Shakedown analysis of porous materials via mixed meshless methods
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TECHNICAL PAPER
Shakedown analysis of porous materials via mixed meshless methods Carlos C. de La Plata Ruiz1 · Jose Luis Silveira2 Received: 4 February 2020 / Accepted: 4 May 2020 / Published online: 15 May 2020 © The Brazilian Society of Mechanical Sciences and Engineering 2020
Abstract In this paper, the elastic shakedown analysis of porous materials is performed by means of meshless methods using mixed approximations. Based on a mixed variational principle for shakedown analysis and using a yield function for porous materials, two meshless methods are adapted to perform mixed approximations of the stress and velocity fields for the solution of the discrete shakedown problem. These two new methods are named mixed moving least squares method and mixed Shepard’s method and are used to solve some numerical examples. The numerical results obtained showed a good agreement with available analytical solutions and published results by the finite element method. The proposed mixed methods can be applied in the analysis of structural and machine parts made of porous materials and subjected to variable loads. Keywords Plasticity · Shakedown analysis · Porous material · Yield function · Meshless method
1 Introduction Shakedown analysis enables to find the safe condition of operation for structures under variable loads and has been used in the design of several applications, such as pressure vessels and components for aerospace industry. The first studies on this subject started in the 1920s. Weichert and Ponter [33] presented a comprehensive historical introduction on shakedown theory and pointed out that the first contribution to an embryonic idea of shakedown was due to Gruning in the mid-1920s. Bleich and Melan’s later contributions gave rise to the lower-bound shakedown theorem [22], and subsequently Koiter established the upper-bound theorem [16] in the context of ideal plasticity. Over the years, shakedown theory has been refined and expanded to solve more complex problems with the aid of numerical methods. In particular, the finite element method
Technical Editor: João Marciano Laredo dos Reis. * Jose Luis Silveira [email protected] Carlos C. de La Plata Ruiz [email protected] 1
Department of Mechanical Engineering, Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil
Department of Mechanical Engineering, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
2
(FEM) is one of the most used techniques. Belytschko [2] was the first to apply the FEM for the solution of a shakedown problem. He obtained the numerical solution for the shakedown of a plate with a circular hole in a state of plane stress, which became a benchmark for this subject. However, despite its successful application, the finite element method is not the only method used for the numerical shakedown solution. Based on the meshless local Petrov–Galerkin (MLPG) method, Chen et al. [7] presented a solution procedure for the lower-bound shakedown analysis and Ruiz and Silveira [27] proposed a generalization
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