This repository takes a stiffness tensor and an indentation direction and calculates the expected indentation modulus.
This can be used to study the effect of
- Changes to the stiffness components.
- Changes to the indentation direction.
- Different indenter shapes.
- Inverse modeling to estimate some or all of the elastic components from a series of indentation experiments via error minimization.
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Clone or download this repository. The best way is to issue
$ git clone --recurse-submodules https://github.com/abrandberg/indentationToElasticModulus.git
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This repository uses the tensor manipulation library MMTensor, originally hosted at
Maarten Moesen (2020). MMTensor 1.0 (https://www.mathworks.com/matlabcentral/fileexchange/32891-mmtensor-1-0).
A copy of this repository is hosted on Github (https://github.com/abrandberg/MM_Tensor) and is included as a submodule in this repository. If you cloned the repository without recursively cloning the submodules (step 1) you can initiate the submodules manually by issuing (while standing in the indentationToElasticModulus repository)
$ git submodule init
$ git submodule update
This will clone the submodule.
The basics of indentation testing is Hertzian contact mechanics. Beyond a general understanding of this topic, I recommend the following articles to understand and extend this repository:
[1] Oliver, W. C., & Pharr, G. M. (1992).
An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments.
Journal of Materials Research, 7(6), 1564–1583. https://doi.org/10.1557/JMR.1992.1564
[2] Vlassak, J. J., & Nix, W. D. (1994).
Measuring the elastic properties of anisotropic materials by means of indentation experiments.
Journal of the Mechanics and Physics of Solids, 42(8), 1223–1245. https://doi.org/10.1016/0022-5096(94)90033-7
[3] Swadener, J. G., & Pharr, G. M. (2001).
Indentation of elastically anisotropic half-spaces by cones and parabolae of revolution.
Philosophical Magazine A, 81(2), 447–466. https://doi.org/10.1080/01418610108214314
[4] Vlassak, J. J., Ciavarella, M., Barber, J. R., & Wang, X. (2003).
The indentation modulus of elastically anisotropic materials for indenters of arbitrary shape.
Journal of the Mechanics and Physics of Solids, 51(9), 1701–1721. https://doi.org/10.1016/S0022-5096(03)00066-8
[5] Delafargue, A., & Ulm, F. J. (2004).
Explicit approximations of the indentation modulus of elastically orthotropic solids for conical indenters.
International Journal of Solids and Structures, 41(26), 7351–7360. https://doi.org/10.1016/j.ijsolstr.2004.06.019
[6] Jäger, A., Bader, T., Hofstetter, K., & Eberhardsteiner, J. (2011).
The relation between indentation modulus, microfibril angle, and elastic properties of wood cell walls.
Composites Part A: Applied Science and Manufacturing, 42(6), 677–685. https://doi.org/10.1016/j.compositesa.2011.02.007
[7] Argatov, I., & Mishuris, G. (2018).
Indentation of an Anisotropic Elastic Half-Space (pp. 323–371). https://doi.org/10.1007/978-3-319-78533-2_12
This repository currently contains an independent implementation in MATLAB of [4] and [5], verified against the curves presented in [6]. The code is heavily un-optimized and seeks to be similar to the actual equations in the papers. That means it is slow. If you want to get involved, I suggest that a good way to get up to speed is to go through the repository and look for optimizations. Examples include performing the integrations of h in the Fourier domain, vectorization of the Green's function, and probably many other things.
If you have a suggestion I propose that you contact me directly and I will try to accomodate you. You can also directly submit a pull request if you implement some additional functionality.
August Brandberg augustbr at k t h . s e.