documents

Overview

The Cook membrane (or cantilever) problem is a classic benchmark test for finite element formulations for solid mechanics. It is typically used to test for and demonstrate the shear-locking (or locking-free) behaviour of a finite element ansatz under quasi-incompressible conditions. As it is so widely referred to in the literature on finite-strain elasticity, we reproduce the example here. However, we consider on the compressible case to avoid many of the complexities that arise in step-44, which provides an efficient approach to deal with the quasi-incompressible case.

A classical approach to solving the cook membrane problem.

In this work we take a classical approach to solving the equations governing quasi-static finite-strain compressible elasticity, with code based on step-44. The formulation adopted here is that seen in many texts on solid mechanics and can be used as the starting point for extension into many topics such as material anisotropy, rate dependence or plasticity, or even as a component of multi-physics problems.

The basic problem configuration is summarised in the following image. A beam of specific dimensions is fixed at one end and a uniform traction load is applied at the other end such that the total force acting on this surface totals 1 Newton. Displacement in the third coordinate direction (out of plane) is prevented in order to impose plane strain conditions.

Note that we perform a three-dimensional computation as, for this particular formulation, the two-dimensional case corresponds to neither plane-strain nor plane-stress conditions.

Compiling and running

Similar to the example programs, run

cmake -DDEAL_II_DIR=/path/to/deal.II .

in this directory to configure the problem.
You can switch between debug and release mode by calling either

make debug

or

make release

The problem may then be run with

make run

Reference for this work

If you use this program as a basis for your own work, please consider citing it in your list of references. The initial version of this work was contributed to the deal.II project by the authors listed in the following citation:

• J-P. V. Pelteret and A. McBride, The deal.II code gallery: Quasi-Static Finite-Strain Compressible Elasticity, 2016. DOI: 10.5281/zenodo.1228964

Acknowledgements

The support of this work by the European Research Council (ERC) through the Advanced Grant 289049 MOCOPOLY is gratefully acknowledged by the first author.

Recommended Literature

• C. Miehe (1994), Aspects of the formulation and finite element implementation of large strain isotropic elasticity International Journal for Numerical Methods in Engineering 37 , 12, 1981-2004. DOI: 10.1002/nme.1620371202;
• G.A. Holzapfel (2001), Nonlinear Solid Mechanics. A Continuum Approach for Engineering, John Wiley & Sons. ISBN: 978-0-471-82319-3;
• P. Wriggers (2008), Nonlinear finite element methods, Springer. DOI: 10.1007/978-3-540-71001-1;
• T.J.R. Hughes (2000), The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover. ISBN: 978-0486411811

The derivation of the finite-element problem, namely the definition and linearisation of the residual and their subsequent discretisation are quite lengthy and involved. Thankfully, the classical approach adopted in this work is well documented and therefore does not need to be reproduced here. We refer the reader to, among many other possible texts, Holzapfel (2001) and Wriggers (2008) for a detailed description of the approach applied in this work. It amounts to a reduction and slight reworking of step-44 (accounting for the removal of the two additional fields used therein). We also refer the reader to step-44 for a brief overview of the continuum mechanics and kinematics related to solid mechanics. We also provide two alternative assembly mechanisms, which can be selected within the parameter file, that use automatic differentiation to assemble the linear system. These not only demonstrate how one might employ such automated techniques to linearise the residual, but also serve as a verification of the correctness of the hand-developed implementation of the tangent stiffness matrix.

Results

These results were produced using the following material properties:

• Shear modulus is 422.5kPa
• Poisson ratio is 0.3

The 32x32x1 discretised reference geometry looks as follows:

And an example of the displaced solution is given in the next image.

Below we briefly document the tip displacement as predicted for different discretisation levels and ansatz for the displacement field. A direct and, by visual inspection, favourable comparison of the following results can be made with those found in Miehe (1994). Specifically, figure 5 of Miehe (1994) plots the vertical displacement at the midway point on the traction surface for the compressible 3d case. Since the material is compressible, shear-locking is not exhibited by the beam for low-order elements.

Number of degrees of freedom

Elements per edge	Q1	Q2
2	54	225
4	150	729
8	486	2601
16	1734	9801
32	6534	38025
64	25350	149769

Tip y-displacement (in mm)

Elements per edge	Q1	Q2
2	8.638	14.30
4	12.07	14.65
8	13.86	14.71
16	14.49	14.73
32	14.67	14.74
64	14.72	14.74