{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,29]],"date-time":"2026-10-29T10:09:59Z","timestamp":1793268599028,"version":"4.5.0"},"reference-count":41,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,1,7]],"date-time":"2019-01-07T00:00:00Z","timestamp":1546819200000},"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>The cohesive zone model (CZM) has found wide acceptance as a tool for the simulation of delamination in composites and debonding in bonded joints and various implementations of the cohesive zone model dedicated to fatigue problems have been proposed in the past decade. In previous works, the authors have developed a model based on cohesive zone to simulate the propagation of fatigue defects where damage acts on cohesive stiffness, with an initial (undamaged) stiffness representative of that of the entire thickness of an adhesive layer. In the case of a stiffness that is order of magnitude higher than the previous one (for instance, in the simulation of the ply-to-ply interface in composites), the model prediction becomes inaccurate. In this work, a new formulation of the model that overcomes this limitation is developed. Finite element simulations have been conducted on a mode I, constant bending (constant G)-loaded double cantilever beam (DCB) joint to assess the response of the new model with respect to the original one for varying initial stiffness K0 and cohesive strength \u03c30. The results showed that the modified model is robust with respect to changes of two orders of magnitude in initial stiffness and of a factor of two in \u03c30.<\/jats:p>","DOI":"10.3390\/ma12010181","type":"journal-article","created":{"date-parts":[[2019,1,9]],"date-time":"2019-01-09T03:06:06Z","timestamp":1547003166000},"page":"181","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":32,"title":["Improvement of a Cohesive Zone Model for Fatigue Delamination Rate Simulation"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0533-1805","authenticated-orcid":false,"given":"Alessandro","family":"Pirondi","sequence":"first","affiliation":[{"name":"Dipartimento di Ingegneria e Architettura, Universit\u00e0 di Parma, Parco Area delle Scienze 181\/A, 43124 Parma, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4661-3799","authenticated-orcid":false,"given":"Fabrizio","family":"Moroni","sequence":"additional","affiliation":[{"name":"Dipartimento di Ingegneria e Architettura, Universit\u00e0 di Parma, Parco Area delle Scienze 181\/A, 43124 Parma, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2019,1,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/S0065-2156(08)70121-2","article-title":"The Mathematical Theory of Equilibrium Cracks in Brittle Fracture","volume":"7","author":"Barenlbatt","year":"1962","journal-title":"Adv. Appl. Mech."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00466-014-1079-0","article-title":"Progressive delamination analysis of composite materials using XFEM and a discrete damage zone model","volume":"55","author":"Wang","year":"2015","journal-title":"Comput. Mech."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.engfracmech.2013.10.003","article-title":"Methods for the prediction of fatigue delamination growth in composites and adhesive bonds\u2014A critical review","volume":"112","author":"Pascoe","year":"2013","journal-title":"Eng. Fract. Mech."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"060803","DOI":"10.1115\/1.4027647","article-title":"Delamination Under Fatigue Loads in Composite Laminates: Review on the Observed Phenomenology and Computational Methods","volume":"66","author":"Bak","year":"2014","journal-title":"ASME Appl. Mech. Rev."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2231","DOI":"10.1016\/S0020-7683(98)00059-6","article-title":"Elastoplastic finite element analysis of three-dimensional fatigue crack growth in aluminum shafts subjected to axial loading","volume":"36","author":"Ortiz","year":"1999","journal-title":"Int. J. Solids Struct."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3927","DOI":"10.1016\/S0020-7683(00)00253-5","article-title":"A cohesive zone model for fatigue crack growth in quasibrittle materials","volume":"38","author":"Yang","year":"2001","journal-title":"Int. J. Solids Struct."},{"key":"ref_7","first-page":"351","article-title":"A cohesive model of fatigue crack growth","volume":"110","author":"Nguyen","year":"2001","journal-title":"Int. J. Struct."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"868","DOI":"10.1016\/j.ijfatigue.2008.11.002","article-title":"A cohesive zone model for fatigue and creep\u2013fatigue crack growth in single crystal superalloys","volume":"31","author":"Bouvard","year":"2009","journal-title":"Int. J. Fat"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2453","DOI":"10.1016\/j.ijsolstr.2009.01.031","article-title":"Cohesive zone model for fatigue crack growth allowing for crack retardation","volume":"46","author":"Ural","year":"2009","journal-title":"Int. J. Solids Struct."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/j.engfracmech.2008.08.011","article-title":"Computational analysis of mixed-mode fatigue crack growth in quasi-brittle materials using extended finite element methods","volume":"76","author":"Xu","year":"2009","journal-title":"Eng. Fract. Mech."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1016\/j.commatsci.2009.04.029","article-title":"On damage accumulations in the cyclic cohesive zone model for xfem analysis of mixed-mode fatigue crack growth","volume":"46","author":"Xu","year":"2009","journal-title":"Comput. Mater. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2164","DOI":"10.1016\/j.proeng.2011.04.358","article-title":"Novel approach for the treatment of cyclic loading using a potential-based cohesive zone model","volume":"10","author":"Scheider","year":"2011","journal-title":"Procedia Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2399","DOI":"10.1016\/j.engfracmech.2011.05.011","article-title":"Modeling of the variability of fatigue crack growth using cohesive zone elements","volume":"78","author":"Beaurepaire","year":"2011","journal-title":"Eng. Fract. Mech."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1246","DOI":"10.1111\/ffe.12061","article-title":"Cohesive zone modelling of low cycle fatigue cracks in cracked and notched specimens","volume":"36","author":"I","year":"2013","journal-title":"Fatigue Fract. Eng. Mater. Struct."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/s10704-014-9942-8","article-title":"Simulation of fatigue crack growth with a cyclic cohesive zone model","volume":"188","author":"Roth","year":"2014","journal-title":"Int. J. Fract."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.ijfatigue.2017.01.044","article-title":"Prediction of size-dependent fatigue failure modes by means of a cyclic cohesive zone model","volume":"100","author":"Roth","year":"2017","journal-title":"Int. J. Fatigue"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1016\/j.engfracmech.2004.06.005","article-title":"A cohesive model for fatigue failure of polymers","volume":"72","author":"Maiti","year":"2005","journal-title":"Eng. Fract. Mech."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/S0013-7944(02)00034-6","article-title":"An irreversible cohesive zone model for interface fatigue crack growth simulation","volume":"70","author":"Roe","year":"2003","journal-title":"Eng. Fract. Mech."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1824","DOI":"10.1002\/nme.1338","article-title":"Numerical simulation of fatigue-driven delamination using interface elements","volume":"63","author":"Robinson","year":"2005","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1136","DOI":"10.1016\/j.ijfatigue.2006.02.003","article-title":"On the numerical simulation of fatigue driven delamination with interface element","volume":"28","author":"Galvanetto","year":"2006","journal-title":"Int. J. Fatigue"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2270","DOI":"10.1016\/j.compositesa.2006.11.009","article-title":"Simulation of delamination in composites under high-cycle fatigue","volume":"38","author":"Turon","year":"2007","journal-title":"Composites A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2415","DOI":"10.1177\/0021998307075439","article-title":"Simulation of fatigue delamination growth in composites with different mode mixtures","volume":"41","author":"Tumino","year":"2007","journal-title":"J. Compos. Mater."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1016\/j.ijadhadh.2010.05.003","article-title":"A generalized damage model for constant amplitude fatigue loading of adhesively bonded joints","volume":"30","author":"Khoramishad","year":"2010","journal-title":"Int. J. Adhes. Adhes."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1146","DOI":"10.1016\/j.ijfatigue.2009.12.013","article-title":"Predicting fatigue damage in adhesively bonded joints using a cohesive zone model","volume":"32","author":"Khoramishad","year":"2010","journal-title":"Int. J. Fatigue"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1774","DOI":"10.1016\/j.ijfatigue.2010.04.006","article-title":"A fatigue degradation law for cohesive interface elements\u2014Development and application to composite materials","volume":"32","author":"Harper","year":"2010","journal-title":"Int. J. Fatigue"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1787","DOI":"10.1016\/j.compositesa.2010.08.015","article-title":"A combined model for initiation and propagation of damage under fatigue loading for cohesive interface elements","volume":"41","author":"May","year":"2010","journal-title":"Composites A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1080\/00218464.2010.484305","article-title":"A Progressive Damage Model for the Prediction of Fatigue Crack Growth in Bonded Joints","volume":"86","author":"Pirondi","year":"2010","journal-title":"J. Adhes."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2340","DOI":"10.1016\/j.compstruct.2011.03.022","article-title":"An advanced model for initiation and propagation of damage under fatigue loading\u2014Part I: Model formulation","volume":"93","author":"May","year":"2011","journal-title":"Compos. Struct."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2350","DOI":"10.1016\/j.compstruct.2011.03.023","article-title":"An advanced model for initiation and propagation of damage under fatigue loading\u2014Part ii: Matrix cracking validation cases","volume":"93","author":"May","year":"2011","journal-title":"Compos. Struct."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1070","DOI":"10.1016\/j.ijsolstr.2010.12.012","article-title":"Simulation and experimental validation of mixed mode delamination in multidirectional CF\/PEEK laminates under fatigue loading","volume":"48","author":"Naghipour","year":"2011","journal-title":"Int. J. Solids Struct."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2483","DOI":"10.1163\/016942411X580180","article-title":"Simulation of mixed-mode I\/II fatigue crack propagation in adhesive joints with a modified cohesive zone model","volume":"25","author":"Pirondi","year":"2011","journal-title":"J. Adhes. Sci. Technol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1016\/j.compositesb.2011.08.020","article-title":"Modeling delamination growth in composites under fatigue loadings of varying amplitudes","volume":"43","author":"Landry","year":"2012","journal-title":"Composites B"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2898","DOI":"10.1016\/j.ijsolstr.2012.03.034","article-title":"A crack tip tracking algorithm for cohesive interface element analysis of fatigue delamination propagation in composite materials","volume":"49","author":"Kawashita","year":"2012","journal-title":"Int. J. Solids Struct."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1123","DOI":"10.1016\/j.ijsolstr.2013.12.009","article-title":"Cohesive zone model for high-cycle fatigue of adhesively bonded joints under mode I loading","volume":"51","year":"2014","journal-title":"Int. J. Solids Struct."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.engfracmech.2015.03.044","article-title":"Cohesive zone model for high-cycle fatigue of composite bonded joints under mixed-mode I+II loading","volume":"140","year":"2015","journal-title":"Eng. Fract. Mech."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1080\/00218464.2015.1127764","article-title":"Fatigue Debonding Three-dimensional Simulation with Cohesive Zone","volume":"92","author":"Pirondi","year":"2016","journal-title":"J. Adhes."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1002\/nme.5117","article-title":"A simulation method for high-cycle fatigue-driven delamination using a cohesive zone model","volume":"106","author":"Bak","year":"2016","journal-title":"Int. J. Numer. Meth. Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/s10704-015-0053-y","article-title":"General remarks on cyclic cohesive zone models","volume":"196","author":"Roth","year":"2015","journal-title":"Int. J. Fract."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Lemaitre, J. (1996). A Course on Damage Mechanics, Springer.","DOI":"10.1007\/978-3-642-18255-6"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/s10704-012-9734-y","article-title":"Artificial compliance inherent to the intrinsic cohesive zone models: Criteria and application to planar meshes","volume":"178","author":"Blal","year":"2012","journal-title":"Int. J. Fract."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1016\/j.compstruct.2016.11.081","article-title":"A benchmark study of simulation methods for high-cycle fatigue-driven delamination based on cohesive zone models","volume":"164","author":"Bak","year":"2017","journal-title":"Compos. Struct."}],"container-title":["Materials"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1996-1944\/12\/1\/181\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:24:05Z","timestamp":1760185445000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1996-1944\/12\/1\/181"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1,7]]},"references-count":41,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2019,1]]}},"alternative-id":["ma12010181"],"URL":"https:\/\/doi.org\/10.3390\/ma12010181","relation":{},"ISSN":["1996-1944"],"issn-type":[{"value":"1996-1944","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1,7]]}}}