposed. The developed algorithm, implemented in the MATLAB software, allows modeling both equally stressed and equally strength structures.<\/jats:p>","DOI":"10.3390\/polym13152408","type":"journal-article","created":{"date-parts":[[2021,7,22]],"date-time":"2021-07-22T22:35:31Z","timestamp":1626993331000},"page":"2408","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Optimization of Thick-Walled Viscoelastic Hollow Polymer Cylinders by Artificial Heterogeneity Creation: Theoretical Aspects"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9133-8546","authenticated-orcid":false,"given":"Anton","family":"Chepurnenko","sequence":"first","affiliation":[{"name":"Strength of Materials Department, Faculty of Civil and Industrial Engineering, Don State Technical University, Rostov-on-Don 344000, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stepan","family":"Litvinov","sequence":"additional","affiliation":[{"name":"Strength of Materials Department, Faculty of Civil and Industrial Engineering, Don State Technical University, Rostov-on-Don 344000, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Besarion","family":"Meskhi","sequence":"additional","affiliation":[{"name":"Department of Life Safety and Environmental Protection, Faculty of Life Safety and Environmental Engineering, Don State Technical University, Rostov-on-Don 344000, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6173-9365","authenticated-orcid":false,"given":"Alexey","family":"Beskopylny","sequence":"additional","affiliation":[{"name":"Department of Transport Systems, Faculty of Roads and Transport Systems, Don State Technical University, Rostov-on-Don 344000, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,22]]},"reference":[{"key":"ref_1","first-page":"379","article-title":"Elastic analysis for thick-walled tubes of functionally graded material subjected to internal pressure","volume":"35","author":"Fukui","year":"1992","journal-title":"JSME Int. J. Ser. 1 Solid Mech. Strength Mater."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1023\/A:1007625401963","article-title":"The pressurized hollow cylinder or disk problem for functionally graded isotropic linearly elastic materials","volume":"55","author":"Horgan","year":"1999","journal-title":"J. Elast."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1016\/S1359-8368(01)00041-5","article-title":"Exact solutions for stresses in functionally graded pressure vessels","volume":"32","author":"Tutuncu","year":"2001","journal-title":"Compos. Part B Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2032","DOI":"10.1016\/j.engstruct.2006.12.003","article-title":"Stresses in thick-walled FGM cylinders with exponentially varying properties","volume":"29","author":"Tutuncu","year":"2007","journal-title":"Eng. Struct."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1649","DOI":"10.1515\/amm-2016-0267","article-title":"Exact and numerical elastic analysis for the FGM thick-walled cylindrical pressure vessels with exponentially varying properties","volume":"61","author":"Nejad","year":"2016","journal-title":"Arch. Metall. Mater."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"593","DOI":"10.4028\/www.scientific.net\/AMM.752-753.593","article-title":"The Method of Separation of Variables in the Problem of Theory of Elasticity for Radially Inhomogeneous Cylinder","volume":"752","author":"Andreev","year":"2015","journal-title":"Appl. Mech. Mater."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1016\/j.mechrescom.2006.01.005","article-title":"Elastic analyses of heterogeneous hollow cylinders","volume":"33","author":"Xiang","year":"2006","journal-title":"Mech. Res. Commun."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1016\/j.ijmecsci.2014.08.028","article-title":"An elasticity solution for functionally graded thick-walled tube subjected to internal pressure","volume":"89","author":"Xin","year":"2014","journal-title":"Int. J. Mech. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Andreev, V.I. (2012). Optimization of thick-walled shells based on solutions of inverse problems of the elastic theory for inhomogeneous bodies. Comput. Aided Optim. Des. Eng., 189\u2013202.","DOI":"10.2495\/OP120171"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"666","DOI":"10.1016\/j.compscitech.2011.01.009","article-title":"Material tailoring for functionally graded hollow cylinders and spheres","volume":"71","author":"Nie","year":"2011","journal-title":"Compos. Sci. Technol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"354","DOI":"10.4028\/www.scientific.net\/AMM.166-169.354","article-title":"About one way of optimization of the thick-walled shells","volume":"166","author":"Andreev","year":"2012","journal-title":"Appl. Mech. Mater."},{"key":"ref_12","first-page":"202","article-title":"Inverse problems of the inhomogeneous theory of elasticity for thick-walled shells","volume":"2","author":"Andreev","year":"2014","journal-title":"Int. J. Comput. Methods Exp. Meas."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"869","DOI":"10.4028\/www.scientific.net\/AMR.887-888.869","article-title":"Model of equal-stressed cylinder based on the Mohr failure criterion","volume":"887","author":"Andreev","year":"2014","journal-title":"Adv. Mater. Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.proeng.2016.08.077","article-title":"Way of optimization of stress state of elements of concrete structures","volume":"153","author":"Andreev","year":"2016","journal-title":"Procedia Eng."},{"key":"ref_15","first-page":"232","article-title":"Equal Strength and Equal Stress Structures. Models and Reality","volume":"102","author":"Andreev","year":"2017","journal-title":"Adv. Eng. Res."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Andreev, V., and Leontyev, A. (2017, January 11\u201315). Creation on the basis of the first theory of strength model equal stressed cylinder exposed to power and temperature loads. Proceedings of the International Conference on Modern Trends in Manufacturing Technologies and Equipment (ICMTMTE 2017), Sevastopol, Russia.","DOI":"10.1051\/matecconf\/201712905006"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Litvinov, S., Beskopylny, A., Trush, L., and Yazyev, S. (2017, January 16\u201317). Optimization of thick-walled spherical shells at thermal and power influences. Proceedings of the International Scientific Conference, Saint-Petersburg, Russia.","DOI":"10.1051\/matecconf\/201710604013"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/s10659-009-9239-8","article-title":"Exact solutions and material tailoring for functionally graded hollow circular cylinders","volume":"99","author":"Nie","year":"2010","journal-title":"J. Elast."},{"key":"ref_19","unstructured":"Andreev, V., and Barmenkova, E. (2016, January 16\u201317). Optimization of the thin-walled rod with an open profile. Proceedings of the 5th International Scientific Conference \u201cIntegration, Partnership and Innovation in Construction Science and Education\u201d, Moscow, Russia."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1016\/j.proeng.2014.12.005","article-title":"Iterative Method of Optimization of Stress State of Column under Eccentric Compression","volume":"91","author":"Andreev","year":"2014","journal-title":"Procedia Eng."},{"key":"ref_21","unstructured":"Yazyev, B.M. (1990). Nonlinear creep of continuously heterogeneous cylinders. [Ph.D. Thesis, Moscow State University of Civil Engineering]."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Kliukas, R., Luko\u0161evi\u010dien\u0117, O., Jaras, A., and Jonaitis, B. (2020). The Mechanical Properties of Centrifuged Concrete in Reinforced Concrete Structures. Appl. Sci., 10.","DOI":"10.3390\/app10103570"},{"key":"ref_23","first-page":"30","article-title":"Setting a diagram approach to calculating vibrated, centrifuged and vibrocentrifuged reinforced concrete columns with a variatropic structure","volume":"1","author":"Mailyan","year":"2021","journal-title":"Russ. J. Build. Constr. Archit."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"022014","DOI":"10.1088\/1757-899X\/753\/2\/022014","article-title":"Influence of type of filler and dispersive reinforcement on the nature of structured formation and deformative properties of vibrocentrifuged concrete","volume":"753","author":"Stelmakh","year":"2020","journal-title":"IOP Conf. Ser. Mater. Sci. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"012060","DOI":"10.1088\/1757-899X\/905\/1\/012060","article-title":"Study of changes in strength properties along section thickness of high-strength centrifuged and vibro-centrifuged concrete","volume":"905","author":"Stelmakh","year":"2020","journal-title":"IOP Conf. Ser. Mater. Sci. Eng."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"488","DOI":"10.1115\/1.2812406","article-title":"Manufacturing and Testing of a Functionally Graded Material","volume":"121","author":"Lambros","year":"1999","journal-title":"J. Eng. Mater. Technol."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1016\/j.jsv.2004.03.068","article-title":"Natural Frequencies of a Functionally Graded Anisotropic Rectangular Plate","volume":"282","author":"Batra","year":"2005","journal-title":"J. Sound Vib."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/s10659-011-9350-5","article-title":"Analytical solution for radial deformations of functionally graded isotropic and incompressible second-order elastic hollow spheres","volume":"107","author":"Iaccarino","year":"2011","journal-title":"J. Elast."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1590\/1679-78252386","article-title":"Stress analysis of thick spherical pressure vessel composed of transversely isotropic functionally graded incompressible hyperelastic materials","volume":"13","author":"Anani","year":"2016","journal-title":"Lat. Am. J. Solids Struct."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1670","DOI":"10.4028\/www.scientific.net\/AMM.423-426.1670","article-title":"Equilibrium of a thick-walled sphere of inhomogeneous nonlinear-elastic material","volume":"423","author":"Andreev","year":"2013","journal-title":"Appl. Mech. Mater."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3182","DOI":"10.4028\/www.scientific.net\/AMM.405-408.3182","article-title":"Elastic-plastic equilibrium of a hollow cylinder from inhomogeneous perfectly plastic material","volume":"405","author":"Andreev","year":"2013","journal-title":"Appl. Mech. Mater."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1697","DOI":"10.1007\/s11012-014-9948-3","article-title":"Nonlinear transient thermal stress analysis of thick-walled FGM cylinder with temperature-dependent material properties","volume":"49","author":"Abbas","year":"2014","journal-title":"Meccanica"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Najibi, A., Alizadeh, P., and Ghazifard, P. (2021). Transient thermal stress analysis for a short thick hollow FGM cylinder with nonlinear temperature-dependent material properties. J. Therm. Anal. Calorim., 1\u201312.","DOI":"10.1007\/s10973-020-10442-2"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/j.conbuildmat.2013.10.051","article-title":"Development of a hollow cylinder test for the elastic modulus distribution and the ultimate strength of bamboo","volume":"51","author":"Lee","year":"2014","journal-title":"Constr. Build. Mater."},{"key":"ref_35","unstructured":"Yazyev, B.M., Andreev, V.I., and Turusov, R.A. (2009). Some Problems and Methods of Mechanics of a Macro-Inhomogeneous Viscoelastic Medium, Rostov State University of Civil Engineering."},{"key":"ref_36","first-page":"191","article-title":"Transient thermo-viscoelastic response of a functionally graded non-axisymmetric cylinder","volume":"46","author":"Ghajar","year":"2015","journal-title":"J. Comput. Appl. Mech."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1686","DOI":"10.1016\/j.proeng.2016.07.156","article-title":"Flat axisymmetrical problem of thermal creepage for thick-walled cylinder made of recyclable PVC","volume":"150","author":"Litvinov","year":"2016","journal-title":"Procedia Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"96","DOI":"10.4028\/www.scientific.net\/KEM.816.96","article-title":"Approbation of the Mathematical Model of Adhesive Strength with Viscoelasticity","volume":"816","author":"Litvinov","year":"2019","journal-title":"Key Eng. Mater."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1186\/1475-925X-13-125","article-title":"Thermal, creep-recovery and viscoelastic behavior of high density polyethylene\/hydroxyapatite nano particles for bone substitutes: Effects of gamma radiation","volume":"13","author":"Alothman","year":"2014","journal-title":"Biomed. Eng. Online"},{"key":"ref_40","unstructured":"Andreev, V.I. (2002). Some Problems and Methods of Mechanics of Inhomogeneous Bodies: Monograph, Publishing House ASV."},{"key":"ref_41","first-page":"26","article-title":"S Clarification of the solution to the problem of the adhesive bond long-term strength on normal separation","volume":"290","author":"Chepurnenko","year":"2020","journal-title":"Struct. Mech. Anal. Constr."}],"container-title":["Polymers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-4360\/13\/15\/2408\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:33:32Z","timestamp":1760164412000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-4360\/13\/15\/2408"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,22]]},"references-count":41,"journal-issue":{"issue":"15","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["polym13152408"],"URL":"https:\/\/doi.org\/10.3390\/polym13152408","relation":{},"ISSN":["2073-4360"],"issn-type":[{"value":"2073-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,22]]}}}