A dual decomposition of the closest point projection in incremental elasto‐plasticity using a mixed shell finite element

Abstract

Mixed assumed stress finite elements (FEs) have shown good advantages over traditional displacement‐based formulations in various contexts. However, their use in incremental elasto‐plasticity is limited by the need for return mapping schemes which preserve the assumed stress interpolation. For elastic‐perfectly plastic materials and small deformation problems, the integration of the constitutive equation furnishes a closest point projection (CPP) involving all the element stress parameters. In this work, a dual decomposition strategy is adopted to split this problem into a series of CPPs at the integration points level and in a nonlinear system of equations over the element, in order to simplify its solution. The strategy is tested with a four nodes mixed shell FE, named MISS‐4, characterized by an equilibrated stress interpolation which improves the accuracy. Two decomposition strategies are tested to express the plastic admissibility either in terms of stress resultants or point‐wise Cauchy stresses. The recovered elasto‐plastic solution preserves all the advantages of MISS‐4, namely it is accurate for coarse meshes in recovering the equilibrium path and evaluating the limit load showing a quadratic rate of convergence, as demonstrated by the numerical results.

A mixed finite-element formulation for the elasto-plastic analysis of shell structures

“…In is worth noting that the proposed decomposed formulation furnishes the same discrete equations as in other works [8,9,10]. Further details on the proposed approach can be found in [11].…”

“…The equilibrium path of the structures is usually evaluated by means of a continuation method [11]. Starting from a known equilibrium point 𝑧𝑧 0 = 𝑧𝑧 (𝑛𝑛) , where 𝑧𝑧 = [𝑑𝑑, 𝜆𝜆], a new state 𝑧𝑧 𝑘𝑘+1 is obtained by solving, through a Newton iteration, the following system representing the equilibrium equations plus the arc-length constraint for an assigned value of Δ𝜉𝜉.…”

“…A sequence of IP state determination and element state determination problems allows the evaluation of a plastically-admissible 𝜷𝜷 𝒆𝒆 for an assigned 𝚫𝚫𝒅𝒅 𝒆𝒆 , as explained in [11].…”

Mechanical-Based Seismic Vulnerability Analysis of Industrial Steel Structures With Masonry Infills

“…The accuracy of the representation of the yield surface can be readily improved by adopting techniques based on the Minkowsky sum of ellipsoids [20]. Plasticity is checked at 3 Gauss-Lobatto points on each FE and the integration of the constitutive equation is performed at the element level in order to preserve the assumed stress interpolation [21,22]. Masonry infills are modelled by adopting a simple single equivalent strut description (see Fig.…”

“…Nonlinear static analyses are performed on the FE model to simulate the seismic response. An arc-length algorithm is employed to trace the equilibrium path [22].…”