# Data for the Assessment of Stress Integration Schemes for Large-Deformation Finite Element Analysis

## Description

Files "results_comp_ext_repository.xlsx", "results_ext_rot_repository.xlsx", "results_simple_shear_repository.xlsx", "results_uniaxial_extension_repository.xlsx" gather the results (stress predictions) of four tests, aimed to check stress integration algorithms in four corresponding deformation paths: “extension-compression”, “extension-rotation”, “simple shear”, "uniaxial extension". They are chosen in order to include the most common motions of the structure particles, i.e. relative motions that yield extension, compression and shear combined with possible rigid rotations of the body. Particularly, the rotation motion is simulated in order to test the incremental objectivity or, better, the priority on rigid motion or weak objectivity of the algorithms. The exact analytical solutions of the three tests are known from literature. In all of the tests, a unit square (1 x 1 m2) with unit thickness undergoes a deformation path under plane stress conditions. It is made of linear elastic material with Young’s modulus E = 1 Pa and Poisson ratio nu = 0. A fixed number of time steps N equal to 50 (the highest employed in the reference papers) is chosen. The simulated “time” parameter t ranges between 0 and 1 in “extension-compression” as well as “extension-rotation” tests, whereas it raises from 0 to 0.9 in “simple shear” test. Therefore, the corresponding time steps t are equal to 1/50 and 9/500, respectively. The various simulation models include different stress integration methods labelled as described in the reference research paper [1]. In addition, data for plane elements (PLANE42, PLANE182) of Ansys code are added. The files include a data sheet and a corresponding plot for each category of algorithms (classical update-Lagrange algorithms, CC methods, and commercial software). The data are organized in columns, where stress components are given for each stress update method. The file "test_Rodriguez.ans" collects the APDL commands required to generate the data for the Ansys plane elements considered. In the file "cc methods_validation_repository.xlsx" the simple shear test is used in order to validate the methods based on a corotated configuration (CC methods) that use Green-Naghdi and Zaremba-Jaumann objective stress rates. Such validation is cited in the reference article. [1] F. Stefani, M. Frascio, C. A. Niccolini Marmont Du Haut Champ, Choice of the stress integration scheme for accurate large-deformation finite element analysis, Proc IMechE Part C: J Mechanical Engineering Science, 2023, Vol. 237(17) 3977–3986

## Files

## Steps to reproduce

The data of the stress integration tests for the stress-update methods in the commercial software category are generated by means of the Ansys software (version 2021 R1). The data can be easily reproduced by running the APDL command file "test_Rodriguez.ans" included in the repository. By changing the input parameters in such file, the analysed element type (PLANE42 or PLANE182) and the stress integration path can be changed. The remaining data have been obtained by means of suitable implementations of the corresponding stress integration methods in the "FEMLub" finite element code. It is based on a large C++ library, which cannot be included in the repository. Nevertheless, the code has been developed as explained in the article [1] referenced in the Description above, in agreement with the cited literature and the proposed modifications.