Qualitative and Numerical Analysis of Quasi-Static Problems in Elastoplasticity

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dc.contributor.authorHan, Weimin
dc.contributor.authorReddy, B Daya
dc.contributor.authorSchroeder, Gregory C
dc.date.accessioned2021-10-08T07:16:04Z
dc.date.available2021-10-08T07:16:04Z
dc.date.issued1997
dc.description.abstractThe quasi-static problem of elastoplasticity with combined kinematic-isotropic hardening is formulated as a time-dependent variational inequality (VI) of the mixed kind; that is, it is an inequality involving a nondifferentiable functional and is imposed on a subset of a space. This VI differs from the standard parabolic VI in that time derivatives of the unknown variable occur in all of its terms. The problem is shown to possess a unique solution. We consider two types of approximations to the VI corresponding to the quasi-static problem of elastoplasticity: semidiscrete approximations, in which only the spatial domain is discretized, by finite elements; and fully discrete approximations, in which the spatial domain is again discretized by finite elements, and the temporal domain is discretized and the time-derivative appearing in the VI is approximated in an appropriate way. Estimates of the errors inherent in the above two types of approximations, in suitable Sobolev norms, are obtained for the quasi-static problem of elastoplasticity; in particular, these estimates express rates of convergence of successive finite element approximations to the solution of the variational inequality in terms of element size h and, where appropriate, of the time step size k. A major difficulty in solving the problems is caused by the presence of the nondifferentiable terms. We consider some regularization techniques for overcoming the difficulty. Besides the usual convergence estimates, we also provide a posteriori error estimates which enable us to estimate the error by using only the solution of a regularized problem.
dc.identifier.apacitationHan, W., Reddy, B. D., & Schroeder, G. C. (1997). Qualitative and Numerical Analysis of Quasi-Static Problems in Elastoplasticity. <i>SIAM Journal on Numerical Analysis</i>, 34(1), 143 - 177. http://hdl.handle.net/11427/34761en_ZA
dc.identifier.chicagocitationHan, Weimin, B Daya Reddy, and Gregory C Schroeder "Qualitative and Numerical Analysis of Quasi-Static Problems in Elastoplasticity." <i>SIAM Journal on Numerical Analysis</i> 34, 1. (1997): 143 - 177. http://hdl.handle.net/11427/34761en_ZA
dc.identifier.citationHan, W., Reddy, B.D. & Schroeder, G.C. 1997. Qualitative and Numerical Analysis of Quasi-Static Problems in Elastoplasticity. <i>SIAM Journal on Numerical Analysis.</i> 34(1):143 - 177. http://hdl.handle.net/11427/34761en_ZA
dc.identifier.issn0036-1429
dc.identifier.issn1095-7170
dc.identifier.ris TY - Journal Article AU - Han, Weimin AU - Reddy, B Daya AU - Schroeder, Gregory C AB - The quasi-static problem of elastoplasticity with combined kinematic-isotropic hardening is formulated as a time-dependent variational inequality (VI) of the mixed kind; that is, it is an inequality involving a nondifferentiable functional and is imposed on a subset of a space. This VI differs from the standard parabolic VI in that time derivatives of the unknown variable occur in all of its terms. The problem is shown to possess a unique solution. We consider two types of approximations to the VI corresponding to the quasi-static problem of elastoplasticity: semidiscrete approximations, in which only the spatial domain is discretized, by finite elements; and fully discrete approximations, in which the spatial domain is again discretized by finite elements, and the temporal domain is discretized and the time-derivative appearing in the VI is approximated in an appropriate way. Estimates of the errors inherent in the above two types of approximations, in suitable Sobolev norms, are obtained for the quasi-static problem of elastoplasticity; in particular, these estimates express rates of convergence of successive finite element approximations to the solution of the variational inequality in terms of element size h and, where appropriate, of the time step size k. A major difficulty in solving the problems is caused by the presence of the nondifferentiable terms. We consider some regularization techniques for overcoming the difficulty. Besides the usual convergence estimates, we also provide a posteriori error estimates which enable us to estimate the error by using only the solution of a regularized problem. DA - 1997 DB - OpenUCT DP - University of Cape Town IS - 1 J1 - SIAM Journal on Numerical Analysis LK - https://open.uct.ac.za PY - 1997 SM - 0036-1429 SM - 1095-7170 T1 - Qualitative and Numerical Analysis of Quasi-Static Problems in Elastoplasticity TI - Qualitative and Numerical Analysis of Quasi-Static Problems in Elastoplasticity UR - http://hdl.handle.net/11427/34761 ER - en_ZA
dc.identifier.urihttp://hdl.handle.net/11427/34761
dc.identifier.vancouvercitationHan W, Reddy BD, Schroeder GC. Qualitative and Numerical Analysis of Quasi-Static Problems in Elastoplasticity. SIAM Journal on Numerical Analysis. 1997;34(1):143 - 177. http://hdl.handle.net/11427/34761.en_ZA
dc.language.isoeng
dc.publisher.departmentDepartment of Mathematics and Applied Mathematics
dc.publisher.facultyFaculty of Science
dc.sourceSIAM Journal on Numerical Analysis
dc.source.journalissue1
dc.source.journalvolume34
dc.source.pagination143 - 177
dc.source.urihttps://dx.doi.org/10.1137/S0036142994265383
dc.subject.otherNumerical analysis
dc.subject.otherFinite element method
dc.subject.otherEuler scheme
dc.subject.otherCrank Nicolson method
dc.subject.otherRegularization method
dc.subject.otherError estimation
dc.subject.otherA posteriori estimation
dc.subject.otherConvergence
dc.subject.otherVariational inequality
dc.subject.otherElastoplasticity
dc.subject.otherAnalyse numérique
dc.subject.otherMéthode élément fini
dc.subject.otherSchéma Euler
dc.subject.otherMéthode Crank Nicolson
dc.subject.otherMéthode régularisation
dc.subject.otherEstimation erreur
dc.subject.otherEstimation a posteriori
dc.subject.otherInégalité variationnelle
dc.subject.otherElastoplasticité
dc.subject.otherAnálisis numérico
dc.titleQualitative and Numerical Analysis of Quasi-Static Problems in Elastoplasticity
dc.typeJournal Article
uct.type.publicationResearch
uct.type.resourceJournal Article
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