Explicit mixed strain–displacement finite elements for compressible and quasi-incompressible elasticity and plasticity

Abstract

K: explicit mixed finite elements, stabilization, incompressibility, plasticity, strain softening, strain localization, mesh independence. AbstractThis paper presents an explicit mixed finite element formulation to address compressible and quasi-incompressible problems in elasticity and plasticity. This implies that the numerical solution only involves diagonal systems of equations. The formulation uses independent and equal interpolation of displacements and strains, stabilized by variational subscales (VMS). A displacement sub-scale is introduced in order to stabilize the mean-stress field. Compared to the standard irreducible formulation, the proposed mixed formulation yields improved strain and stress fields. The paper investigates the effect of this enhancement on the accuracy in problems involving strain softening and localization leading to failure, using low order finite elements with linear continuous strain and displacement fields (P 1P 1 triangles in 2D and tetrahedra in 3D) in conjunction with associative frictional Mohr-Coulomb and Drucker-Prager plastic models. The performance of the strain/displacement formulation under compressible and nearly incompressible deformation patterns is assessed and compared to analytical solutions for plane stress and plane strain situations. Benchmark numerical examples show the capacity of the mixed formulation to predict correctly failure mechanisms with localized patterns of strain, virtually free from any dependence of the mesh directional bias. No auxiliary crack tracking technique is necessary.

Elastoplasticity with linear tetrahedral elements: A variational multiscale method

“…More recently, Cervera et al pursued a mixed finite element implementation based on the OSS approach, with nodal finite element spaces for the displacements and strains, in both dynamic and static regimes, using Mohr‐Coulomb and Drucker‐Prager plasticity models. With the exception of the work of Cervera et al, all these works presented only static computations.…”

“…It is important to note that the OSS approach requires storing an additional pressure gradient field, with an increase in computational cost with respect to the HFB approach . More recently, Cervera et al pursued a mixed finite element implementation based on the OSS approach, with nodal finite element spaces for the displacements and strains, in both dynamic and static regimes, using Mohr‐Coulomb and Drucker‐Prager plasticity models. With the exception of the work of Cervera et al, all these works presented only static computations.…”

The shifted boundary method for solid mechanics

“…Our first step is then to describe a general mixed form of the shifted boundary method that applies to the entire surrogate domain . Note that other choices of mixed formulations are applicable for the same purpose, 39‐44,47,48,50‐53 and the discussion to follow in Section 3.4 is not confined to the specific mixed formulation considered here. Let and be the discrete, globally continuous trial and test function spaces for the displacement and strain .…”

An Overview of Mixed Finite Elements for the Analysis of Inelastic Bidimensional Structures

“…Especially the strictness of the latter has motivated the twofold focus of researchers to modify mixed formulations in order to circumvent stability issues or to explore the stability of newly proposed elements. Among stabilization techniques proposed to achieve stable elements irrespective of the inf-sup condition, it is worth mentioning Galerkin least square stabilization [42], bubble enrichment of displacement field [4,53,19] and variational multi-scale formulations [21,22,23]. On the other hand, an analytical proof of mixed element stability would be an indisputable achievement.…”