Subdifferential-based implicit return-mapping operators in computational plasticity

• Published in: Zeitschrift für angewandte Mathematik und Mechanik 2016-11, Vol.96 (11), p.1318-1338
• Main Authors: Sysala, Stanislav, Cermak, Martin, Koudelka, Tomáš, Kruis, Jaroslav, Zeman, Jan, Blaheta, Radim
• Format: Article
• Language: English
• Subjects: 35Q74; 74C05; 74S05; 90C25; Construction; Elastoplasticity; Hydrostatics; implicit return-mapping scheme; Initial value problems; limit analysis; Mathematical models; Matlab; multivalued flow direction; Nonlinear equations; Nonlinear programming; Nonlinearity; nonsmooth yield surface; semismooth Newton method
• ISSN: 0044-2267; 1521-4001
• DOI: 10.1002/zamm.201500305

In this paper we explore a numerical solution to elastoplastic constitutive initial value problems. An improved form of the implicit return‐mapping scheme for nonsmooth yield surfaces is proposed that systematically builds upon a subdifferential formulation of the flow rule. The main advantage of this approach is that the treatment of singular points – apices or edges at which the flow direction is multivalued – only involves a uniquely defined set of non‐linear equations, similarly to smooth yield surfaces. This paper focuses on isotropic models containing: a) yield surfaces with one or two apices (singular points) on the hydrostatic axis, b) plastic pseudo‐potentials that are independent of the Lode angle, and c) possibly nonlinear isotropic hardening. We show that for some models the improved integration scheme also enables us to a priori decide about a type of the return and to investigate the existence, uniqueness, and semismoothness of discretized constitutive operators. The semismooth Newton method is also introduced for solving the incremental boundary‐value problems. The paper contains numerical examples related to slope stability with publicly available Matlab implementations. The authors explore a numerical solution to elastoplastic constitutive initial value problems. An improved form of the implicit return‐mapping scheme for nonsmooth yield surfaces is proposed that systematically builds upon a subdifferential formulation of the flow rule. The main advantage of this approach is that the treatment of singular points – apices or edges at which the flow direction is multivalued – only involves a uniquely defined set of non‐linear equations, similarly to smooth yield surfaces. This paper focuses on isotropic models containing: a) yield surfaces with one or two apices (singular points) on the hydrostatic axis, b) plastic pseudo‐potentials that are independent of the Lode angle, and c) possibly nonlinear isotropic hardening.