In this project you will write MATLAB code defining several functions from the plastic behavior of materials, and use curve-fitting approaches to extract parameters from experimental measurements.
Successful completion will demonstrate competence in MATLAB data manipulation, analysis, and visualization. These skills are invaluable in analyzing the output of scientific computing software.
You should submit your report as a single pdf file containing your Matlab scripts as embedded text (do not submit separate m-files) via Assignment Upload by 11:59pm on 10 April 2018. Late submissions will not be accepted.
Your response to each deliverable will be assessed along the specific deliverables listed below. Plots possessing unlabeled axes (variable and units!), illegibly small text, or inappropriately scaled axes will be penalized. Points will be deducted for inadequate interpretation of results or a failure to supply the MATLAB code. The total points available for each analysis are listed below.
[10 pts] Multiaxial loading. Two of the equations to convert a uniaxial yield stress \(\sigma_\text{Y}\) into the yield “surface” for a general stress state are the Tresca criterion, \[\max{\left|\sigma_i - \sigma_j\right|} \ge \sigma_\text{Y},\] and the von Mises criterion, \[\left(\sigma_1-\sigma_2\right)^2 + \left(\sigma_2-\sigma_3\right)^2 + \left(\sigma_3-\sigma_1\right)^2 \ge 2\sigma_\text{Y}^2\] where yield occurs when the criterion is true. Make two three dimensional plots showing the yield surface (where the inequality is a strict equality) in \(\sigma_1\), \(\sigma_2\), \(\sigma_3\): one plot for the Tresca criterion (use \(\sigma_\text{Y}\)=100 MPa) and one plot for the von Mises criterion (use \(\sigma_\text{Y}\) = 100 MPa). Comment on the differences between the two surfaces.
[10 pts] Solute/dislocation interaction. The hydrostatic pressure of an edge dislocation is given by the equation \(p = -(Kb/4\pi r)\sin\theta\) for bulk modulus \(K\), Burgers vector \(b\), distance from slip plane \(r\) and angle from slip plane \(\theta\). If a solute is introduced at some location, that changes the volume by \(\Delta V = a_0^3 \epsilon\), the interaction energy is given by \(p\Delta V\); this interaction depends on the solute position, because the pressure is position dependent. Consider a solute of magnesium in aluminum (\(G\) = 29 GPa, \(\nu\) = 0.33, \(a_0\) = 0.405 nm, and for Mg in Al, \(\epsilon\) = 0.2). If the solute is placed at \(\sqrt{3} a_0\) above the slip system, (1) make a plot of the interaction energy as the dislocation glides in its slip plane, (2) plot the interaction force in the glide plane as a function of position, and (3) determine the maximum interaction force. Comment on the size of the interaction energies and forces.
[20 pts] Work-hardening model. Kocks (http://dx.doi.org/doi:10.1115/1.3443340) suggested a physically-motivated model for the evolution of dislocation density with two contributions: dislocation production due to jogs, that is proportional to the inverse distance between dislocations divided by a mean-free path \(\Lambda\) of dislocation motion, and dislocation annihilation that assumes a constant density of “recovery sites,” so that the rate of annihilation is proportional to the dislocation density, \[\frac{d\rho}{d\varepsilon_\text{T}} = \frac{\rho^{1/2}}{\Lambda} - \frac{2\rho}{\varepsilon_\text{r}}\] for true strain \(\varepsilon_\text{T}\), and where \(\varepsilon_\text{r}\) (a unitless strain) is an empirical parameter governing the recovery of dislocations. Combined with the equation for work-hardening, \[\sigma_\text{T}(\rho) = \sigma_0 + \bar m\alpha Gb\left(\sqrt{\rho} - \sqrt{\rho_0}\right)\] with \(\bar m = 3.06\) (FCC-averaged Schmid factor) and \(\alpha = 0.2\) for the dislocation hardening, we can determine a true-stress / true-strain relationship for work-hardening that includes two empirical parameters.
/class/mse404pla.