{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,24]],"date-time":"2026-07-24T20:07:43Z","timestamp":1784923663648,"version":"3.55.0"},"reference-count":79,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,12,23]],"date-time":"2019-12-23T00:00:00Z","timestamp":1577059200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Fatigue in any material is a result of continuous irreversible degradation process. Traditionally, fatigue life is predicted by extrapolating experimentally curve fitted empirical models. In the current study, unified mechanics theory is used to predict life of Ti-6Al-4V under monotonic tensile, compressive and cyclic load conditions. The unified mechanics theory is used to derive a constitutive model for fatigue life prediction using a three-dimensional computational model. The proposed analytical and computational models have been used to predict the low cycle fatigue life of Ti-6Al-4V alloys. It is shown that the unified mechanics theory can be used to predict fatigue life of Ti-6Al-4V alloys by using simple predictive models that are based on fundamental equation of the material, which is based on thermodynamics associated with degradation of materials.<\/jats:p>","DOI":"10.3390\/e22010024","type":"journal-article","created":{"date-parts":[[2019,12,24]],"date-time":"2019-12-24T05:56:15Z","timestamp":1577166975000},"page":"24","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":29,"title":["Low Cycle Fatigue Life Prediction Using Unified Mechanics Theory in Ti-6Al-4V Alloys"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8280-2044","authenticated-orcid":false,"given":"Noushad","family":"Bin Jamal M","sequence":"first","affiliation":[{"name":"Department of Applied Mechanics, Indian Institute of Technology, Madras 600036, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Aman","family":"Kumar","sequence":"additional","affiliation":[{"name":"Department of Applied Mechanics, Indian Institute of Technology, Madras 600036, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chebolu","family":"Lakshmana Rao","sequence":"additional","affiliation":[{"name":"Department of Applied Mechanics, Indian Institute of Technology, Madras 600036, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4378-0476","authenticated-orcid":false,"given":"Cemal","family":"Basaran","sequence":"additional","affiliation":[{"name":"Civil, Structural and Environmental Engineering, University at Buffalo, State University of New York, New York, NY 10031, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Mouritz, A.P. (2012). Introduction to Aerospace Materials, Woodhead Publishing Limited.","DOI":"10.2514\/4.869198"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/j.msea.2012.04.036","article-title":"Deformation and fracture behavior of laser processed dense and porous Ti6Al4V alloy under static and dynamic loading","volume":"549","author":"Biswas","year":"2012","journal-title":"Mater. Sci. Eng. A"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"844","DOI":"10.1016\/j.actamat.2012.10.043","article-title":"Perspectives on titanium science and technology","volume":"61","author":"Banerjee","year":"2013","journal-title":"Acta Mater."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1016\/j.ijfatigue.2012.03.008","article-title":"On the effect of deep-rolling and laser-peening on the stress-controlled low- and high-cycle fatigue behavior of Ti-6Al-4V at elevated temperatures up to 550 \u00b0C","volume":"44","author":"Altenberger","year":"2012","journal-title":"Int. J. Fatigue"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1111\/j.1460-2695.1981.tb01130.x","article-title":"Fatigue Life Behavior of Copper Single Crystals. Part I: Observations of Crack Nucleation","volume":"4","author":"CHENG","year":"1981","journal-title":"Fatigue Fract. Eng. Mater. Struct."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.ijfatigue.2019.05.035","article-title":"Low cycle fatigue properties of Ti-6Al-4V alloy fabricated by high-power laser directed energy deposition: Experimental and prediction","volume":"127","author":"Ren","year":"2019","journal-title":"Int. J. Fatigue"},{"key":"ref_7","first-page":"97","article-title":"Dislocation Model for Fatigue Crack Initiation","volume":"48","author":"Tanaka","year":"1981","journal-title":"Am. Soc. Mech. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/j.ijmecsci.2018.02.047","article-title":"An intrinsic dissipation model for high-cycle fatigue life prediction","volume":"140","author":"Guo","year":"2018","journal-title":"Int. J. Mech. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Sosnovskiy, L.A., and Senko, V.I. (2005, January 5\u201311). Tribo-fatigue. Proceedings of the ASME International Mechanical Engineering Congress and Exposition, Tribology, Orlando, FL, USA.","DOI":"10.1115\/IMECE2005-80352"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1109\/TCAD.2018.2803623","article-title":"Entropy Production-Based Full-Chip Fatigue Analysis: From Theory to Mobile Applications","volume":"38","author":"Wang","year":"2019","journal-title":"IEEE Trans. Comput. Des. Integr. Circuits Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1115\/1.3153663","article-title":"A Relnterpretation of the Palmgren- miner Rule for Fatigue Life","volume":"47","author":"Hashin","year":"2016","journal-title":"J. Appl. Mech."},{"key":"ref_12","first-page":"767","article-title":"A stress\u2013strain function for the fatigue of metals (stress-strain function for metal fatigue including mean stress effect)","volume":"5","author":"Smith","year":"1970","journal-title":"J. Mater."},{"key":"ref_13","unstructured":"Lemaitre, J., and Desmorat, R. (2005). Engineering damage mechanics: ductile, creep, fatigue and brittle failures, Springer."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1016\/S0142-1123(98)00032-2","article-title":"Continuum damage mechanics analysis of fatigue crack initiation","volume":"20","author":"Bhattacharya","year":"1998","journal-title":"Int. J. Fatigue"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1111\/ffe.12506","article-title":"Thermodynamic entropy generation in the course of the fatigue crack initiation","volume":"40","author":"Ontiveros","year":"2017","journal-title":"Fatigue Fract. Eng. Mater. Struct."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3493","DOI":"10.1016\/0956-7151(94)90481-2","article-title":"Atomic force microscopy and modeling of fatigue crack initiation in metals","volume":"42","author":"Harvey","year":"1994","journal-title":"Acta Metall. Mater."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10921-015-0317-5","article-title":"Analysis and Modeling of Thermal Signatures for Fatigue Damage Characterization in Ti\u20136Al\u20134V Titanium Alloy","volume":"35","author":"Kumar","year":"2016","journal-title":"J. Nondestruct. Eval."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Sosnovskiy, L.A., and Sherbakov, S.S. (2016). Mechanothermodynamic entropy and analysis of damage state of complex systems. Entropy, 18.","DOI":"10.3390\/e18070268"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/j.ijsolstr.2015.06.032","article-title":"Energy dissipation in the course of the fatigue degradation: Mathematical derivation and experimental quantification","volume":"77","author":"Kahirdeh","year":"2015","journal-title":"Int. J. Solids Struct."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1177\/1056789509343082","article-title":"An experimental approach to evaluate the critical damage","volume":"20","author":"Amiri","year":"2011","journal-title":"Int. J. Damage Mech."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1007\/s00161-012-0235-z","article-title":"Mechanothermodynamical system and its behavior","volume":"24","author":"Sosnovskiy","year":"2012","journal-title":"Contin. Mech. Thermodyn."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Zhang, M.H., Shen, X.H., He, L., and Zhang, K.-S. (2018). Application of Differential Entropy in Characterizing the Deformation Inhomogeneity and Life Prediction of Low-Cycle Fatigue of Metals. Materials, 11.","DOI":"10.3390\/ma11101917"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Sosnovskiy, L.A., and Sherbakov, S.S. (2017). A Model of Mechanothermodynamic Entropy in Tribology. Entropy, 19.","DOI":"10.3390\/e19030115"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Sosnovskiy, L., and Sherbakov, S. (2016). Mechanothermodynamics, Springer.","DOI":"10.1007\/978-3-319-24981-0"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Young, C., and Subbarayan, G. (2019). Maximum Entropy Models for Fatigue Damage in Metals with Application to Low-Cycle Fatigue of Aluminum 2024-T351. Entropy, 21.","DOI":"10.3390\/e21100967"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"9573524","DOI":"10.1155\/2016\/9573524","article-title":"A Review on Fatigue Life Prediction Methods for Metals","volume":"2016","author":"Santecchia","year":"2016","journal-title":"Adv. Mater. Sci. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Haddad, W.M. (2017). Thermodynamics: The unique universal science. Entropy, 19.","DOI":"10.3390\/e19110621"},{"key":"ref_28","unstructured":"Haddad, W.M., Chellabonia, V., and Nersesov, S.G. (2005). Thermodynamics: A Dynamical Systems Approach, Princeton University Press."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1115\/1.2792650","article-title":"A thermodynamic framework for damage mechanics of solder joints","volume":"120","author":"Basaran","year":"1998","journal-title":"J. Electron. Packag. Trans. ASME"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1177\/1056789504041058","article-title":"An irreversible thermodynamics theory for damage mechanics of solids","volume":"13","author":"Basaran","year":"2004","journal-title":"Int. J. Damage Mech."},{"key":"ref_31","first-page":"61","article-title":"Implementation of a thermodynamic framework for damage mechanics of solder interconnect in microelectronic packaging","volume":"11","author":"Basaran","year":"2002","journal-title":"ASME Int. Mech. Eng. Congr. Expo. Proc."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1738","DOI":"10.1016\/j.ijsolstr.2003.08.018","article-title":"A thermodynamic model for electrical current induced damage","volume":"40","author":"Basaran","year":"2003","journal-title":"Int. J. Solids Struct."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3744","DOI":"10.1016\/j.ijsolstr.2004.11.022","article-title":"A thermodynamics based damage mechanics constitutive model for low cycle fatigue analysis of microelectronics solder joints incorporating size effects","volume":"42","author":"Gomez","year":"2005","journal-title":"Int. J. Solids Struct."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1016\/j.mechmat.2005.11.008","article-title":"Damage mechanics constitutive model for Pb\/Sn solder joints incorporating nonlinear kinematic hardening and rate dependent effects using a return mapping integration algorithm","volume":"38","author":"Gomez","year":"2006","journal-title":"Mech. Mater."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1115\/1.1536171","article-title":"A damage mechanics-based fatigue life prediction model for solder joints","volume":"125","author":"Tang","year":"2003","journal-title":"J. Electron. Packag. Trans. ASME"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1627","DOI":"10.1179\/1743284715Y.0000000074","article-title":"Experimental verification of thermodynamic fatigue life prediction model using entropy as damage metric","volume":"31","author":"Temfack","year":"2015","journal-title":"Mater. Sci. Technol."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Wang, J., and Yao, Y. (2017). An entropy based low-cycle fatigue life prediction model for solder materials. Entropy, 19.","DOI":"10.3390\/e19100503"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1115\/1.1939822","article-title":"A damage-mechanics-based constitutive model for solder joints","volume":"127","author":"Basaran","year":"2005","journal-title":"J. Electron. Packag. Trans. ASME"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1115\/1.1400752","article-title":"Thermomechanical analysis of solder joints under thermal and vibration loading","volume":"124","author":"Basaran","year":"2002","journal-title":"J. Electron. Packag. Trans. ASME"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"123520-1","DOI":"10.1063\/1.2943261","article-title":"Thermomigration induced degradation in solder alloys","volume":"103","author":"Basaran","year":"2008","journal-title":"J. Appl. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/j.mechmat.2007.06.006","article-title":"Damage mechanics of electromigration induced failure","volume":"40","author":"Basaran","year":"2008","journal-title":"Mech. Mater."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1504\/IJMSI.2007.013864","article-title":"Damage mechanics of electromigration in microelectronics copper interconnects","volume":"1","author":"Basaran","year":"2007","journal-title":"Int. J. Mater. Struct. Integr."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2003","DOI":"10.1177\/0021998308094542","article-title":"Time dependent behavior of a particle filled composite PMMA\/ATH at elevated temperatures","volume":"42","author":"Basaran","year":"2008","journal-title":"J. Compos. Mater."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1099","DOI":"10.1016\/j.ijsolstr.2006.06.001","article-title":"A thermodynamics based damage mechanics model for particulate composites","volume":"44","author":"Basaran","year":"2007","journal-title":"Int. J. Solids Struct."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/j.mechmat.2008.10.013","article-title":"A computational damage mechanics model for thermomigration","volume":"41","author":"Li","year":"2009","journal-title":"Mech. Mater."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.commatsci.2004.10.007","article-title":"Electromigration induced stress analysis using fully coupled mechanical-diffusion equations with nonlinear material properties","volume":"34","author":"Lin","year":"2005","journal-title":"Comput. Mater. Sci."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1177\/0037549708094856","article-title":"Simulating Damage Mechanics of Electromigration and Thermomigration","volume":"84","author":"Shidong","year":"2008","journal-title":"Simulation"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1016\/j.commatsci.2013.01.016","article-title":"Electromigration damage mechanics of lead-free solder joints under pulsed DC: A computational model","volume":"71","author":"Yao","year":"2013","journal-title":"Comput. Mater. Sci."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"103708","DOI":"10.1063\/1.4821015","article-title":"Computational damage mechanics of electromigration and thermomigration","volume":"114","author":"Yao","year":"2013","journal-title":"J. Appl. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"165103-1","DOI":"10.1063\/1.4934740","article-title":"Irreversible entropy model for damage diagnosis in resistors","volume":"118","author":"Cuadras","year":"2015","journal-title":"J. Appl. Phys."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1016\/j.jpowsour.2016.10.077","article-title":"Entropy characterisation of overstressed capacitors for lifetime prediction","volume":"336","author":"Cuadras","year":"2016","journal-title":"J. Power Sources"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"145702-1","DOI":"10.1063\/1.4996629","article-title":"Determination of LEDs degradation with entropy generation rate","volume":"122","author":"Cuadras","year":"2017","journal-title":"J. Appl. Phys."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"6995","DOI":"10.3390\/e17106995","article-title":"A thermodynamic entropy approach to reliability assessment with applications to corrosion fatigue","volume":"17","author":"Imanian","year":"2015","journal-title":"Entropy"},{"key":"ref_54","first-page":"1","article-title":"A thermodynamic entropy-based damage assessment with applications to prognostics and health management","volume":"17","author":"Imanian","year":"2018","journal-title":"Struct. Heal. Monit."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1016\/j.tafmec.2018.05.013","article-title":"On the evaluation of fracture fatigue entropy","volume":"96","author":"Jang","year":"2018","journal-title":"Theor. Appl. Fract. Mech."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1016\/j.ijfatigue.2014.09.014","article-title":"Entropic characterization of metal fatigue with stress concentration","volume":"70","author":"Liakat","year":"2015","journal-title":"Int. J. Fatigue"},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Osara, J.A., and Bryant, M.D. (2019). A Thermodynamic Model for Lithium-Ion Battery Degradation: Application of the Degradation-Entropy Generation Theorem. Inventions, 4.","DOI":"10.3390\/inventions4020023"},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Osara, J.A., and Bryant, M.D. (2019). Thermodynamics of Fatigue: Degradation-Entropy Generation Methodology for System and Process Characterization and Failure Analysis. Entropy, 21.","DOI":"10.3390\/e21070685"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1163\/157361106777641387","article-title":"Damage Mechanics Modeling of Concurrent Thermal and Vibration Loading on Electronics Packaging","volume":"2","author":"Gomez","year":"2006","journal-title":"Multidiscip. Model. Mater. Struct."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"24","DOI":"10.3390\/e14010024","article-title":"On the role of entropy generation in processes involving fatigue","volume":"14","author":"Amiri","year":"2012","journal-title":"Entropy"},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Wang, J., and Yao, Y. (2019). An entropy-based failure prediction model for the creep and fatigue of metallic materials. Entropy, 21.","DOI":"10.3390\/e21111104"},{"key":"ref_62","doi-asserted-by":"crossref","unstructured":"Sun, F., Zhang, W., Wang, N., and Zhang, W. (2019). A copula entropy approach to dependence measurement for multiple degradation processes. Entropy, 21.","DOI":"10.3390\/e21080724"},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Yun, H., and Modarres, M. (2019). Measures of Entropy to Characterize Fatigue Damage in Metallic Materials. Entropy, 21.","DOI":"10.3390\/e21080804"},{"key":"ref_64","doi-asserted-by":"crossref","unstructured":"Li, E.H., Li, Y.Z., Li, T.T., Li, J.X., Zhai, Z.Z., and Li, T. (2019). Intelligent analysis algorithm for satellite health under time-varying and extremely high thermal loads. Entropy, 21.","DOI":"10.3390\/e21100983"},{"key":"ref_65","doi-asserted-by":"crossref","unstructured":"Sosnovskiy, L.A., and Sherbakov, S.S. (2019). On the Development of Mechanothermodynamics as a New Branch of Physics. Entropy, 21.","DOI":"10.3390\/e21121188"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"979","DOI":"10.1016\/j.mechmat.2011.09.002","article-title":"Damage characterization in non-isothermal stretching of acrylics. Part I: Theory","volume":"43","author":"Gunel","year":"2011","journal-title":"Mech. Mater."},{"key":"ref_67","first-page":"423","article-title":"On the thermodynamic entropy of fatigue fracture","volume":"466","author":"Naderi","year":"2010","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"875","DOI":"10.1016\/j.ijsolstr.2009.12.005","article-title":"An experimental approach to low-cycle fatigue damage based on thermodynamic entropy","volume":"47","author":"Naderi","year":"2010","journal-title":"Int. J. Solids Struct."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"011002","DOI":"10.1115\/1.3068296","article-title":"Influence of thermomigration on lead-free solder joint mechanical properties","volume":"131","author":"Abdulhamid","year":"2009","journal-title":"J. Electron. Packag. Trans. ASME"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1002\/andp.18842580616","article-title":"Ableitung des Stefan\u2019schen Gesetzes, betreffend die Abh\u00e4ngigkeit der W\u00e4rmestrahlung von der Temperatur aus der electromagnetischen Lichttheorie","volume":"258","author":"Boltzmann","year":"1884","journal-title":"Ann. Phys."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"1971","DOI":"10.3390\/e17041971","article-title":"Translation of Ludwig Boltzmann\u2019s paper \u201con the relationship between the second fundamental theorem of the mechanical theory of heat and probability calculations regarding the conditions for thermal equilibrium\u201d Sitzungberichte der kaiserlichen akademie d","volume":"17","author":"Sharp","year":"2015","journal-title":"Entropy"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1002\/andp.19013090310","article-title":"On the Law of Distribution of Energy in the Normal Spectrum","volume":"4","author":"Planck","year":"1901","journal-title":"Ann. Phys."},{"key":"ref_73","doi-asserted-by":"crossref","unstructured":"Lemaitre, J., and Chaboche, J.-L. (1990). Mechanics of Solid Materials, Cambridge University Press.","DOI":"10.1017\/CBO9781139167970"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1016\/j.ijplas.2011.10.007","article-title":"Thermo-mechanical strain gradient plasticity with energetic and dissipative length scales","volume":"30\u201331","author":"Voyiadjis","year":"2012","journal-title":"Int. J. Plast."},{"key":"ref_75","doi-asserted-by":"crossref","unstructured":"Murakami, S. (2012). Continuum Damage Mechanics, Springer.","DOI":"10.1007\/978-94-007-2666-6"},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/j.ijfatigue.2017.02.013","article-title":"Fatigue behavior of Ti-6Al-4V ELI including mean stress effects","volume":"99","author":"Carrion","year":"2017","journal-title":"Int. J. Fatigue"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1115\/1.3173673","article-title":"Mechanical modeling of material damage","volume":"55","author":"Murakami","year":"1988","journal-title":"J. Appl. Mech. Trans. ASME"},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"885","DOI":"10.1016\/0001-6160(75)90205-9","article-title":"Pressure dependence of yielding and associated volume expansion in tempered martensite","volume":"23","author":"Spitzig","year":"1975","journal-title":"Acta Metall."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"5057","DOI":"10.1016\/S0045-7825(00)00364-9","article-title":"Strength difference in compression and tension and pressure dependence of yielding in elasto-plasticity","volume":"190","author":"Mahnken","year":"2001","journal-title":"Comput. Methods Appl. Mech. Eng."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/1\/24\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:45:03Z","timestamp":1760190303000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/1\/24"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,23]]},"references-count":79,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["e22010024"],"URL":"https:\/\/doi.org\/10.3390\/e22010024","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints201911.0317.v1","asserted-by":"object"}]},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,12,23]]}}}