## 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,
+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
### 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
+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.
![Problem geometry](./doc/problem_setup.png)
-Note that we perform a three-dimensional computation as, for this particular
-formulation, the two-dimensional case corresponds to neither plane-strain
+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.
-
+
## Requirements
* Version 8.2.1 or greater of `deal.II`
```
+## 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.437601](http://doi.org/10.5281/zenodo.437601) <img style="float: right;" src="https://zenodo.org/badge/DOI/10.5281/zenodo.437601.svg">
+
+### 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](http://doi.org/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](http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471823198.html);
* P. Wriggers (2008), Nonlinear finite element methods, Springer. DOI: [10.1007/978-3-540-71001-1](http://doi.org/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](http://store.doverpublications.com/0486411818.html)
-The derivation of the finite-element problem, namely the definition and
-linearisation of the residual and their subsequent discretisation are quite
+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
+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.
## Results
-These results were produced using the following material properties:
-* Shear modulus is 422.5kPa
+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:
+The 32x32x1 discretised reference geometry looks as follows:
![Problem geometry](./doc/ref_grid.png)
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
+A direct and, by visual inspection, favourable comparison of the following
results can be made with those found in Miehe (1994).
Since the material is compressible, shear-locking is not exhibited by the
beam for low-order elements.