{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,10]],"date-time":"2026-10-10T02:38:49Z","timestamp":1791599929512,"version":"4.3.4"},"reference-count":49,"publisher":"MDPI AG","issue":"22","license":[{"start":{"date-parts":[[2025,11,14]],"date-time":"2025-11-14T00:00:00Z","timestamp":1763078400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001321","name":"National Research Foundation of South Africa","doi-asserted-by":"publisher","award":["PMDS22051210010"],"award-info":[{"award-number":["PMDS22051210010"]}],"id":[{"id":"10.13039\/501100001321","id-type":"DOI","asserted-by":"publisher"}]},{"name":"University of Cape Town"},{"name":"Science Faculty of UCT"},{"name":"Erasmus+\/International Credit Mobility KA171 program"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>Using molecular dynamics simulations of a planar graphene sheet, we investigate the temperature dependence of its mechanical behavior under uniaxial tensile stress applied either in the armchair or zigzag direction. Stress\u2013strain curves are calculated for different temperatures, and the corresponding dependence of various elastic parameters is discussed. Fracture stress and strain, as well as the Young\u2019s modulus, decrease almost linearly with temperature, in accordance with previous investigations. An almost linear variation in the third-order elastic modulus with temperature is demonstrated, revealing opposite trends for uniaxial loadings in the armchair or zigzag direction. The detailed dependence of the distributions of bond lengths and bond angles both on strain and temperature is presented for the first time, along with approximate analytical expressions. The latter accurately describe the numerically obtained distributions.<\/jats:p>","DOI":"10.3390\/ma18225179","type":"journal-article","created":{"date-parts":[[2025,11,14]],"date-time":"2025-11-14T14:37:52Z","timestamp":1763131072000},"page":"5179","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Temperature-Dependent Mechanical and Structural Properties of Uniaxially Strained Planar Graphene"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6582-9117","authenticated-orcid":false,"given":"San\u00e9","family":"Erasmus","sequence":"first","affiliation":[{"name":"Nonlinear Dynamics and Chaos Group, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7700, South Africa"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2276-8251","authenticated-orcid":false,"given":"Charalampos","family":"Skokos","sequence":"additional","affiliation":[{"name":"Nonlinear Dynamics and Chaos Group, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7700, South Africa"},{"name":"Max Planck Institute for the Physics of Complex Systems, N\u00f6thnitzer Str. 38, D-01187 Dresden, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7763-718X","authenticated-orcid":false,"given":"George","family":"Kalosakas","sequence":"additional","affiliation":[{"name":"Department of Materials Science, University of Patras, GR-26504 Rio, Greece"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1126\/science.1157996","article-title":"Measurement of the elastic properties and intrinsic strength of monolayer graphene","volume":"321","author":"Lee","year":"2008","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1038\/s41467-019-14130-0","article-title":"Elastic straining of free-standing monolayer graphene","volume":"11","author":"Cao","year":"2020","journal-title":"Nat. Commun."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"109208","DOI":"10.1016\/j.ijmecsci.2024.109208","article-title":"Mechanical response of monolayer graphene via a multi-probe approach","volume":"273","author":"Varillas","year":"2024","journal-title":"Int. J. Mech. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"046808","DOI":"10.1103\/PhysRevLett.102.046808","article-title":"Finite Temperature Lattice Properties of Graphene beyond the Quasiharmonic Approximation","volume":"102","author":"Zakharchenko","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3012","DOI":"10.1021\/nl901448z","article-title":"Size and chirality dependent elastic properties of graphene nanoribbons under uniaxial tension","volume":"9","author":"Zhao","year":"2009","journal-title":"Nano Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"085432","DOI":"10.1103\/PhysRevB.82.085432","article-title":"Graphene nanoribbons subjected to axial stress","volume":"82","author":"Peeters","year":"2010","journal-title":"Phys. Rev. B"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"4445","DOI":"10.1016\/j.matdes.2010.04.016","article-title":"Temperature-dependent elastic properties of single layer graphene sheets","volume":"31","author":"Shen","year":"2010","journal-title":"Mater. Des."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1016\/j.matdes.2009.06.032","article-title":"Characterizing mechanical properties of graphite using molecular dynamics simulation","volume":"31","author":"Tsai","year":"2010","journal-title":"Mater. Des."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"064321","DOI":"10.1063\/1.3488620","article-title":"Temperature and strain-rate dependent fracture strength of graphene","volume":"108","author":"Zhao","year":"2010","journal-title":"J. Appl. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"184704","DOI":"10.1063\/1.3589163","article-title":"A simple molecular mechanics potential for \u00b5m scale graphene simulations from the adaptive force matching method","volume":"134","author":"Wei","year":"2011","journal-title":"J. Chem. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1016\/j.commatsci.2011.08.019","article-title":"Ultimate strength, ripples, sound velocities, and density of phonon states of strained graphene","volume":"53","author":"Dmitriev","year":"2012","journal-title":"Comput. Mater. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"134307","DOI":"10.1063\/1.4798384","article-title":"In-plane force fields and elastic properties of graphene","volume":"113","author":"Kalosakas","year":"2013","journal-title":"J. Appl. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"46001","DOI":"10.1209\/0295-5075\/103\/46001","article-title":"Effect of Stone-Thrower-Wales defect on structural stability of graphene at zero and finite temperatures","volume":"103","author":"Baimova","year":"2013","journal-title":"Europhys. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/j.commatsci.2013.01.032","article-title":"Mechanical properties of graphene: Effects of layer number, temperature and isotope","volume":"71","author":"Zhang","year":"2013","journal-title":"Comput. Mater. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"025033","DOI":"10.1088\/2053-1583\/3\/2\/025033","article-title":"Uniaxial compression of suspended single and multilayer graphenes","volume":"3","author":"Sgouros","year":"2016","journal-title":"2D Mater."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Sgouros, A.P., Kalosakas, G., Papagelis, K., and Galiotis, C. (2018). Compressive response and buckling of graphene nanoribbons. Sci. Rep., 8.","DOI":"10.1038\/s41598-018-27808-0"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1016\/j.physe.2018.11.025","article-title":"Elastic constants of graphene: Comparison of empirical potentials and DFT calculations","volume":"108","author":"Lebedeva","year":"2019","journal-title":"Phys. E Low-Dimens. Syst. Nanostruct."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"103123","DOI":"10.1016\/j.mechmat.2019.103123","article-title":"Improving bending rigidity of graphene nanoribbons by twisting","volume":"137","author":"Savin","year":"2019","journal-title":"Mech. Mater."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"045709","DOI":"10.1088\/1361-6528\/abc036","article-title":"Simulated mechanical properties of finite-size graphene nanoribbons","volume":"32","author":"Aparicio","year":"2021","journal-title":"Nanotechnology"},{"key":"ref_20","unstructured":"Eving\u00fcr, G.A., and Pekcan, \u00d6 (2021). Nanostructures failures and fully atomistic molecular dynamics simulations. Elasticity of Materials, IntechOpen. Chapter 2."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Kalosakas, G., Lathiotakis, N.N., and Papagelis, K. (2021). Width Dependent Elastic Properties of Graphene Nanoribbons. Materials, 14.","DOI":"10.3390\/ma14175042"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"7134","DOI":"10.1039\/D2CP00038E","article-title":"Inducing a topological transition in graphene nanoribbon superlattices by external strain","volume":"24","author":"Flores","year":"2022","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Fthenakis, Z.G., Petsalakis, I.D., Tozzini, V., and Lathiotakis, N.N. (2022). Evaluating the performance of ReaxFF potentials for sp2 carbon systems (graphene, carbon nanotubes, fullerenes) and a new ReaxFF potential. Front. Chem., 10.","DOI":"10.3389\/fchem.2022.951261"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1007\/s10999-022-09612-x","article-title":"Fracture strength of Graphene at high temperatures: Data driven investigations supported by MD and analytical approaches","volume":"18","author":"Verma","year":"2022","journal-title":"Int. J. Mech. Mater. Des."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"118416","DOI":"10.1016\/j.carbon.2023.118416","article-title":"Insights into the auxetic behavior of graphene: A study on the temperature dependence of Poisson\u2019s ratio and in-plane moduli","volume":"215","author":"Batra","year":"2023","journal-title":"Carbon"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"L023001","DOI":"10.1103\/PhysRevE.109.L023001","article-title":"Buckling kinetics of graphene membranes under uniaxial compression","volume":"109","author":"Sgouros","year":"2024","journal-title":"Phys. Rev. E"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"235406","DOI":"10.1103\/PhysRevB.64.235406","article-title":"C2F, BN, and C nanoshell elasticity from ab initio computations","volume":"64","author":"Kudin","year":"2001","journal-title":"Phys. Rev. B"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"064120","DOI":"10.1103\/PhysRevB.76.064120","article-title":"Ab initio calculation of ideal strength and phonon instability of graphene under tension","volume":"76","author":"Liu","year":"2007","journal-title":"Phys. Rev. B"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"054318","DOI":"10.1063\/1.3211944","article-title":"Linear correlation between binding energy and Young\u2019s modulus in graphene nanoribbons","volume":"106","author":"Christofides","year":"2009","journal-title":"J. Appl. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"245502","DOI":"10.1103\/PhysRevLett.105.245502","article-title":"Failure Mechanisms of Graphene under Tension","volume":"105","author":"Marianetti","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"134110","DOI":"10.1103\/PhysRevB.84.134110","article-title":"Ripple edge engineering of graphene nanoribbons","volume":"84","author":"Wagner","year":"2011","journal-title":"Phys. Rev. B"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"913","DOI":"10.1080\/01495739.2012.720214","article-title":"Effect of temperature on Young\u2019s modulus of graphene","volume":"35","author":"Mirnezhad","year":"2012","journal-title":"J. Therm. Stress."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"194901","DOI":"10.1063\/1.4766203","article-title":"Temperature dependent elastic constants and ultimate strength of graphene and graphyne","volume":"137","author":"Shao","year":"2012","journal-title":"J. Chem. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1142\/S1758825109000228","article-title":"Nonlinear mechanics of single-atomic-layer graphene sheets","volume":"1","author":"Lu","year":"2009","journal-title":"Int. J. Appl. Mech."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"175401","DOI":"10.1088\/1361-648X\/aa63d5","article-title":"Structural deformations of two-dimensional planar structures under uniaxial strain: The case of graphene","volume":"29","author":"Fthenakis","year":"2017","journal-title":"J. Phys. Condens. Matter"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Genoese, A., Genoese, A., Rizzi, N.L., and Salerno, G. (2019). Buckling Analysis of Single-Layer Graphene Sheets Using Molecular Mechanics. Front. Mater., 6.","DOI":"10.3389\/fmats.2019.00026"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"235502","DOI":"10.1103\/PhysRevLett.102.235502","article-title":"Nonlinear elasticity of monolayer graphene","volume":"102","author":"Cadelano","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.commatsci.2011.09.018","article-title":"Continuum-molecular modelling of graphene","volume":"53","author":"Samadikhah","year":"2012","journal-title":"Comput. Mater. Sci."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1021\/nl303168w","article-title":"Bending Rigidity and Gaussian Bending Stiffness of Single-Layered Graphene","volume":"13","author":"Wei","year":"2013","journal-title":"Nano Lett."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"109334","DOI":"10.1016\/j.diamond.2022.109334","article-title":"Temperature-dependent multiscale modeling of graphene sheet under finite deformation","volume":"129","author":"Tanhadoust","year":"2022","journal-title":"Diam. Relat. Mater."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1016\/j.apm.2023.12.021","article-title":"Influence of temperature on the elastic properties of graphene and graphene-like nanosheets based on the asymptotic homogenization method","volume":"127","author":"Tahani","year":"2024","journal-title":"Appl. Math. Model."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Kalosakas, G., Lathiotakis, N.N., and Papagelis, K. (2022). Uniaxially strained graphene: Structural characteristics and G-mode splitting. Materials, 15.","DOI":"10.3390\/ma15010067"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"345703","DOI":"10.1088\/1361-6528\/ab9042","article-title":"Thermal conductivity of graphene under biaxial strain: An analysis of spectral phonon properties","volume":"31","author":"Dheeraj","year":"2020","journal-title":"Nanotechnology"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"063150","DOI":"10.1063\/5.0007761","article-title":"Chaotic dynamics of graphene and graphene nanoribbons","volume":"30","author":"Hillebrand","year":"2020","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"858","DOI":"10.1038\/nmat2011","article-title":"Intrinsic ripples in graphene","volume":"6","author":"Fasolino","year":"2007","journal-title":"Nat. Mater."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1038\/nature05545","article-title":"The structure of suspended graphene sheets","volume":"446","author":"Meyer","year":"2007","journal-title":"Nature"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1186\/s11671-015-1135-5","article-title":"Effect of Intrinsic Ripples on Elasticity of the Graphene Monolayer","volume":"10","author":"Lee","year":"2015","journal-title":"Nanoscale Res. Lett."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.apnum.2013.01.003","article-title":"New families of symplectic splitting methods for numerical integration in dynamical astronomy","volume":"68","author":"Blanes","year":"2013","journal-title":"Appl. Numer. Math."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"447","DOI":"10.3934\/mine.2019.3.447","article-title":"Computational efficiency of numerical integration methods for the tangent dynamics of many-body Hamiltonian systems in one and two spatial dimensions","volume":"1","author":"Danieli","year":"2019","journal-title":"Math. Eng."}],"container-title":["Materials"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1996-1944\/18\/22\/5179\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,14]],"date-time":"2025-11-14T14:46:07Z","timestamp":1763131567000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1996-1944\/18\/22\/5179"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,14]]},"references-count":49,"journal-issue":{"issue":"22","published-online":{"date-parts":[[2025,11]]}},"alternative-id":["ma18225179"],"URL":"https:\/\/doi.org\/10.3390\/ma18225179","relation":{},"ISSN":["1996-1944"],"issn-type":[{"value":"1996-1944","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,14]]}}}