Precipitate Dislocation Model from Brown and Ham

This example shows how uncertainties in empirical model parameters affect the uncertainty distribution of the model's output, for a model of a physical process. The example implements the equation for a materials precipitate "cutting" dislocation model from Brown and Ham¹ and shows how metallurgists can gain insight into the uncertainty of a model of a metal alloy's strength. The example highlights how the Signaloid C0 processor allows you to take unmodified programs and track uncertainty through them², getting all the benefits that you would usually only be able to obtain from a hand-crafted (and time-consuming) Monte Carlo evaluation.

Inputs

The inputs and their ranges are:

• gamma: 0.15 to 0.25
• phi: 0.30 to 0.45
• Rs: 1 10^-8 to 3 10^-8
• G: 6 10^10 to 8 10^10
• b: 2.54 10^-9 to 2.54 10^-9 (i.e., constant)
• M: 2.9 to 3.2.

The parameter gamma is the APB energy with units J/m^2, phi is the precipitate volume fraction, Rs is mean particle radius on plane with units m, G is the shear modulus with units Pa, b is the magnitude of the Burgers vector with units m, and M is the Taylor factor.

Outputs

The output is the cutting stress, σc where

                    ⎛    _________________    ⎞
       ⎛ M ⋅ γ  ⎞   ⎜   ╱8.0 ⋅ γ ⋅ φ ⋅ Rs     ⎟
  σ  = ⎜─────── ⎟ ⋅ ⎜  ╱ ───────────────── - φ⎟
   c   ⎝2.0 ⋅ b ⎠   ⎝╲╱  π ⋅ G ⋅ pow(b, 2)    ⎠

Repository Tree Structure

The repository contains three different variants of a simple program implementing the Brown and Ham model. The first variant (v1) has all the model parameters as point-valued numbers, with the Taylor factor (M) computed as the mean value of a number of empirical values. The second variant (v2) has the Taylor factor (M) as the distribution constructed directly from the empirical values, and the third variant (v3) has all parameters except the Burgers vector (b) as distributions.

.
├── README.md
├── v1
│   ├── README.md
│   └── src
│       ├── README.md
│       └── Brown-and-Ham-no-distributions.c
├── v2
│   ├── src
│   │   ├── Brown-and-Ham-with-only-Taylor-Factor-as-distribution.c
│   │   └── README.md
│   └── README.md
└── v3
    ├── README.md
    └── src
        ├── Brown-and-Ham-with-all-parameters-as-distributions.c
        └── README.md

Acknowledgements

We learned about the Brown and Ham model from Prof. Hector Basoalto³ of the University of Sheffield. We are most grateful to him and his team for guiding us through the ideas and evaluating our initial implementation in this example.

Footnotes

1. Brown, L. M., and R. K. Ham. "Dislocation-particle interactions." Strengthening methods in crystals (1971): 9–135. ↩

2. Running this example on the Signaloid C0-Cloud processor uses less than 1% of the free monthly credits on the Signaloid C0-Cloud Free Tier plan. ↩

3. M.J. Anderson, F. Schulz, Y. Lu, H.S. Kitaguchi, P. Bowen, C. Argyrakis, and H.C. Basoalto. 2020. On the modelling of precipitation kinetics in a turbine disc nickel based superalloy. Acta Materialia 191 (2020), 81–100. https://doi.org/10.1016/j. actamat.2020.03.058. ↩