Time integration schemes for piecewise linear plasticity

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dc.contributor.advisorMartin, JBen_ZA
dc.contributor.authorRencontre, LJen_ZA
dc.date.accessioned2016-10-10T14:52:16Z
dc.date.available2016-10-10T14:52:16Z
dc.date.issued1991en_ZA
dc.description.abstractThe formulation of a generalized trapezoidal rule for the integration of the constitutive equations for a convex elastic-plastic solid is presented. This rule, which is based on an internal variable description, is consistent with a generalized trapezoidal rule for creep. It is shown that by suitable linear extrapolation, the standard backward difference algorithm can lead to this generalized trapezoidal rule or to a generalized midpoint rule. In either case, the generalized rules retain the symmetry of the consistent tangent modulus. It is also shown that the generalized trapezoidal and midpoint rules are fully equivalent in the sense that they lead to the establishment of the same minimum principle for the increment. The generalized trapezoidal rule thus inherits the notion of B-stability and both rules offer the opportunity to exploit the second order rate of convergence for a = ½. However, in the generalized trapezoidal rule, the equilibrium. and constitutive equations are fully satisfied at the end of the time increment. This may be more convenient than the generalized midpoint rule, in which equilibrium and plastic consistency are satisfied at the generalized midpoint. A backward difference return algorithm for piecewise linear yield surfaces is then formulated, with attention restricted to an associated flow rule and isotropic material behavior. Both the Tresca and Mohr-Coulomb yield surfaces with perfectly plastic and linear hardening rules are considered in detail. The algorithm has the advantage of being fully linked to the governing principles and avoids the inherent problems associated with corners on the yield surface. It is fully consistent in that no heuristic assumptions are made. The algorithm is extended to include the generalized trapezoidal rule in such a way that the general structure of the backward difference algorithm is maintained. This allows both for the computational advantages of the generalized trapezoidal rule to be utilized, and for a basis for comparison between this algorithm and existing backward difference algorithms to be established. Using this fully consistent algorithm, the return paths in stress space for the Tresca and Mohr-Coulomb yield surfaces with perfectly plastic and linear hardening rules are identified. These return paths thus provide a basis against which heuristically developed algorithms can be compared.en_ZA
dc.identifier.apacitationRencontre, L. (1991). <i>Time integration schemes for piecewise linear plasticity</i>. (Thesis). University of Cape Town ,Faculty of Engineering & the Built Environment ,Centre for Research in Computational and Applied Mechanics (CERECAM). Retrieved from http://hdl.handle.net/11427/22103en_ZA
dc.identifier.chicagocitationRencontre, LJ. <i>"Time integration schemes for piecewise linear plasticity."</i> Thesis., University of Cape Town ,Faculty of Engineering & the Built Environment ,Centre for Research in Computational and Applied Mechanics (CERECAM), 1991. http://hdl.handle.net/11427/22103en_ZA
dc.identifier.citationRencontre, L. 1991. Time integration schemes for piecewise linear plasticity. University of Cape Town.en_ZA
dc.identifier.ris TY - Thesis / Dissertation AU - Rencontre, LJ AB - The formulation of a generalized trapezoidal rule for the integration of the constitutive equations for a convex elastic-plastic solid is presented. This rule, which is based on an internal variable description, is consistent with a generalized trapezoidal rule for creep. It is shown that by suitable linear extrapolation, the standard backward difference algorithm can lead to this generalized trapezoidal rule or to a generalized midpoint rule. In either case, the generalized rules retain the symmetry of the consistent tangent modulus. It is also shown that the generalized trapezoidal and midpoint rules are fully equivalent in the sense that they lead to the establishment of the same minimum principle for the increment. The generalized trapezoidal rule thus inherits the notion of B-stability and both rules offer the opportunity to exploit the second order rate of convergence for a = ½. However, in the generalized trapezoidal rule, the equilibrium. and constitutive equations are fully satisfied at the end of the time increment. This may be more convenient than the generalized midpoint rule, in which equilibrium and plastic consistency are satisfied at the generalized midpoint. A backward difference return algorithm for piecewise linear yield surfaces is then formulated, with attention restricted to an associated flow rule and isotropic material behavior. Both the Tresca and Mohr-Coulomb yield surfaces with perfectly plastic and linear hardening rules are considered in detail. The algorithm has the advantage of being fully linked to the governing principles and avoids the inherent problems associated with corners on the yield surface. It is fully consistent in that no heuristic assumptions are made. The algorithm is extended to include the generalized trapezoidal rule in such a way that the general structure of the backward difference algorithm is maintained. This allows both for the computational advantages of the generalized trapezoidal rule to be utilized, and for a basis for comparison between this algorithm and existing backward difference algorithms to be established. Using this fully consistent algorithm, the return paths in stress space for the Tresca and Mohr-Coulomb yield surfaces with perfectly plastic and linear hardening rules are identified. These return paths thus provide a basis against which heuristically developed algorithms can be compared. DA - 1991 DB - OpenUCT DP - University of Cape Town LK - https://open.uct.ac.za PB - University of Cape Town PY - 1991 T1 - Time integration schemes for piecewise linear plasticity TI - Time integration schemes for piecewise linear plasticity UR - http://hdl.handle.net/11427/22103 ER - en_ZA
dc.identifier.urihttp://hdl.handle.net/11427/22103
dc.identifier.vancouvercitationRencontre L. Time integration schemes for piecewise linear plasticity. [Thesis]. University of Cape Town ,Faculty of Engineering & the Built Environment ,Centre for Research in Computational and Applied Mechanics (CERECAM), 1991 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/22103en_ZA
dc.language.isoengen_ZA
dc.publisher.departmentCentre for Research in Computational and Applied Mechanics (CERECAM)en_ZA
dc.publisher.facultyFaculty of Engineering and the Built Environment
dc.publisher.institutionUniversity of Cape Town
dc.subject.otherComputational and Applied Mechanicsen_ZA
dc.titleTime integration schemes for piecewise linear plasticityen_ZA
dc.typeDoctoral Thesis
dc.type.qualificationlevelDoctoral
dc.type.qualificationnamePhDen_ZA
uct.type.filetypeText
uct.type.filetypeImage
uct.type.publicationResearchen_ZA
uct.type.resourceThesisen_ZA
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