Artificial Instabilities of Finite Elements for Nonlinear Elasticity Assignment

Assignment Task

Some locking-free finite element formulations are stable in the linear regime but exhibit artifi- cial instabilities in the presence of large deformations in the context of geometrically non-linear analysis. Those artificial instabilities often show up as hourglassing. Bieber et al. [2023] have studied the effect of locking-free formulations on geometric stiffness intensively. The problem shown in the figure below is established as one of the benchmarks to demonstrate the hourglas sing effects in the case of large deformations. The line load is placed asymmetrically, and the load-controlled method is used to perform the non-linear analysis.

The goal of this work is to implement a non-linear, 4-node planar solid element, solve the problem depicted in the figure, and identify the critical deformation state for a single ele- ment.

The specific tasks are:

1. Extend the provided Maple script for a linear-elastic 4-node planar solid element to include geometric non-linearities.

2. Solve the compression of a rubber block (assume: St. Venant-Kirchhoff material law) problem depicted above using load-controlled method.

3. Implement a non-linear Q1/E4 element and identify the state of deformation where the artificial instability occurs for it by performing eigen value analysis on the kers and kg. stiffness matrices (refer to Wall et al. (2000).

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