{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,22]],"date-time":"2026-09-22T05:20:33Z","timestamp":1790054433398,"version":"4.0.1"},"reference-count":27,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,31]],"date-time":"2021-01-31T00:00:00Z","timestamp":1612051200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003093","name":"Ministry of Higher Education, Malaysia","doi-asserted-by":"publisher","award":["5540269"],"award-info":[{"award-number":["5540269"]}],"id":[{"id":"10.13039\/501100003093","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A new mathematical model is developed for the analytical study of two cracks in the upper plane of dissimilar materials under various mechanical loadings, i.e., shear, normal, tearing and mixed stresses with different geometry conditions. This problem is developed into a new mathematical model of hypersingular integral equations (HSIEs) by using the modified complex potentials (MCPs) function and the continuity conditions of the resultant force and displacement with the crack opening displacement (COD) function as the unknown. The newly obtained mathematical model of HSIEs are solved numerically by utilizing the appropriate quadrature formulas. Numerical computations and graphical demonstrations are carried out to observe the profound effect of the elastic constants ratio, mode of stresses and geometry conditions on the dimensionless stress intensity factors (SIFs) at the crack tips.<\/jats:p>","DOI":"10.3390\/sym13020235","type":"journal-article","created":{"date-parts":[[2021,1,31]],"date-time":"2021-01-31T04:53:00Z","timestamp":1612068780000},"page":"235","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Numerical Solution for Crack Phenomenon in Dissimilar Materials under Various Mechanical Loadings"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0332-7853","authenticated-orcid":false,"given":"Khairum Bin","family":"Hamzah","sequence":"first","affiliation":[{"name":"Fakulti Teknologi Kejuruteraan Mekanikal dan Pembuatan, Universiti Teknikal Malaysia Melaka, Hang Tuah Jaya, Durian Tunggal, Melaka 76100, Malaysia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9493-2631","authenticated-orcid":false,"given":"Nik Mohd Asri","family":"Nik Long","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, Universiti Putra Malaysia, Serdang, Selangor 43400, Malaysia"},{"name":"Laboratory of Computational Sciences and Mathematical Physics, Institute for Mathematical Research, Universiti Putra Malaysia, Serdang, Selangor 43400, Malaysia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8614-8281","authenticated-orcid":false,"given":"Norazak","family":"Senu","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, Universiti Putra Malaysia, Serdang, Selangor 43400, Malaysia"},{"name":"Laboratory of Computational Sciences and Mathematical Physics, Institute for Mathematical Research, Universiti Putra Malaysia, Serdang, Selangor 43400, Malaysia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6594-7102","authenticated-orcid":false,"given":"Zainidin K.","family":"Eshkuvatov","sequence":"additional","affiliation":[{"name":"Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, Kuala Terengganu 21030, Malaysia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,31]]},"reference":[{"key":"ref_1","unstructured":"Muskhelishvili, N.I. (1953). Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff International Publishing."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1135","DOI":"10.1002\/nme.1620290603","article-title":"Hypersingular integrals in boundary element fracture analysis","volume":"29","author":"Gray","year":"1990","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2611","DOI":"10.1016\/j.ijsolstr.2009.02.008","article-title":"Hypersingular integral equation for multiple curved cracks problem in plane elasticity","volume":"46","author":"Eshkuvatov","year":"2009","journal-title":"Int. J. Solids Struct."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/j.engfracmech.2019.01.025","article-title":"Weight Functions and Stress Intensity Factors for Two Unequal-Length Collinear Cracks in an Infinite Sheet","volume":"209","author":"Liu","year":"2019","journal-title":"Eng. Fract. Mech."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1016\/0013-7944(90)90117-Y","article-title":"The stress intensity factor and stress concentration for a finite plate with a single crack emanating from a hole","volume":"36","author":"Lai","year":"1990","journal-title":"Eng. Fract. Mech."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/j.crme.2019.01.004","article-title":"Automated numerical simulation of the propagation of multiple cracks in a finite plane using the distributed dislocation method","volume":"347","author":"Zhang","year":"2019","journal-title":"Comptes Rendus Mec."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1083","DOI":"10.1016\/j.enganabound.2004.02.010","article-title":"A boundary integral method for multiple circular inclusions in an elastic half-plane","volume":"28","author":"Legros","year":"2004","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/j.tafmec.2016.07.008","article-title":"Interaction between a screw dislocation and oblique edge crack in a half-infinite MEE solid","volume":"86","author":"Liu","year":"2016","journal-title":"Theor. Appl. Fract. Mech."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1007\/s10409-018-0803-y","article-title":"Numerical solutions for cracks in an elastic half-plane","volume":"35","author":"Elfakhakhre","year":"2019","journal-title":"Acta Mech. Sin."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0013-7944(86)90197-9","article-title":"Multiple crack problems for two bonded half planes in plane and antiplane elasticity","volume":"25","author":"Chen","year":"1986","journal-title":"Eng. Fract. Mech."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/0167-8442(92)90027-U","article-title":"Stress-intensity factors for curved circular crack in bonded dissimilar materials","volume":"17","author":"Chen","year":"1992","journal-title":"Theor. Appl. Fract. Mech."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1016\/0013-7944(93)90230-P","article-title":"Arbitrary array of cracks in bonded half planes subjected to various loadings","volume":"46","author":"Isida","year":"1993","journal-title":"Eng. Fract. Mech."},{"key":"ref_13","first-page":"147","article-title":"Integral method for contact problem of bonded plane material with arbitrary cracks","volume":"36","author":"Zhou","year":"2008","journal-title":"Comput. Model. Eng. Sci"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1650070","DOI":"10.1142\/S1758825116500708","article-title":"A combined boundary integral method for analysis of crack problems in multilayered elastic media","volume":"8","author":"Long","year":"2016","journal-title":"Int. J. Appl. Mech."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.apm.2019.04.002","article-title":"Stress intensity factor for multiple cracks in bonded dissimilar materials using hypersingular integral equations","volume":"73","author":"Hamzah","year":"2019","journal-title":"Appl. Math. Model."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1016\/j.apm.2019.07.063","article-title":"Stress intensity factor for bonded dissimilar materials weakened by multiple cracks","volume":"77","author":"Hamzah","year":"2020","journal-title":"Appl. Math. Model."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"179","DOI":"10.13189\/ujme.2019.070405","article-title":"Stress intensity factors for a crack in bonded dissimilar materials subjected to various stresses","volume":"7","author":"Hamzah","year":"2019","journal-title":"Univers. J. Mech. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2239","DOI":"10.1080\/14786430903032555","article-title":"Numerical solution for curved crack problem in elastic half-plane using hypersingular integral equation","volume":"89","author":"Chen","year":"2009","journal-title":"Philos. Mag."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/0377-0427(94)90287-9","article-title":"Numerical evaluation of hypersingular integrals","volume":"50","author":"Monegato","year":"1994","journal-title":"J. Comput. Appl. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1027","DOI":"10.1002\/nme.1620330509","article-title":"Derivation of a new analytical solution for a general two-dimensional finite-part integral applicable in fracture mechanics","volume":"33","author":"Mayrhofer","year":"1992","journal-title":"Int. J. Numer. Method Eng."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Mason, T.C., and Handscomb, D.C. (2003). Chebyshev Polynomials, Chapman and Hall\/CRC.","DOI":"10.1201\/9781420036114"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Kythe, P.K., and Schaferkotter, M.R. (2004). Handbook of Computational Methods for Integration, Chapman and Hall\/CRC.","DOI":"10.1201\/9780203490303"},{"key":"ref_23","first-page":"737","article-title":"Accurate critical stress intensity factor Griffith crack theory measurements by numerical techniques","volume":"2013","author":"Petersen","year":"2013","journal-title":"SAMPE J. Soc. Adv. Mater. Process. Eng."},{"key":"ref_24","first-page":"89","article-title":"Fundamentals of interface mechanics","volume":"8","author":"Wang","year":"2003","journal-title":"Ref. Modul. Mater. Sci. Mater. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2111","DOI":"10.1007\/s00707-013-1080-2","article-title":"Thermoelastic interaction of two offset interfacial cracks in bonded dissimilar half-planes with a functionally graded interlayer","volume":"225","author":"Choi","year":"2014","journal-title":"Acta Mech."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1007\/s004190050129","article-title":"Fredholm integral equation for the multiple circular arc crack problem in plane elasticity","volume":"67","author":"Chen","year":"1997","journal-title":"Arch. Appl. Mech."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4019","DOI":"10.1016\/S0045-7825(00)00374-1","article-title":"Numerical solution of three-dimensional crack problem by using hypersingular integral equation","volume":"190","author":"Chen","year":"2001","journal-title":"Comput. Methods Appl. Mech. Eng."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/2\/235\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:17:47Z","timestamp":1760159867000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/2\/235"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,31]]},"references-count":27,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["sym13020235"],"URL":"https:\/\/doi.org\/10.3390\/sym13020235","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,31]]}}}