A matrix structural theory of piecewise linear elastoplasticity with interacting yield planes

Maier, G.

doi:10.1007/bf02133524

Cited by 307 publications

(130 citation statements)

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“…Various approaches have however been used for deriving global variational formulations for plasticity, each of which when discretized in time, leads to a constrained minimization problem in every step. These are base on (1) Complementarity and Mathematical Programming Approach (Maier [4]), (2) Variational Inequality Approach (Duvaut [5]) and (3) Convex Analysis and Monotone Operator Approach (Romano [6]). The most common procedure is to use the Backward Euler method to approximate the rate quantities in the variational statement leading, to a constrained minimization problem in each time increment (see Simo [1] for a detailed presentation).…”

Section: Variational Principles For Plasticitymentioning

confidence: 99%

Lagrangian Approach to Structural Collapse Simulation

Sivaselvan

Reinhorn

2006

J. Eng. Mech.

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SUMMARYComputer analysis of structures has traditionally been carried out using the displacement method combined with an incremental iterative scheme for nonlinear problems. In this paper, a Lagrangian approach is developed, which is a mixed method, where besides displacements, the stress-resultants and other variables of state are primary unknowns. The method can potentially be used for the analysis of structures to collapse as demonstrated by numerical examples. The evolution of the structural state in time is provided a weak formulation using Hamilton's principle. It is shown that a certain class of structures, known as reciprocal structures has a mixed Lagrangian formulation in terms of displacements and internal forces. The form of the Lagrangian is invariant under finite displacements and can be used in geometric nonlinear analysis. For numerical solution, a discrete variational integrator is derived starting from the weak formulation. This integrator inherits the energy and momentum conservation characteristics for conservative systems and the contractivity of dissipative systems. The integration of each step is a constrained minimization problem and is solved using an Augmented Lagrangian algorithm. In contrast to the displacement-based method, the Lagrangian method clearly separates the modeling of components from the numerical solution. Phenomenological models of components essential to simulate collapse can therefore be incorporated without having to implement model-specific incremental state determination algorithms. The state determination is performed at the global level by the optimization method.

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Section: Variational Principles For Plasticitymentioning

confidence: 99%

Lagrangian Approach to Structural Collapse Simulation

Sivaselvan

Reinhorn

2006

J. Eng. Mech.

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“…Maier and his colleagues from the Italian school [37,119,120,121] were the pioneers in this approach. Kaneko [85,86,87] and Wakefield and Tin-Loi [201,202] focused on the linear complementarity approach.…”

Section: Structural Mechanics Problemsmentioning

confidence: 99%

Engineering and Economic Applications of Complementarity Problems

Ferris¹,

1997

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Abstract. This paper gives an extensive documentation of applications of finite-dimensional nonlinear complementarity problems in engineering and equilibrium modeling. For most applications, we describe the problem briefly, state the defining equations of the model, and give functional expressions for the complementarity formulations. The goal of this documentation is threefold: (i) to summarize the essential applications of the nonlinear complementarity problem known to date, (ii) to provide a basis for the continued research on the nonlinear complementarity problem, and (iii) to supply a broad collection of realistic complementarity problems for use in algorithmic experimentation and other studies.

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“…Although other parametric problems sometimes occur (see, for example, Pang, Kaneko, & Hallman, 1979), problems of the form (5.1) are by far the most common. Classes of such problems occur in, for example, the fields of portfolio analysis (see Markowitz, 1952), structural engineering (see Maier, 1970), and multiple criterion decision processes (see Rhode & Weber, 1984). Suppose we have found a local solution x for the value t = 1 e [t ,i), that the components jcpx of JC are contained in / and fixed at one of the bounds (5.1c), and that K{1) satisfies SOC.…”

Section: Parametric Problemsmentioning

confidence: 99%