{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:32:41Z","timestamp":1760239961577,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2019,2,10]],"date-time":"2019-02-10T00:00:00Z","timestamp":1549756800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Chongqing Technology and Business University, China","award":["950117041"],"award-info":[{"award-number":["950117041"]}]},{"name":"National Key Research &amp; Development Program of China","award":["2016YFE0205600"],"award-info":[{"award-number":["2016YFE0205600"]}]},{"name":"MoST Science &amp; Technology Partnership Program","award":["KY201802006"],"award-info":[{"award-number":["KY201802006"]}]},{"name":"Program of Chongqing Municipal Education Commission, China","award":["KJZH17123"],"award-info":[{"award-number":["KJZH17123"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>To implement a sensor structure analysis and design (as well as other engineering applications), a two-parameter approach using elastic\u2013plastic fracture mechanics (EPFM) could be applied to analyze a structure more accurately than a one-parameter approach, especially for structures with low crack constraint. The application of the J-A two-parameter approach on sensors and other structures depends on the obtainment of a constraint parameter A. To conveniently and effectively obtain the A parameter values, the authors have developed a T-stress-based estimate method under a small-scale yielding (SSY) condition. Under a uniaxial external loading condition, a simplified format of the T-stress-based estimate has been proposed by the authors to obtain the parameter A much more conveniently and effectively. Generally, sensors and other practical engineering structures endure biaxial external loading instead of the uniaxial one. In the current work, the simplified formation of the estimate method is extended to a biaxial loading condition. By comparing the estimated A parameter values with their numerical solutions from a finite element analysis (FEA) results, the extension of the simplified formation of T-stress-based estimate method to biaxial loading was discussed and validated. The comparison procedure was completed using a wide variety of materials and geometrical properties on three types of specimens: single edge cracked plate (SECP), center cracked plate (CCP), and double edge cracked plate (DECP).<\/jats:p>","DOI":"10.3390\/s19030717","type":"journal-article","created":{"date-parts":[[2019,2,12]],"date-time":"2019-02-12T03:18:20Z","timestamp":1549941500000},"page":"717","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Simplified SSY Estimate Method to Determine EPFM Constraint Parameter for Sensor Design"],"prefix":"10.3390","volume":"19","author":[{"given":"Ping","family":"Ding","sequence":"first","affiliation":[{"name":"National Research Base of Intelligent Manufacturing Service, Chongqing Technology and Business University, Chongqing 400067, China"},{"name":"Department of Mechanical and Aerospace Engineering, Carleton University, Ottawa, ON K1S 5B6, Canada"}]},{"given":"Xin","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Aerospace Engineering, Carleton University, Ottawa, ON K1S 5B6, Canada"}]}],"member":"1968","published-online":{"date-parts":[[2019,2,10]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Li, C., S\u00e1nchez, R.V., Zurita, G., Cerrada, M., and Cabrera, D. (2016). Fault diagnosis for rotating machinery using vibration measurement deep statistical feature learning. Sensors, 16.","key":"ref_1","DOI":"10.3390\/s16060895"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3565","DOI":"10.3233\/JIFS-169534","article-title":"A comparison of fuzzy clustering algorithms for bearing fault diagnosis","volume":"34","author":"Li","year":"2018","journal-title":"J. Intell. Fuzzy Syst."},{"doi-asserted-by":"crossref","unstructured":"Li, C., Valente de Oliveira, J.L., Lozada, M.C., Cabrera, D., Sanchez, V., and Zurita, G. (2018). A systematic review of fuzzy formalisms for bearing fault diagnosis. IEEE Trans. Fuzzy Syst.","key":"ref_3","DOI":"10.1109\/TFUZZ.2018.2878200"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/0022-5096(68)90014-8","article-title":"Singular behavior at the end of a tensile crack in a hardening material","volume":"16","author":"Hutchinson","year":"1968","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0022-5096(68)90013-6","article-title":"Plane strain deformation near a crack tip in a power law hardening material","volume":"16","author":"Rice","year":"1968","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_6","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_7","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1115\/1.4011454","article-title":"On the stress distribution at the base of a stationary crack","volume":"24","author":"Williams","year":"1957","journal-title":"J. Appl. Mech."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1115\/1.2897135","article-title":"Two parameter characterization of elastic-plastic crack-tip fields","volume":"58","author":"Betegon","year":"1991","journal-title":"J. Appl. Mech."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/0022-5096(91)90029-N","article-title":"J-dominance of short cracks in tension and bending","volume":"39","author":"Hancock","year":"1991","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"989","DOI":"10.1016\/0022-5096(91)90049-T","article-title":"Family of crack-tip fields characterized by a triaxiality parameter\u2013I. Structure of fields","volume":"39","author":"Shih","year":"1991","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"939","DOI":"10.1016\/0022-5096(92)90057-9","article-title":"Family of crack-tip fields characterized by a triaxiality parameter\u2013II. Fracture applications","volume":"40","author":"Shih","year":"1992","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0013-7944(93)90002-A","article-title":"Higher-order asymptotic fields in a power-law hardening material","volume":"45","author":"Yang","year":"1993","journal-title":"Eng. Fract. Mech."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/0013-7944(94)00139-9","article-title":"An algorithm and a computer program for the three-term asymptotic expansion of elastic-plastic crack tip stress and displacement fields","volume":"50","author":"Nikishkov","year":"1995","journal-title":"Eng. Fract. Mech."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"685","DOI":"10.1016\/0013-7944(95)00024-P","article-title":"Calculation of the second fracture parameter for finite cracked bodies using a three-term elastic-plastic asymptotic expansion","volume":"52","author":"Nikishkov","year":"1995","journal-title":"Eng. Fract. Mech."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/BF00019710","article-title":"Some evaluation of the elastic T-term using Eshelby\u2019s method","volume":"30","author":"Kfouri","year":"1986","journal-title":"Int. J. Fract."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1339","DOI":"10.1016\/S0013-7944(01)00149-7","article-title":"Elastic T-stress for cracks in test specimens subjected to non-uniform stress distributions","volume":"69","author":"Wang","year":"2002","journal-title":"Eng. Fract. Mech."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1016\/j.engfracmech.2011.11.010","article-title":"An estimation method for the determination of the second elastic-plastic fracture mechanics parameters","volume":"79","author":"Ding","year":"2012","journal-title":"Eng. Fract. Mech."},{"doi-asserted-by":"crossref","unstructured":"Ding, P., and Wang, X. (2018, January 15\u201317). A Simplified Small Scale Yielding Estimate Method for Determination of Second Elastic-Plastic Fracture Mechanics Parameter. Proceedings of the 2018 International Conference on Sensing, Diagnostics, Prognostics, and Control, Xi\u2019an, China.","key":"ref_18","DOI":"10.1109\/SDPC.2018.8664755"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/j.ijpvp.2013.09.001","article-title":"Solutions of the second elastic-plastic fracture mechanics parameter in test specimens under biaxial loading","volume":"111\u2013112","author":"Ding","year":"2013","journal-title":"Int. J. Press. Vessels Pip."},{"unstructured":"(2016). ABAQUS Manual, Version 10.1, Hibbitt, Karlsson & Sorensen, Inc.","key":"ref_20"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1007\/BF00276359","article-title":"A general treatment of crack tip contour integrals","volume":"35","author":"Moran","year":"1987","journal-title":"Int. J. Fract."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/3\/717\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:30:59Z","timestamp":1760185859000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/3\/717"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,2,10]]},"references-count":21,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,2]]}},"alternative-id":["s19030717"],"URL":"https:\/\/doi.org\/10.3390\/s19030717","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2019,2,10]]}}}