On volumetric locking of low‐order solid and solid‐shell elements for finite elastoviscoplastic deformations and selective reduced integration

Abstract

As known from nearly incompressible elasticity, selective reduced integration (SRI) is a simple and effective method of overcoming the volumetric locking problem in 2D and 3D solid elements. This method of finite elastoviscoplasticity is discussed as are its well-known limitations. In this context, an isochoric-volumetric decoupled material behavior is assumed and thus the additive deviatoric-volumetric decoupling of the consistent algorithmic moduli tensor is essential. By means of several numerical examples, the performance of elements using selective reduced integration is demonstrated and compared to the performance of other elements such as the enhanced assumed strain elements. It is shown that a minor modification, with little numerical effort, leads to rather robust element behaviour. The application of this process to so-called solid-shell elements for thin-walled structures is also discussed. IntroductionIt is a well-known fact, that the three-dimensional standard linear eight-node displacement element (here denoted as Q1) exhibits severe locking for nearly or fully incompressible material. Rubber elasticity or metal plasticity are examples for nearly incompressible material behavior. Rubber is characterized by a large ratio of the bulk modulus to the shear modulus. In metal plasticity the plastic deformation is isochoric, i.e. fully incompressible, and the compressible part is a result from elastic deformations only, which remain small in many applications. In both cases reliable results with the standard Q1 element cannot be expected. Due to the volumetric locking behavior of the Q1 element, additional measures are necessary to overcome this problem. Various methods are known from literature and are currently discussed. Among others, a brief review of some of these methods is given in the following as a reference for the developments shown in this contribution. Our focus is particularly on the capabilities concerning finite non-linear deformations:

A physically and geometrically nonlinear scaled‐boundary‐based finite element formulation for fracture in elastomers

“…Standard isoparametric FEs (Q1P1) suffer from locking in case of nearly incompressible material behavior, for example, of rubber-like materials, as has been shown in previous studies by, for example, Doll et al [39]. So-called Q1P0 FE formulations overcome locking problems by a kinematically decoupled ansatz for volumetric and isochoric deformations.…”

Enhanced transverse shear strain shell formulation applied to large elasto‐plastic deformation problems

“…This example has been treated in a number of references, including shell and solid-shell formulations and adopting a variety of mesh topologies (see, for instance References [28,61,78,[83][84][85][86]). About Figure 4, and following the previous references, the total length of the plate is 2L = 508, with thickness a = 2.54 consistent unities.…”

“…For each simulation just two Gauss points along thickness direction are employed, as for a higher interpolation order the same results were obtained. For comparison, Reference [78] adopts five Gauss points along thickness direction, while References [85,86] use six integration points. About boundary and loading conditions, displacements along the OZ direction are restrained on the outer edges, while a deformation dependent pressure load p = f × p 0 is applied on one side of the shell, for a nominal load level of p 0 = 10 −2 .…”

A partial hybrid stress solid-shell element for the analysis of laminated composites

“…Wriggers and Reese [76] mentioned in their work that reliable 3D finite elements for shell-type structures with finite strains can be obtained using the EAS method of Simo and Rifai [63]. Significant works on the solid-shell elements include the intensive work of Schweizerhof and co-workers [19,23,25,26], Klinkel and Wagner [33], Klinkel et al [32], Wagner et al [72], Miehe [40], Vu-Quoc and Tan [70,71], to exploit the combination of the EAS method and the assumed natural strain (ANS) method to develop fully integrated loworder solid-shell formulations. Solid-shell formulations by Alves de Sousa et al [2,3,5], Cardoso et al [13], Legay and Combescure [36], Reese [58], Schwarze and Reese [61,62], Li et al [37], employed the reduced integration schemes to enhance their accuracy and efficiency.…”