{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T18:17:12Z","timestamp":1774030632095,"version":"3.50.1"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2021,11,8]],"date-time":"2021-11-08T00:00:00Z","timestamp":1636329600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2021,11,8]],"date-time":"2021-11-08T00:00:00Z","timestamp":1636329600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"national natural science foundation of china","doi-asserted-by":"publisher","award":["51839009"],"award-info":[{"award-number":["51839009"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"national natural science foundation of china","doi-asserted-by":"publisher","award":["51679017"],"award-info":[{"award-number":["51679017"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012231","name":"graduate scientific research and innovation foundation of chongqing,china","doi-asserted-by":"crossref","award":["CYB20033"],"award-info":[{"award-number":["CYB20033"]}],"id":[{"id":"10.13039\/501100012231","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Engineering with Computers"],"published-print":{"date-parts":[[2023,4]]},"DOI":"10.1007\/s00366-021-01527-z","type":"journal-article","created":{"date-parts":[[2021,11,8]],"date-time":"2021-11-08T19:02:43Z","timestamp":1636398163000},"page":"1395-1414","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":43,"title":["Drucker-Prager plasticity model in the framework of OSB-PD theory with shear deformation"],"prefix":"10.1007","volume":"39","author":[{"given":"Ting","family":"Zhang","sequence":"first","affiliation":[]},{"given":"Xiao-Ping","family":"Zhou","sequence":"additional","affiliation":[]},{"given":"Qi-Hu","family":"Qian","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,11,8]]},"reference":[{"key":"1527_CR1","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1016\/j.jmps.2019.06.006","volume":"130","author":"LW Zhang","year":"2019","unstructured":"Zhang LW, Xie Y, Lyu D, Li S (2019) Multiscale modeling of dislocation patterns and simulation of nanoscale plasticity in body-centered cubic (BCC) single crystals. J Mech Phys Solids 130:297\u2013319. https:\/\/doi.org\/10.1016\/j.jmps.2019.06.006","journal-title":"J Mech Phys Solids"},{"key":"1527_CR2","doi-asserted-by":"publisher","first-page":"1091","DOI":"10.1016\/j.jrmge.2018.09.002","volume":"10","author":"S Oberhollenzer","year":"2018","unstructured":"Oberhollenzer S, Tschuchnigg F, Schweiger HF (2018) Finite element analyses of slope stability problems using non-associated plasticity. J Rock Mech Geotech Eng 10:1091\u20131101. https:\/\/doi.org\/10.1016\/j.jrmge.2018.09.002","journal-title":"J Rock Mech Geotech Eng"},{"key":"1527_CR3","doi-asserted-by":"publisher","first-page":"707","DOI":"10.1680\/geot.8.P.155","volume":"62","author":"MF Randolph","year":"2012","unstructured":"Randolph MF, Goh SH, Lee FH, Yi JT (2012) A numerical study of cone penetration in fine-grained soils allowing for consolidation effects. G\u00e9otechnique 62:707\u2013719","journal-title":"G\u00e9otechnique"},{"key":"1527_CR4","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1016\/j.tafmec.2019.02.019","volume":"101","author":"M Liao","year":"2019","unstructured":"Liao M, Zhang P (2019) An improved approach for computation of stress intensity factors using the finite element method. Theor Appl Fract Mech 101:185\u2013190. https:\/\/doi.org\/10.1016\/j.tafmec.2019.02.019","journal-title":"Theor Appl Fract Mech"},{"key":"1527_CR5","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1002\/(SICI)1097-0207(19990910)46:13.0.CO;2-J","volume":"46","author":"N Mo\u00ebs","year":"1999","unstructured":"Mo\u00ebs N, JohnBelytschko DT (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131\u2013150. https:\/\/doi.org\/10.1002\/(SICI)1097-0207(19990910)46:13.0.CO;2-J","journal-title":"Int J Numer Methods Eng"},{"key":"1527_CR6","doi-asserted-by":"publisher","first-page":"1051","DOI":"10.1016\/j.cma.2018.08.007","volume":"346","author":"K Agathos","year":"2018","unstructured":"Agathos K, Bordas SPA, Chatzi E (2018) Improving the conditioning of XFEM\/GFEM for fracture mechanics problems through enrichment quasi-orthogonalization. Comput Methods Appl Mech Eng 346:1051\u20131073","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1527_CR7","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1111\/ffe.12918","volume":"42","author":"JW Chen","year":"2019","unstructured":"Chen JW, Zhou XP, Berto F (2019) The improvement of crack propagation modelling in triangular 2D structures using the extended finite element method. Fatigue Fract Eng Mater Struct 42:397\u2013414. https:\/\/doi.org\/10.1111\/ffe.12918","journal-title":"Fatigue Fract Eng Mater Struct"},{"key":"1527_CR8","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1016\/j.enggeo.2019.01.012","volume":"250","author":"X Zhou","year":"2019","unstructured":"Zhou X, Chen J (2019) Extended finite element simulation of step-path brittle failure in rock slopes with non-persistent en-echelon joints. Eng Geol 250:65\u201388","journal-title":"Eng Geol"},{"key":"1527_CR9","doi-asserted-by":"publisher","first-page":"534","DOI":"10.1007\/BF02472449","volume":"25","author":"E Schlangen","year":"1992","unstructured":"Schlangen E, Van MJGM (1992) Simple lattice model for numerical simulation of fracture of concrete materials and structures. Mater Struct 25:534\u2013542","journal-title":"Mater Struct"},{"key":"1527_CR10","doi-asserted-by":"publisher","first-page":"1755","DOI":"10.1142\/S0129183106010182","volume":"17","author":"K Kadau","year":"2011","unstructured":"Kadau K, Germann TC, Lomdahl PS (2011) Molecular dynamics comes of age: 320 billion atom simulation on BlueGene\/L. Int J Mod Phys C 17:1755\u20131761","journal-title":"Int J Mod Phys C"},{"key":"1527_CR11","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1016\/S0022-5096(99)00029-0","volume":"48","author":"SA Silling","year":"2000","unstructured":"Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48:175\u2013209. https:\/\/doi.org\/10.1016\/S0022-5096(99)00029-0","journal-title":"J Mech Phys Solids"},{"key":"1527_CR12","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1007\/s10659-007-9125-1","volume":"88","author":"SA Silling","year":"2007","unstructured":"Silling SA, Epton M, Weckner O et al (2007) Peridynamic states and constitutive modeling. J Elast 88:151\u2013184. https:\/\/doi.org\/10.1007\/s10659-007-9125-1","journal-title":"J Elast"},{"key":"1527_CR13","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1615\/IntJMultCompEng.2014011338","volume":"13","author":"P Seleson","year":"2015","unstructured":"Seleson P, Ha YD, Beneddine S (2015) Concurrent coupling of bond-based peridynamics and the navier equation of classical elasticity by blending. Int J Multiscale Comput Eng 13:91\u2013113","journal-title":"Int J Multiscale Comput Eng"},{"key":"1527_CR14","doi-asserted-by":"publisher","first-page":"334","DOI":"10.1016\/j.jmps.2018.03.028","volume":"116","author":"L Wang","year":"2018","unstructured":"Wang L, Abeyaratne R (2018) A one-dimensional peridynamic model of defect propagation and its relation to certain other continuum models. J Mech Phys Solids 116:334\u2013349","journal-title":"J Mech Phys Solids"},{"key":"1527_CR15","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1016\/j.ijsolstr.2017.10.022","volume":"134","author":"Y Wang","year":"2018","unstructured":"Wang Y, Zhou X, Wang Y, Shou Y (2018) A 3-D conjugated bond-pair-based peridynamic formulation for initiation and propagation of cracks in brittle solids. Int J Solids Struct 134:89\u2013115. https:\/\/doi.org\/10.1016\/j.ijsolstr.2017.10.022","journal-title":"Int J Solids Struct"},{"key":"1527_CR16","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1016\/j.cma.2013.05.001","volume":"263","author":"R Beckmann","year":"2013","unstructured":"Beckmann R, Mella R, Wenman MR (2013) Mesh and timestep sensitivity of fracture from thermal strains using peridynamics implemented in Abaqus. Comput Methods Appl Mech Eng 263:71\u201380","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1527_CR17","doi-asserted-by":"publisher","first-page":"499","DOI":"10.1007\/s00466-017-1469-1","volume":"61","author":"QV Le","year":"2018","unstructured":"Le QV, Bobaru F (2018) Surface corrections for peridynamic models in elasticity and fracture. Comput Mech 61:499\u2013518. https:\/\/doi.org\/10.1007\/s00466-017-1469-1","journal-title":"Comput Mech"},{"key":"1527_CR18","doi-asserted-by":"publisher","first-page":"130","DOI":"10.1016\/j.ijimpeng.2017.08.008","volume":"111","author":"X Lai","year":"2018","unstructured":"Lai X, Liu L, Li S et al (2018) A non-ordinary state-based peridynamics modeling of fractures in quasi-brittle materials. Int J Impact Eng 111:130\u2013146","journal-title":"Int J Impact Eng"},{"key":"1527_CR19","doi-asserted-by":"publisher","first-page":"238","DOI":"10.1016\/j.engfracmech.2016.11.004","volume":"169","author":"A Yaghoobi","year":"2017","unstructured":"Yaghoobi A, Chorzepa MG (2017) Fracture analysis of fiber reinforced concrete structures in the micropolar peridynamic analysis framework. Eng Fract Mech 169:238\u2013250. https:\/\/doi.org\/10.1016\/j.engfracmech.2016.11.004","journal-title":"Eng Fract Mech"},{"key":"1527_CR20","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1016\/j.cma.2017.03.043","volume":"322","author":"SA Silling","year":"2017","unstructured":"Silling SA (2017) Stability of peridynamic correspondence material models and their particle discretizations. Comput Methods Appl Mech Eng 322:42\u201357","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1527_CR21","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1007\/s10659-018-09723-7","volume":"137","author":"LJ Wang","year":"2019","unstructured":"Wang LJ, Xu JF, Wang JX (2019) Elastodynamics of linearized isotropic state-based peridynamic media. J Elast 137:157\u2013176","journal-title":"J Elast"},{"key":"1527_CR22","doi-asserted-by":"publisher","first-page":"526","DOI":"10.1680\/jgeot.17.P.274","volume":"69","author":"F Zhu","year":"2019","unstructured":"Zhu F, Zhao J (2019) A peridynamic investigation on crushing of sand particles. Geotechnique 69:526\u2013540. https:\/\/doi.org\/10.1680\/jgeot.17.P.274","journal-title":"Geotechnique"},{"key":"1527_CR23","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2020.113098","volume":"366","author":"S Liu","year":"2020","unstructured":"Liu S, Fang G, Liang J et al (2020) A new type of peridynamics: element-based peridynamics. Comput Methods Appl Mech Eng 366:113098. https:\/\/doi.org\/10.1016\/j.cma.2020.113098","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1527_CR24","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1016\/j.mechrescom.2019.01.005","volume":"95","author":"G Fang","year":"2019","unstructured":"Fang G, Liu S, Fu M et al (2019) A method to couple state-based peridynamics and finite element method for crack propagation problem. Mech Res Commun 95:89\u201395","journal-title":"Mech Res Commun"},{"key":"1527_CR25","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechmat.2019.103133","volume":"137","author":"Y Wang","year":"2019","unstructured":"Wang Y, Zhou X, Zhang T (2019) Size effect of thermal shock crack patterns in ceramics: Insights from a nonlocal numerical approach. Mech Mater 137:103133. https:\/\/doi.org\/10.1016\/j.mechmat.2019.103133","journal-title":"Mech Mater"},{"key":"1527_CR26","doi-asserted-by":"publisher","first-page":"118","DOI":"10.1016\/j.ijengsci.2017.09.004","volume":"121","author":"QZ Zhu","year":"2017","unstructured":"Zhu QZ, Ni T (2017) Peridynamic formulations enriched with bond rotation effects. Int J Eng Sci 121:118\u2013129. https:\/\/doi.org\/10.1016\/j.ijengsci.2017.09.004","journal-title":"Int J Eng Sci"},{"key":"1527_CR27","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1016\/j.engfracmech.2018.08.028","volume":"204","author":"Y Wang","year":"2018","unstructured":"Wang Y, Zhou X, Kou M (2018) Numerical studies on thermal shock crack branching instability in brittle solids. Eng Fract Mech 204:157\u2013184. https:\/\/doi.org\/10.1016\/j.engfracmech.2018.08.028","journal-title":"Eng Fract Mech"},{"key":"1527_CR28","doi-asserted-by":"publisher","DOI":"10.2172\/1018475","author":"JA Mitchell","year":"2011","unstructured":"Mitchell JA (2011) A nonlocal, ordinary, state-based plasticity model for peridynamics. United States. https:\/\/doi.org\/10.2172\/1018475","journal-title":"United States"},{"key":"1527_CR29","doi-asserted-by":"crossref","unstructured":"Lammi CJ, Vogler TJ (2014) A nonlocal peridynamic plasticity model for the dynamic flow and fracture of concrete. Sandia National Lab.(SNL-CA), Livermore, CA. United States.","DOI":"10.2172\/1159446"},{"key":"1527_CR30","doi-asserted-by":"publisher","first-page":"192","DOI":"10.1016\/j.jmps.2015.09.016","volume":"86","author":"E Madenci","year":"2016","unstructured":"Madenci E, Oterkus S (2016) Ordinary state-based peridynamics for plastic deformation according to von Mises yield criteria with isotropic hardening. J Mech Phys Solids 86:192\u2013219. https:\/\/doi.org\/10.1016\/j.jmps.2015.09.016","journal-title":"J Mech Phys Solids"},{"key":"1527_CR31","doi-asserted-by":"publisher","first-page":"182","DOI":"10.1016\/j.ijmecsci.2019.03.033","volume":"156","author":"H Pashazad","year":"2019","unstructured":"Pashazad H, Kharazi M (2019) A peridynamic plastic model based on von Mises criteria with isotropic, kinematic and mixed hardenings under cyclic loading. Int J Mech Sci 156:182\u2013204. https:\/\/doi.org\/10.1016\/j.ijmecsci.2019.03.033","journal-title":"Int J Mech Sci"},{"key":"1527_CR32","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2019.106782","volume":"223","author":"ZM Liu","year":"2020","unstructured":"Liu ZM, Bie YH, Cui ZQ, Cui XY (2020) Ordinary state-based peridynamics for nonlinear hardening plastic materials\u2019 deformation and its fracture process. Eng Fract Mech 223:106782. https:\/\/doi.org\/10.1016\/j.engfracmech.2019.106782","journal-title":"Eng Fract Mech"},{"key":"1527_CR33","doi-asserted-by":"publisher","first-page":"1236","DOI":"10.1002\/zamm.201600242","volume":"97","author":"E Madenci","year":"2017","unstructured":"Madenci E (2017) Peridynamic integrals for strain invariants of homogeneous deformation. ZAMM-Zeitschrift fur Angew Math und Mech 97:1236\u20131251. https:\/\/doi.org\/10.1002\/zamm.201600242","journal-title":"ZAMM-Zeitschrift fur Angew Math und Mech"},{"key":"1527_CR34","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-8465-3","volume-title":"Peridynamic theory and its applications","author":"E Madenci","year":"2014","unstructured":"Madenci E, Oterkus E (2014) Peridynamic theory and its applications. Springer, Berlin"},{"key":"1527_CR35","doi-asserted-by":"publisher","first-page":"1650009","DOI":"10.1142\/s2424913016500090","volume":"01","author":"H Ren","year":"2016","unstructured":"Ren H, Zhuang X, Rabczuk T (2016) A new peridynamic formulation with shear deformation for elastic solid. J Micromech Mol Phys 01:1650009. https:\/\/doi.org\/10.1142\/s2424913016500090","journal-title":"J Micromech Mol Phys"},{"key":"1527_CR36","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1115\/1.4011929","volume":"26","author":"DC Drucker","year":"1959","unstructured":"Drucker DC (1959) A definition of stable inelastic material. Trans ASME J Appl Mech 26:101\u2013106","journal-title":"Trans ASME J Appl Mech"},{"key":"1527_CR37","doi-asserted-by":"publisher","first-page":"675","DOI":"10.1615\/intjmultcompeng.2011002407","volume":"9","author":"J Foster","year":"2011","unstructured":"Foster J, Silling SA, Chen WN (2011) An energy based failure criterion for use with peridynamic states. Int J Multiscale Comput Eng 9:675\u2013688. https:\/\/doi.org\/10.1615\/intjmultcompeng.2011002407","journal-title":"Int J Multiscale Comput Eng"},{"key":"1527_CR38","doi-asserted-by":"publisher","DOI":"10.1155\/2013\/631074","volume":"2013","author":"F Shen","year":"2013","unstructured":"Shen F, Zhang Q, Huang D (2013) Damage and failure process of concrete structure under uniaxial compression based on Peridynamics modeling. Math Probl Eng 2013:631074. https:\/\/doi.org\/10.1155\/2013\/631074","journal-title":"Math Probl Eng"},{"key":"1527_CR39","doi-asserted-by":"publisher","first-page":"517","DOI":"10.1016\/j.cma.2018.07.009","volume":"341","author":"Y Zhang","year":"2018","unstructured":"Zhang Y, Qiao P (2018) An axisymmetric ordinary state-based peridynamic model for linear elastic solids. Comput Methods Appl Mech Eng 341:517\u2013550. https:\/\/doi.org\/10.1016\/j.cma.2018.07.009","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1527_CR40","doi-asserted-by":"publisher","DOI":"10.1016\/j.euromechsol.2019.103810","volume":"77","author":"T Zhang","year":"2019","unstructured":"Zhang T, Zhou XP (2019) A modified axisymmetric ordinary state-based peridynamics with shear deformation for elastic and fracture problems in brittle solids. Eur J Mech A\/Solids 77:103810. https:\/\/doi.org\/10.1016\/j.euromechsol.2019.103810","journal-title":"Eur J Mech A\/Solids"},{"key":"1527_CR41","doi-asserted-by":"publisher","first-page":"194","DOI":"10.1016\/j.tafmec.2010.08.001","volume":"53","author":"B Kilic","year":"2010","unstructured":"Kilic B, Madenci E (2010) An adaptive dynamic relaxation method for quasi-static simulation using the peridynamic theory. Theor Appl Fract Mech 53:194\u2013204","journal-title":"Theor Appl Fract Mech"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-021-01527-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-021-01527-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-021-01527-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,28]],"date-time":"2023-04-28T21:09:36Z","timestamp":1682716176000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-021-01527-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,8]]},"references-count":41,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,4]]}},"alternative-id":["1527"],"URL":"https:\/\/doi.org\/10.1007\/s00366-021-01527-z","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,11,8]]},"assertion":[{"value":"7 July 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 October 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 November 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}