{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,8]],"date-time":"2025-11-08T18:06:57Z","timestamp":1762625217403,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,1,25]],"date-time":"2024-01-25T00:00:00Z","timestamp":1706140800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This work investigates temperature\u2019s effect on the critical energy release rate using damage mechanics material models and the element deletion method. The energy release rate describes the decrease in total potential energy per increase in crack surface area. The critical energy release rate is widely used as the failure criterion for various elastic and plastic materials. In real-life scenarios, fractures may occur at different temperatures. The temperature dependency of the critical energy release rate for aluminum 2024-T351 and titanium Ti-6Al-4V is studied in this work. We utilized test-data-based advanced material models of these two alloys, considering the strain rate, temperature, and state of stress for plasticity and failure. These material models are used to simulate a three-dimensional fracture specimen to find the critical energy release rate at different temperatures. A new method to calculate the critical energy release rate with the element deletion method is introduced and verified with the virtual crack opening method. This method enables the calculation of the energy release rate in a classical damage mechanics simulation for dynamic cack propagation. The simulation result indicates that the critical energy release rate increases with rising temperatures for these alloys.<\/jats:p>","DOI":"10.3390\/sym16020142","type":"journal-article","created":{"date-parts":[[2024,1,25]],"date-time":"2024-01-25T03:59:42Z","timestamp":1706155182000},"page":"142","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Temperature Effects on Critical Energy Release Rate for Aluminum and Titanium Alloys"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9573-5816","authenticated-orcid":false,"given":"Teng","family":"Long","sequence":"first","affiliation":[{"name":"Department of Mechanical and Materials Engineering, University of Cincinnati, 2901 Woodside Drive, Cincinnati, OH 45221, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Leyu","family":"Wang","sequence":"additional","affiliation":[{"name":"Center for Collision Safety and Analysis, College of Science, George Mason University, 4400 University Drive, Fairfax, VA 22030, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"James D.","family":"Lee","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Aerospace Engineering, George Washington University, 2121 I St NW, Washington, DC 20052, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cing-Dao","family":"Kan","sequence":"additional","affiliation":[{"name":"Center for Collision Safety and Analysis, College of Science, George Mason University, 4400 University Drive, Fairfax, VA 22030, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,25]]},"reference":[{"key":"ref_1","unstructured":"Haight, S., Wang, L., Bois, P.D., Carney, K., and Kan, C.-D. (2023, December 01). Development of a Titanium Alloy ti-6al-4v Material Model Used in ls-Dyna, DOT\/FAA\/TC-15\/23, Available online: https:\/\/www.tc.faa.gov\/its\/worldpac\/techrpt\/tc15-23.pdf."},{"key":"ref_2","unstructured":"Park, C.-K., Carney, K., Bois, P.D., Cordasco, D., and Aluminum, C.-D.K. (2023, December 01). 2024-T351 Input Parameters for *MAT_224 in LS-DYNA, DOT\/FAA\/TC-19\/41 2020, Available online: https:\/\/www.tc.faa.gov\/its\/worldpac\/techrpt\/tc19-41-p1.pdf."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1361\/154770206X86563","article-title":"An overview of the space shuttle Columbia accident from recovery through reconstruction","volume":"6","author":"McDanels","year":"2006","journal-title":"J. Fail. Anal. Prev."},{"key":"ref_4","unstructured":"Cikanek, H. (July, January 29). Characteristics of space shuttle main engine failures. Proceedings of the 23rd Joint Propulsion Conference, San Diego, CA, USA."},{"key":"ref_5","first-page":"219","article-title":"Stresses in a plate due to the presence of cracks and sharp corners","volume":"55","author":"Inglis","year":"1913","journal-title":"Trans. Inst. Naval. Archit."},{"key":"ref_6","unstructured":"Wang, L. (2015). Energy Release Rate in Fracture Mechanics. [Ph.D. Thesis, The George Washington University]."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1115\/1.4011547","article-title":"Analysis of stresses and strains near the end of a crack traversing a plate","volume":"24","author":"Irwin","year":"1957","journal-title":"J. Appl. Mech."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1115\/1.3601206","article-title":"A path independent integral and the approximate analysis of strain concentration by notches and cracks","volume":"35","author":"Rice","year":"1968","journal-title":"J. Appl. Mech."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"100202","DOI":"10.1016\/j.finmec.2023.100202","article-title":"Numerical verification of energy release rate and J-Integral in large strain formulation","volume":"11","author":"Long","year":"2023","journal-title":"Forces Mech."},{"key":"ref_10","first-page":"299","article-title":"Ls-dyna keyword user\u2019s manual","volume":"970","author":"Hallquist","year":"2007","journal-title":"Livermore Softw. Technol. Corp."},{"key":"ref_11","first-page":"163","article-title":"VI. the phenomena of rupture and flow in solids, Philosophical transactions of the royal society of London","volume":"221","author":"Griffith","year":"1921","journal-title":"Ser. A Contain. Pap. A Math. Phys. Character"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"823","DOI":"10.1007\/BF00548176","article-title":"Surface energies of silicate glasses calculated from their wettability data","volume":"12","author":"Rhee","year":"1977","journal-title":"J. Mater. Sci."},{"key":"ref_13","unstructured":"(2023, December 01). Available online: https:\/\/www.fracturemechanics.org\/griffith.html."},{"key":"ref_14","unstructured":"Sun, C.-T., and Jin, Z. (2011). Fracture Mechanics, Academic Press."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Anderson, T.L. (2017). Fracture Mechanics: Fundamentals and Applications, CRC Press.","DOI":"10.1201\/9781315370293"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Janssen, M., Zuidema, J., and Wanhill, R. (2004). Fracture Mechanics: Fundamentals and Applications, CRC Press.","DOI":"10.1201\/9781482265583"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/S1359-835X(00)00121-4","article-title":"Measurement of the critical energy release rate Giic for filament wound grp pipes","volume":"32","author":"Zou","year":"2001","journal-title":"Compos. Part A Appl. Sci. Manuf."},{"key":"ref_18","first-page":"327","article-title":"Technique for estimating fracture resistance of cultured neocartilage","volume":"12","author":"Cook","year":"2001","journal-title":"J. Mater. Sci. Mater. Med."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/j.engfracmech.2017.08.026","article-title":"Calculation of the critical energy release rate Gc of the cement line in cortical bone combining experimental tests and finite element models","volume":"184","author":"Giner","year":"2017","journal-title":"Eng. Fract. Mech."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"114314","DOI":"10.1063\/1.4869207","article-title":"An atomistic methodology of energy re- lease rate for graphene at nanoscale","volume":"115","author":"Zhang","year":"2014","journal-title":"J. Appl. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1016\/j.engfracmech.2012.10.026","article-title":"Moving cracks in viscoelastic materials: Temperature and energy-release-rate measurements","volume":"98","author":"Carbone","year":"2013","journal-title":"Eng. Fract. Mech."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1007\/s10704-009-9441-5","article-title":"On the dynamics of localization and fragmentation-iv. expansion of al 6061-o tubes","volume":"163","author":"Zhang","year":"2010","journal-title":"Int. J. Fract."},{"key":"ref_23","first-page":"101664","article-title":"Tensile and ductile fracture properties of as-printed 316l stainless steel thin walls obtained by directed energy deposition","volume":"37","author":"Margerit","year":"2021","journal-title":"Addit. Manuf."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"214010","DOI":"10.1088\/0022-3727\/42\/21\/214010","article-title":"Dynamic fragmentation of ductile materials","volume":"42","author":"Zhang","year":"2009","journal-title":"J. Phys. D Appl. Phys."},{"key":"ref_25","unstructured":"Landis, C.M., Ravi-Chandar, K., and Hughes, T.J. (2017). Texas Univ at Austin Austin United States, An Integrated Experimental and Modeling Study of Ductile Fracture."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Cui, Y., Xiang, D., Shu, L., Liao, Z., Zhang, Z., and Liu, W. (2021). Effects of impact speeds, fall postures and cortical thicknesses on femur fracture by incremental element deletion based finite element analysis. Authorea.","DOI":"10.22541\/au.161490602.24025811\/v1"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1858","DOI":"10.1016\/j.jmatprotec.2010.06.021","article-title":"Prediction of shear-induced fracture in sheet metal forming","volume":"210","author":"Li","year":"2010","journal-title":"J. Mater. Process. Technol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"011010","DOI":"10.1115\/1.4025625","article-title":"Evaluation of ductile fracture models in finite element simulation of metal cutting processes","volume":"136","author":"Liu","year":"2014","journal-title":"J. Manuf. Sci. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/j.conbuildmat.2012.11.089","article-title":"Prediction of the fracture performance of defect-free steel bars for civil engineering applications using finite element simulation","volume":"41","author":"Adewole","year":"2013","journal-title":"Constr. Build. Mater."},{"key":"ref_30","unstructured":"Wang, L., Dicecca, F., Haight, S., Carney, K., DuBois, P., Emmerling, W., and Kan, C. (2016, January 12\u201314). Test and simulation comparison using titanium material models based on mat224. Proceedings of the 14th International LS-DYNA Users Conference, Detroit, MI, USA."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"041006","DOI":"10.1115\/1.4054895","article-title":"An experimental investigation of the influence of the state of stress on the ductile fracture of 2024-t351 aluminum","volume":"144","author":"Seidt","year":"2022","journal-title":"J. Eng. Mater. Technol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"151","DOI":"10.17159\/2411-9717\/1340\/2021","article-title":"The crack growth resistance behaviour of aluminium alloy 2024-T3 at slow strain rates after exposure to standard corrosive environments","volume":"121","author":"Pretorius","year":"2021","journal-title":"J. S. Afr. Inst. Min. Metall."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1115\/1.3443358","article-title":"Initial crack extension in two intermediate-strength aluminum alloys","volume":"98","author":"Griffis","year":"1976","journal-title":"J. Eng. Mater. Technol."},{"key":"ref_34","unstructured":"Ojo, S.A. (2020). Use of Compact Specimens to Determine Fracture Tough-Ness and Fatigue Crack Growth Anisotropy of Ded Additive Manufactured ti-6al-4v. [Ph.D. Thesis, University of Akron]."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1007\/s10704-010-9519-0","article-title":"Analysis and computations of oscillating crack propagation in a heated strip","volume":"167","author":"Menouillard","year":"2011","journal-title":"Int. J. Fract."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"420","DOI":"10.1007\/s11668-023-01594-y","article-title":"Effect of temperature on the energy release rate variation in repaired laminate composites","volume":"23","author":"Belhouari","year":"2023","journal-title":"J. Fail. Anal. Prev."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Freund, L. (1990). Dynamic Fracture Mechanics, Cambridge University Press.","DOI":"10.1017\/CBO9780511546761"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"298","DOI":"10.4028\/www.scientific.net\/AMM.566.298","article-title":"Velocity-toughening and crack speed oscillations in the dynamic fracture of PMMA plates","volume":"566","author":"Zhang","year":"2014","journal-title":"Appl. Mech. Mater."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.enggeo.2016.12.008","article-title":"Effects of strain rate on fracture toughness and energy re- lease rate of gas shales","volume":"218","author":"Mahanta","year":"2017","journal-title":"Eng. Geol."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"4419","DOI":"10.1016\/j.ijsolstr.2014.09.010","article-title":"Energy release rates in rubber during dynamic crack propagation","volume":"51","author":"Kroon","year":"2014","journal-title":"Int. J. Solids Struct."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"102894","DOI":"10.1016\/j.tafmec.2021.102894","article-title":"Loading rate dependency of strain energy release rate in mode i delamination of composite laminates","volume":"112","author":"Ekhtiyari","year":"2021","journal-title":"Theor. Appl. Fract. Mech."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.ijimpeng.2005.07.013","article-title":"Numerical simulation of the post-failure motion of steel plates subjected to blast loading","volume":"32","author":"Balden","year":"2005","journal-title":"Int. J. Impact Eng."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"5168","DOI":"10.1002\/pc.26214","article-title":"Experimental and numerical study on the ballistic performance of ultrahigh molecular weight polyethylene laminate","volume":"42","author":"Zhu","year":"2021","journal-title":"Polym. Compos."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/2\/142\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:48:59Z","timestamp":1760104139000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/2\/142"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,25]]},"references-count":43,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["sym16020142"],"URL":"https:\/\/doi.org\/10.3390\/sym16020142","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,1,25]]}}}