Erratum to "An iterative method for shakedown analysis" [Comput. Methods Appl. Mech. Engrg. 191 (2002) 5761-5792] (DOI:10.1016/S0045-7825(02)00496-6) (original) (raw)
Static Shakedown Theorem Accounting for Material Damage and Creep
Journal of the Mechanical Behavior of Materials, 2004
Conditions for shakedown of structures (solids) of elastic plastic creeping materials with anisotropic plastic damage, strain hardening and softening, as subjected to thermo mechanical loading, are in question. The current yield surfaces are assumed to enclose the one of the minimal diameter (/«-surface). This is valid at the stage of strain hardening, and also at the part of the softening stage. The shakedown conditions are of the classical (Melan) type; however the residual stress may depend on time. The sufficient condition is formulated relative to the /»-surface.
Efficient Shakedown Solutions in Complex Loading Domains
Advances in Direct Methods for Materials and Structures, 2017
To estimate the life of a structure, or a component, which are subjected to a cyclic loading history, the structural engineer must be able to provide safety margins. This is only possible by performing a shakedown analysis which belongs to the class of direct methods. Most of the existing numerical procedures addressing a shakedown analysis are based on the two theorems of plasticity and are formulated within the framework of mathematical programming. A different approach has recently appeared in the literature. It is rather more physical than mathematical as it exploits the physics of the asymptotic steady state cycle. It has been called RSDM-S and has its roots in a previously published procedure (RSDM) which assumes the decomposition of the residual stresses into Fourier series whose coefficients are found by iterations. RSDM-S is a descending sequence of loading factors which stops when only the constant term of the series remains. The method may be implemented in any existing FE code. It is used herein to establish shakedown boundaries for two-dimensional general loadings consisting of mechanical or thermomechanical loads.
Theorems of restricted dynamic shakedown
International Journal of Mechanical Sciences, 1993
Dynamic shakedown for a rate-independent material with internal variables is addressed in the hypothesis that the load values are restricted to those of a specified load history of finite or even infinite duration, thus ruling out the possibility-typical of classical shakedown theory-of indefinite load repetitions.
Decomposition Methods and Strain Driven Algorithms for Limit and Shakedown Analysis
A mathematical programming formulation of strain-driven path-following strategies to perform shakedown and limit analysis for perfectly elastoplastic materials in a FEM context, is presented. From the optimization point of view, standard arc-length strain driven elastoplastic analysis, recently extended to shakedown, are identified as particular decomposition strategies used to solve a proximal point algorithm applied to the static shakedown theorem that is then solved by means of a convergent sequence of safe states. The mathematical programming approach allows: a direct comparison with other nonlinear programming methods, simpler convergence proofs and duality to be exploited. Due to the unified approach in terms of total stresses, the strain driven algorithms become more effective and less nonlinear with respect to a self equilibrated stress formulation and easier to implement in existing codes performing elastoplastic analysis.
An edge-based smoothed finite element method for primalâdual shakedown analysis of structures
International Journal for Numerical Methods in Engineering, 2009
An edge-based smoothed finite element method (ES-FEM) using three-node linear triangular elements was recently proposed to significantly improve the accuracy and convergence rate of the standard finite element formulation for static, free and forced vibration analyses of solids. In this paper, ES-FEM is further extended for limit and shakedown analyses of structures. A primal-dual algorithm based upon the von Mises yield criterion and a non-linear optimization procedure is used to compute both the upper and lower bounds of the plastic collapse limit and the shakedown limit. In the ES-FEM, compatible strains are smoothed over the smoothing domains associated with edges of elements. Using constant smoothing function, only one Gaussian point is required for each smoothing domain ensuring that the total number of variables in the resulting optimization problem is kept to a minimum compared with standard finite element formulation. Three benchmark problems are presented to show the stability and accuracy of solutions obtained by the present method. AN EDGE-BASED SMOOTHED FINITE ELEMENTS METHOD 919 not necessary. Only the maximum loads (limits) count and the envelopes of load domain should be taken into consideration.
International Journal of Solids and Structures, 2006
The class of generalized standard materials is not relevant to model the nonassociative constitutive equations. The bipotential approach, based on a possible generalization of FenchelÕs inequality, allows the recovery of the flow rule normality in a weak form of an implicit relation. This defines the class of implicit standard materials. For such behaviours, this leads to a weak extension of the classical bound theorems of the shakedown analysis. In the present paper, we recall the relevant features of this theory. Considering an elastoplastic material with nonlinear kinematic hardening rule, we apply it to the problem of a sample in plane strain conditions under constant traction and alternating torsion in order to determine analytically the interaction curve bounding the shakedown domain. The aim of the paper is to prove the exactness of the solution for this example by comparing it to step-by-step computations of the elastoplastic response of the body under repeated cyclic loads of increasing level. A reliable criterion to stop the computations is proposed. The analytical and numerical solutions are compared and found to be closed one of each other. Moreover, the method allows uncovering an additional Ô2 cycle shakedown curveÕ that could be useful for the shakedown design of structure.
Shakedown analysis with multidimensional loading spaces
Computational Mechanics, 2012
A numerical method for the computation of shakedown loads of structures subjected to varying thermal and mechanical loading is proposed for the case of multidimensional loading spaces. The shakedown loading factors are determined based on the lower bound direct method using the von Mises yield criterion. The resulting nonlinear convex optimization problem is solved by use of the interior-point method. Although the underlying theory allows for the consideration of arbitrary numbers of loadings in principle, until now applications have been restricted to special cases, where either one or two loads vary independently. In this article, former formulations of the optimization problem are generalized for the case of arbitrary numbers of loadings. The method is implemented into an interior-point algorithm specially designed for this method. For illustration, numerical results are presented for a three-dimensional loading space applied to a square plate with a central circular hole.